Good “casual” advanced math books

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I'm curious if there are any good math books out there that take a "casual approach" to higher level topics. I'm very interested in advanced math, but have lost the time as of late to study textbooks rigorously, and I find them too dense to parse casually.



By "casual", I mean something that goes over maybe the history of a certain field and its implications in math and society, going over how it grew and what important contributions occurred at different points. Perhaps even going over the abstract meaning of famous results in the field, or high level overviews of proofs and their innovations. Conversations between mathematicians at the time, stories about how proofs came to be, etc.



I suppose the best place to start would be math history books, but I was curious what else there may be. Are there any good books out there like this? What are your recommendations?










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  • 9




    $begingroup$
    Dieudonne has a few maths history books on algebraic geometry and algebraic geometry, which explain the context in which these fields developed. The Grothendieck-Serre correspondence contains an exchange of letters of who might be the most influential postwar mathematicians. Villani has his book "the birth of a theorem" or something like that.
    $endgroup$
    – Einfacher Schreiberling
    Feb 11 at 18:48






  • 1




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    This is not history of mathematics, and I don't know if the material can be considered "advanced" in any sense but looks fun: bookstore.ams.org/mbk-46
    $endgroup$
    – Qfwfq
    Feb 11 at 18:59






  • 3




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    I really liked Love & Math by Edward Frenkel.
    $endgroup$
    – Fred Rohrer
    Feb 11 at 19:01






  • 5




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    Not a book, but the ICM surveys (impa.br/icm2018) are often great for getting a very basic idea of what's going on in various fields outside of one's own.
    $endgroup$
    – Sam Hopkins
    Feb 12 at 0:35






  • 4




    $begingroup$
    I haven't looked at it very closely, but perhaps the Princeton Companion to Mathematics? It's more encyclopedic in nature, but I believe it's intended to be fairly accessible. It might be something to dip in and out of.
    $endgroup$
    – Greg Friedman
    Feb 13 at 0:59















101












$begingroup$


I'm curious if there are any good math books out there that take a "casual approach" to higher level topics. I'm very interested in advanced math, but have lost the time as of late to study textbooks rigorously, and I find them too dense to parse casually.



By "casual", I mean something that goes over maybe the history of a certain field and its implications in math and society, going over how it grew and what important contributions occurred at different points. Perhaps even going over the abstract meaning of famous results in the field, or high level overviews of proofs and their innovations. Conversations between mathematicians at the time, stories about how proofs came to be, etc.



I suppose the best place to start would be math history books, but I was curious what else there may be. Are there any good books out there like this? What are your recommendations?










share|cite|improve this question











$endgroup$







  • 9




    $begingroup$
    Dieudonne has a few maths history books on algebraic geometry and algebraic geometry, which explain the context in which these fields developed. The Grothendieck-Serre correspondence contains an exchange of letters of who might be the most influential postwar mathematicians. Villani has his book "the birth of a theorem" or something like that.
    $endgroup$
    – Einfacher Schreiberling
    Feb 11 at 18:48






  • 1




    $begingroup$
    This is not history of mathematics, and I don't know if the material can be considered "advanced" in any sense but looks fun: bookstore.ams.org/mbk-46
    $endgroup$
    – Qfwfq
    Feb 11 at 18:59






  • 3




    $begingroup$
    I really liked Love & Math by Edward Frenkel.
    $endgroup$
    – Fred Rohrer
    Feb 11 at 19:01






  • 5




    $begingroup$
    Not a book, but the ICM surveys (impa.br/icm2018) are often great for getting a very basic idea of what's going on in various fields outside of one's own.
    $endgroup$
    – Sam Hopkins
    Feb 12 at 0:35






  • 4




    $begingroup$
    I haven't looked at it very closely, but perhaps the Princeton Companion to Mathematics? It's more encyclopedic in nature, but I believe it's intended to be fairly accessible. It might be something to dip in and out of.
    $endgroup$
    – Greg Friedman
    Feb 13 at 0:59













101












101








101


136



$begingroup$


I'm curious if there are any good math books out there that take a "casual approach" to higher level topics. I'm very interested in advanced math, but have lost the time as of late to study textbooks rigorously, and I find them too dense to parse casually.



By "casual", I mean something that goes over maybe the history of a certain field and its implications in math and society, going over how it grew and what important contributions occurred at different points. Perhaps even going over the abstract meaning of famous results in the field, or high level overviews of proofs and their innovations. Conversations between mathematicians at the time, stories about how proofs came to be, etc.



I suppose the best place to start would be math history books, but I was curious what else there may be. Are there any good books out there like this? What are your recommendations?










share|cite|improve this question











$endgroup$




I'm curious if there are any good math books out there that take a "casual approach" to higher level topics. I'm very interested in advanced math, but have lost the time as of late to study textbooks rigorously, and I find them too dense to parse casually.



By "casual", I mean something that goes over maybe the history of a certain field and its implications in math and society, going over how it grew and what important contributions occurred at different points. Perhaps even going over the abstract meaning of famous results in the field, or high level overviews of proofs and their innovations. Conversations between mathematicians at the time, stories about how proofs came to be, etc.



I suppose the best place to start would be math history books, but I was curious what else there may be. Are there any good books out there like this? What are your recommendations?







soft-question big-list textbook-recommendation books






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edited Feb 12 at 4:27


























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  • 9




    $begingroup$
    Dieudonne has a few maths history books on algebraic geometry and algebraic geometry, which explain the context in which these fields developed. The Grothendieck-Serre correspondence contains an exchange of letters of who might be the most influential postwar mathematicians. Villani has his book "the birth of a theorem" or something like that.
    $endgroup$
    – Einfacher Schreiberling
    Feb 11 at 18:48






  • 1




    $begingroup$
    This is not history of mathematics, and I don't know if the material can be considered "advanced" in any sense but looks fun: bookstore.ams.org/mbk-46
    $endgroup$
    – Qfwfq
    Feb 11 at 18:59






  • 3




    $begingroup$
    I really liked Love & Math by Edward Frenkel.
    $endgroup$
    – Fred Rohrer
    Feb 11 at 19:01






  • 5




    $begingroup$
    Not a book, but the ICM surveys (impa.br/icm2018) are often great for getting a very basic idea of what's going on in various fields outside of one's own.
    $endgroup$
    – Sam Hopkins
    Feb 12 at 0:35






  • 4




    $begingroup$
    I haven't looked at it very closely, but perhaps the Princeton Companion to Mathematics? It's more encyclopedic in nature, but I believe it's intended to be fairly accessible. It might be something to dip in and out of.
    $endgroup$
    – Greg Friedman
    Feb 13 at 0:59












  • 9




    $begingroup$
    Dieudonne has a few maths history books on algebraic geometry and algebraic geometry, which explain the context in which these fields developed. The Grothendieck-Serre correspondence contains an exchange of letters of who might be the most influential postwar mathematicians. Villani has his book "the birth of a theorem" or something like that.
    $endgroup$
    – Einfacher Schreiberling
    Feb 11 at 18:48






  • 1




    $begingroup$
    This is not history of mathematics, and I don't know if the material can be considered "advanced" in any sense but looks fun: bookstore.ams.org/mbk-46
    $endgroup$
    – Qfwfq
    Feb 11 at 18:59






  • 3




    $begingroup$
    I really liked Love & Math by Edward Frenkel.
    $endgroup$
    – Fred Rohrer
    Feb 11 at 19:01






  • 5




    $begingroup$
    Not a book, but the ICM surveys (impa.br/icm2018) are often great for getting a very basic idea of what's going on in various fields outside of one's own.
    $endgroup$
    – Sam Hopkins
    Feb 12 at 0:35






  • 4




    $begingroup$
    I haven't looked at it very closely, but perhaps the Princeton Companion to Mathematics? It's more encyclopedic in nature, but I believe it's intended to be fairly accessible. It might be something to dip in and out of.
    $endgroup$
    – Greg Friedman
    Feb 13 at 0:59







9




9




$begingroup$
Dieudonne has a few maths history books on algebraic geometry and algebraic geometry, which explain the context in which these fields developed. The Grothendieck-Serre correspondence contains an exchange of letters of who might be the most influential postwar mathematicians. Villani has his book "the birth of a theorem" or something like that.
$endgroup$
– Einfacher Schreiberling
Feb 11 at 18:48




$begingroup$
Dieudonne has a few maths history books on algebraic geometry and algebraic geometry, which explain the context in which these fields developed. The Grothendieck-Serre correspondence contains an exchange of letters of who might be the most influential postwar mathematicians. Villani has his book "the birth of a theorem" or something like that.
$endgroup$
– Einfacher Schreiberling
Feb 11 at 18:48




1




1




$begingroup$
This is not history of mathematics, and I don't know if the material can be considered "advanced" in any sense but looks fun: bookstore.ams.org/mbk-46
$endgroup$
– Qfwfq
Feb 11 at 18:59




$begingroup$
This is not history of mathematics, and I don't know if the material can be considered "advanced" in any sense but looks fun: bookstore.ams.org/mbk-46
$endgroup$
– Qfwfq
Feb 11 at 18:59




3




3




$begingroup$
I really liked Love & Math by Edward Frenkel.
$endgroup$
– Fred Rohrer
Feb 11 at 19:01




$begingroup$
I really liked Love & Math by Edward Frenkel.
$endgroup$
– Fred Rohrer
Feb 11 at 19:01




5




5




$begingroup$
Not a book, but the ICM surveys (impa.br/icm2018) are often great for getting a very basic idea of what's going on in various fields outside of one's own.
$endgroup$
– Sam Hopkins
Feb 12 at 0:35




$begingroup$
Not a book, but the ICM surveys (impa.br/icm2018) are often great for getting a very basic idea of what's going on in various fields outside of one's own.
$endgroup$
– Sam Hopkins
Feb 12 at 0:35




4




4




$begingroup$
I haven't looked at it very closely, but perhaps the Princeton Companion to Mathematics? It's more encyclopedic in nature, but I believe it's intended to be fairly accessible. It might be something to dip in and out of.
$endgroup$
– Greg Friedman
Feb 13 at 0:59




$begingroup$
I haven't looked at it very closely, but perhaps the Princeton Companion to Mathematics? It's more encyclopedic in nature, but I believe it's intended to be fairly accessible. It might be something to dip in and out of.
$endgroup$
– Greg Friedman
Feb 13 at 0:59










18 Answers
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What's one person's "higher level topics" is another person's "elementary math", so you should be more specific about the desired level.



But still you may try these books:



  1. Michio Kuga, Galois' dream,


  2. David Mumford, Caroline Series, David Wright, Indra's Pearls,


  3. Hermann Weyl, Symmetry.


  4. Marcel Berger, Geometry revealed,


  5. D. Hilbert and Cohn-Vossen, Geometry and imagination,


  6. T. W. Korner, Fourier Analysis,


  7. T. W. Korner, The pleasures of counting.


  8. A. A. Kirillov, What are numbers?


  9. V. Arnold, Huygens and Barrow, Newton and Hooke.


  10. Mark Levi, Classical mechanics with Calculus of variations
    and optimal control.


  11. Shlomo Sternberg, Group theory and physics,


  12. Shlomo Sternberg, Celestial mechanics.


All these books are written in a leisurely informal style, with a lot of
side remarks and historical comments, and almost no prerequisites. But the level of sophistication varies widely. Also don't miss:



Roger Penrose, The road to reality. A complete guide to the laws of the universe. It is on physics, but contains a lot of mathematics).






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  • $begingroup$
    Kirillov's is "What are numbers?"
    $endgroup$
    – Amir Asghari
    Feb 11 at 22:04






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    To the reader: the title of Kuga's book might be misleading: the book is not about classical Galois theory (as one might expect), but about some of its more recent extensions, such as fundamental groups of covering spaces or differential Galois theory.
    $endgroup$
    – Alex M.
    Feb 11 at 22:07






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    @Amir Asghari: thanks for the correction. I only have the Russian original.
    $endgroup$
    – Alexandre Eremenko
    Feb 12 at 2:58






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    @Alex M. It is not about Galois theory. It is about Galois dream. About the theory he wanted to create (as we know from his letters).
    $endgroup$
    – Alexandre Eremenko
    Feb 12 at 3:24


















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Elementary Applied Topology by Ghrist does a fantastic job surveying recent trends in the application of (co)homological methods to practical science and engineering. It goes all the way from Euler characteristic to sheaf cohomology.



Oh, and the illustrations, depending on how far you get with deciphering them, are pretty cool too:



enter image description here






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    For a casual and beautiful promenade, there is A Singular Mathematical Promenade by E. Ghys.



    For French readers, the collection
    Leçons de mathématiques d'aujourd'hui (there are now five volumes) presents a panorama of various subjects and research domains in (mostly pure) mathematics accessible to graduate or advanced undergraduate students.






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      Computing the Continuous Discretely by Beck and Robins.



      Good intro to the interplay of analysis (Fourier analysis and number theory), geometry, and combinatorics.



      Google books, pdf






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        A little more advanced, yet also presented with perceptive intuitions and illustrations, the books noted in mathoverflow.net/questions/31879/…
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        – Tom Copeland
        Feb 18 at 11:17


















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      Regarding differential geometry and topology, there is the 3 volume "A Mathematical Gift - The Interplay Between Topology, Functions, Geometry, and Algebra" by K. Ueno, K. Shiga, S. Morita, T. Sunada. The level is "relaxed undergraduate mathematics". The book attempts to bring to the front the intuition behind some of the concepts encountered in geometry. (All the 4 authors are mathematicians working in Japanese universities.)






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        I find the Carus Mathematical Monographs to be in this category.



        Also, by Julian Havil - "The Irrationals" and "Exploring Gamma".






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          Many Carus monographs are excellent. MY favorite one is P. Doyle and L. Snell, Random walks and electric networks.
          $endgroup$
          – Alexandre Eremenko
          Feb 12 at 3:22






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          The title is "Gamma. Exploring Euler's constant".
          $endgroup$
          – Alexandre Eremenko
          Feb 13 at 18:19


















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        I think that the "Number Theory" series by Kato, Kurokawa, Kurihara and Saito fits here. They are beautifully written and require only some undergradute algebra and analysis. They were published (translated) in the AMS Translations of Mathematical Monographs:



        Number theory. 1. Fermat's dream



        Number theory. 2. Introduction to class field theory



        Number theory. 3. Iwasawa theory and modular forms






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          Based on your first paragraph, I would highly recommend the series What's Happening in the Mathematical Sciences. These provide excellent summaries of a wide variety of cutting-edge mathematics topics. The authors are mathematicians and so the accuracy of the discussion is very high and there is enough detail to satisfy the casual interest of a mathematician, but they also don't get bogged down in too much detail.



          There have even been times when I wanted to gain a thorough understanding of a new and unfamiliar toipc, and I found that the introduction in What's Happening was better than any other introduction I could find. Of course I then I had to turn to more technical texts for more detail, but the overall perspective provided by the What's Happening article was invaluable.






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            James Gleick - Chaos: Making a New Science is a popular history I still remember from 25 years ago.






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              I think “Proofs from THE BOOK” of Aigner and Ziegler may be of some interest. Although the book is not primarily historical, it contains such aspects as well. The book is casual. I do not know if it is advanced or too elementary for you.






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                For a casual book on cryptography and the mathematics behind it, I'll recommend Simon Singh's 'The Code Book'. The concept of modular mathematics and public key cryptography are explained beautifully.






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                  Birth of a Theorem: A Mathematical Adventure, by Cedric Villani. I've linked to a review.



                  Also: Diaconis & Skyrms, Ten Great Ideas About Chance. I quote from the Preface:



                  This is a history book, a probability book, and a philosophy book. We give the history of what we see as great ideas in the development of probability, but we also pursue the philosophical import of these ideas....



                  At the beginning of this book we are thinking along with the pioneers, and the tools involved are simple. By the end, we are up to the present, and some technicalities have to be at arms length. We try to ease the flow of exposition by putting some details in appendices, which you can consult as you wish. We also try to provide ample resources for the reader who finds something interesting enough to dig deeper.






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                    I personally enjoyed The KAM Story by H. Scott Dumas. It gives an overview of the history of KAM theory. Very enjoyable and yet informative.






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                      In 'How to bake a π?', Eugenia Cheng provides a nice view of mathematics in general and category theory in particular that may fit your definition of 'casual'. Either way it is a very nice reading.






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                        I would add 3 favorites:



                        The first two (more or less continuing one another) are by W Narkiewicz and are quite comprehensive combining history, some proofs, excellent references and state of the art results up to ~2010:



                        The Development of Prime Number Theory : From Euclid to Hardy and Littlewood



                        Rational Number Theory in the 20th Century: From PNT to FLT



                        Finally a superb exposition with proofs, explanations, history, from D Choimet and H Queffelec:



                        Twelve Landmarks of Twentieth-Century Analysis






                        share|cite|improve this answer











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                        • $begingroup$
                          Narkiewicz has just published "The Story of Algebraic Numbers in the First Half of the 20th Century From Hilbert to Tate".
                          $endgroup$
                          – EFinat-S
                          Feb 12 at 15:38


















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                        I enjoyed Allen Hatcher's "Algebraic Topology" very much.
                        It's free. You can find it online as a PDF on his university's webspace: https://pi.math.cornell.edu/~hatcher/AT/AT.pdf
                        Hope it will stay up there for long. He's been retired for a while now.



                        That was maybe the most enjoyable math book I've read.






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                        • 11




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                          I would not classify AT as a "casual" reading. I actually suffered reading that book when I took a course in AT. Also, he published in 2017, so I do not think he is retired.
                          $endgroup$
                          – EFinat-S
                          Feb 12 at 15:46






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                          Okay, "casual" is probably not the right word, at least for the middle and later parts of the book. But I think for the beginning chapters it might be applicable. And when one's interest has been captivated, the following suffering is ameliorated by a sense of purpose. Hatcher has retired from teaching I think many years ago, but is still active in research and writing. It states so on his uni web page: pi.math.cornell.edu/~hatcher But I guess it's not to be expected then that his public works would disappear from the web anyway.
                          $endgroup$
                          – Moritz Raguschat
                          Feb 16 at 15:05


















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                        Silvanus P. Thompson's Calculus Made Easy is a good read. Per the reviews, the paperback and Kindle versions should be avoided in favor of the hardcover, although my softcover copy predates this by 2-3 decades.






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                          3












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                          Manfred Schroeder's book entitled Number Theory in Science and Communication: With Applications in Cryptography, Physics, Digital Information, Computing, and Self-Similarity is a wonderful introduction to applied discrete mathematics, or concrete mathematics, to use Knuth's phrase.



                          https://www.springer.com/gp/book/9783540852971



                          "A light-hearted and readable volume with a wide range of applications to which the author has been a productive contributor – useful mathematics outside the formalities of theorem and proof." Martin Gardner






                          share|cite|improve this answer











                          $endgroup$











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                            18 Answers
                            18






                            active

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                            18 Answers
                            18






                            active

                            oldest

                            votes









                            active

                            oldest

                            votes






                            active

                            oldest

                            votes









                            54












                            $begingroup$

                            What's one person's "higher level topics" is another person's "elementary math", so you should be more specific about the desired level.



                            But still you may try these books:



                            1. Michio Kuga, Galois' dream,


                            2. David Mumford, Caroline Series, David Wright, Indra's Pearls,


                            3. Hermann Weyl, Symmetry.


                            4. Marcel Berger, Geometry revealed,


                            5. D. Hilbert and Cohn-Vossen, Geometry and imagination,


                            6. T. W. Korner, Fourier Analysis,


                            7. T. W. Korner, The pleasures of counting.


                            8. A. A. Kirillov, What are numbers?


                            9. V. Arnold, Huygens and Barrow, Newton and Hooke.


                            10. Mark Levi, Classical mechanics with Calculus of variations
                              and optimal control.


                            11. Shlomo Sternberg, Group theory and physics,


                            12. Shlomo Sternberg, Celestial mechanics.


                            All these books are written in a leisurely informal style, with a lot of
                            side remarks and historical comments, and almost no prerequisites. But the level of sophistication varies widely. Also don't miss:



                            Roger Penrose, The road to reality. A complete guide to the laws of the universe. It is on physics, but contains a lot of mathematics).






                            share|cite|improve this answer











                            $endgroup$












                            • $begingroup$
                              Kirillov's is "What are numbers?"
                              $endgroup$
                              – Amir Asghari
                              Feb 11 at 22:04






                            • 1




                              $begingroup$
                              To the reader: the title of Kuga's book might be misleading: the book is not about classical Galois theory (as one might expect), but about some of its more recent extensions, such as fundamental groups of covering spaces or differential Galois theory.
                              $endgroup$
                              – Alex M.
                              Feb 11 at 22:07






                            • 2




                              $begingroup$
                              @Amir Asghari: thanks for the correction. I only have the Russian original.
                              $endgroup$
                              – Alexandre Eremenko
                              Feb 12 at 2:58






                            • 2




                              $begingroup$
                              @Alex M. It is not about Galois theory. It is about Galois dream. About the theory he wanted to create (as we know from his letters).
                              $endgroup$
                              – Alexandre Eremenko
                              Feb 12 at 3:24















                            54












                            $begingroup$

                            What's one person's "higher level topics" is another person's "elementary math", so you should be more specific about the desired level.



                            But still you may try these books:



                            1. Michio Kuga, Galois' dream,


                            2. David Mumford, Caroline Series, David Wright, Indra's Pearls,


                            3. Hermann Weyl, Symmetry.


                            4. Marcel Berger, Geometry revealed,


                            5. D. Hilbert and Cohn-Vossen, Geometry and imagination,


                            6. T. W. Korner, Fourier Analysis,


                            7. T. W. Korner, The pleasures of counting.


                            8. A. A. Kirillov, What are numbers?


                            9. V. Arnold, Huygens and Barrow, Newton and Hooke.


                            10. Mark Levi, Classical mechanics with Calculus of variations
                              and optimal control.


                            11. Shlomo Sternberg, Group theory and physics,


                            12. Shlomo Sternberg, Celestial mechanics.


                            All these books are written in a leisurely informal style, with a lot of
                            side remarks and historical comments, and almost no prerequisites. But the level of sophistication varies widely. Also don't miss:



                            Roger Penrose, The road to reality. A complete guide to the laws of the universe. It is on physics, but contains a lot of mathematics).






                            share|cite|improve this answer











                            $endgroup$












                            • $begingroup$
                              Kirillov's is "What are numbers?"
                              $endgroup$
                              – Amir Asghari
                              Feb 11 at 22:04






                            • 1




                              $begingroup$
                              To the reader: the title of Kuga's book might be misleading: the book is not about classical Galois theory (as one might expect), but about some of its more recent extensions, such as fundamental groups of covering spaces or differential Galois theory.
                              $endgroup$
                              – Alex M.
                              Feb 11 at 22:07






                            • 2




                              $begingroup$
                              @Amir Asghari: thanks for the correction. I only have the Russian original.
                              $endgroup$
                              – Alexandre Eremenko
                              Feb 12 at 2:58






                            • 2




                              $begingroup$
                              @Alex M. It is not about Galois theory. It is about Galois dream. About the theory he wanted to create (as we know from his letters).
                              $endgroup$
                              – Alexandre Eremenko
                              Feb 12 at 3:24













                            54












                            54








                            54





                            $begingroup$

                            What's one person's "higher level topics" is another person's "elementary math", so you should be more specific about the desired level.



                            But still you may try these books:



                            1. Michio Kuga, Galois' dream,


                            2. David Mumford, Caroline Series, David Wright, Indra's Pearls,


                            3. Hermann Weyl, Symmetry.


                            4. Marcel Berger, Geometry revealed,


                            5. D. Hilbert and Cohn-Vossen, Geometry and imagination,


                            6. T. W. Korner, Fourier Analysis,


                            7. T. W. Korner, The pleasures of counting.


                            8. A. A. Kirillov, What are numbers?


                            9. V. Arnold, Huygens and Barrow, Newton and Hooke.


                            10. Mark Levi, Classical mechanics with Calculus of variations
                              and optimal control.


                            11. Shlomo Sternberg, Group theory and physics,


                            12. Shlomo Sternberg, Celestial mechanics.


                            All these books are written in a leisurely informal style, with a lot of
                            side remarks and historical comments, and almost no prerequisites. But the level of sophistication varies widely. Also don't miss:



                            Roger Penrose, The road to reality. A complete guide to the laws of the universe. It is on physics, but contains a lot of mathematics).






                            share|cite|improve this answer











                            $endgroup$



                            What's one person's "higher level topics" is another person's "elementary math", so you should be more specific about the desired level.



                            But still you may try these books:



                            1. Michio Kuga, Galois' dream,


                            2. David Mumford, Caroline Series, David Wright, Indra's Pearls,


                            3. Hermann Weyl, Symmetry.


                            4. Marcel Berger, Geometry revealed,


                            5. D. Hilbert and Cohn-Vossen, Geometry and imagination,


                            6. T. W. Korner, Fourier Analysis,


                            7. T. W. Korner, The pleasures of counting.


                            8. A. A. Kirillov, What are numbers?


                            9. V. Arnold, Huygens and Barrow, Newton and Hooke.


                            10. Mark Levi, Classical mechanics with Calculus of variations
                              and optimal control.


                            11. Shlomo Sternberg, Group theory and physics,


                            12. Shlomo Sternberg, Celestial mechanics.


                            All these books are written in a leisurely informal style, with a lot of
                            side remarks and historical comments, and almost no prerequisites. But the level of sophistication varies widely. Also don't miss:



                            Roger Penrose, The road to reality. A complete guide to the laws of the universe. It is on physics, but contains a lot of mathematics).







                            share|cite|improve this answer














                            share|cite|improve this answer



                            share|cite|improve this answer








                            edited Feb 12 at 3:17


























                            community wiki





                            5 revs
                            Alexandre Eremenko












                            • $begingroup$
                              Kirillov's is "What are numbers?"
                              $endgroup$
                              – Amir Asghari
                              Feb 11 at 22:04






                            • 1




                              $begingroup$
                              To the reader: the title of Kuga's book might be misleading: the book is not about classical Galois theory (as one might expect), but about some of its more recent extensions, such as fundamental groups of covering spaces or differential Galois theory.
                              $endgroup$
                              – Alex M.
                              Feb 11 at 22:07






                            • 2




                              $begingroup$
                              @Amir Asghari: thanks for the correction. I only have the Russian original.
                              $endgroup$
                              – Alexandre Eremenko
                              Feb 12 at 2:58






                            • 2




                              $begingroup$
                              @Alex M. It is not about Galois theory. It is about Galois dream. About the theory he wanted to create (as we know from his letters).
                              $endgroup$
                              – Alexandre Eremenko
                              Feb 12 at 3:24
















                            • $begingroup$
                              Kirillov's is "What are numbers?"
                              $endgroup$
                              – Amir Asghari
                              Feb 11 at 22:04






                            • 1




                              $begingroup$
                              To the reader: the title of Kuga's book might be misleading: the book is not about classical Galois theory (as one might expect), but about some of its more recent extensions, such as fundamental groups of covering spaces or differential Galois theory.
                              $endgroup$
                              – Alex M.
                              Feb 11 at 22:07






                            • 2




                              $begingroup$
                              @Amir Asghari: thanks for the correction. I only have the Russian original.
                              $endgroup$
                              – Alexandre Eremenko
                              Feb 12 at 2:58






                            • 2




                              $begingroup$
                              @Alex M. It is not about Galois theory. It is about Galois dream. About the theory he wanted to create (as we know from his letters).
                              $endgroup$
                              – Alexandre Eremenko
                              Feb 12 at 3:24















                            $begingroup$
                            Kirillov's is "What are numbers?"
                            $endgroup$
                            – Amir Asghari
                            Feb 11 at 22:04




                            $begingroup$
                            Kirillov's is "What are numbers?"
                            $endgroup$
                            – Amir Asghari
                            Feb 11 at 22:04




                            1




                            1




                            $begingroup$
                            To the reader: the title of Kuga's book might be misleading: the book is not about classical Galois theory (as one might expect), but about some of its more recent extensions, such as fundamental groups of covering spaces or differential Galois theory.
                            $endgroup$
                            – Alex M.
                            Feb 11 at 22:07




                            $begingroup$
                            To the reader: the title of Kuga's book might be misleading: the book is not about classical Galois theory (as one might expect), but about some of its more recent extensions, such as fundamental groups of covering spaces or differential Galois theory.
                            $endgroup$
                            – Alex M.
                            Feb 11 at 22:07




                            2




                            2




                            $begingroup$
                            @Amir Asghari: thanks for the correction. I only have the Russian original.
                            $endgroup$
                            – Alexandre Eremenko
                            Feb 12 at 2:58




                            $begingroup$
                            @Amir Asghari: thanks for the correction. I only have the Russian original.
                            $endgroup$
                            – Alexandre Eremenko
                            Feb 12 at 2:58




                            2




                            2




                            $begingroup$
                            @Alex M. It is not about Galois theory. It is about Galois dream. About the theory he wanted to create (as we know from his letters).
                            $endgroup$
                            – Alexandre Eremenko
                            Feb 12 at 3:24




                            $begingroup$
                            @Alex M. It is not about Galois theory. It is about Galois dream. About the theory he wanted to create (as we know from his letters).
                            $endgroup$
                            – Alexandre Eremenko
                            Feb 12 at 3:24











                            26












                            $begingroup$

                            Elementary Applied Topology by Ghrist does a fantastic job surveying recent trends in the application of (co)homological methods to practical science and engineering. It goes all the way from Euler characteristic to sheaf cohomology.



                            Oh, and the illustrations, depending on how far you get with deciphering them, are pretty cool too:



                            enter image description here






                            share|cite|improve this answer











                            $endgroup$

















                              26












                              $begingroup$

                              Elementary Applied Topology by Ghrist does a fantastic job surveying recent trends in the application of (co)homological methods to practical science and engineering. It goes all the way from Euler characteristic to sheaf cohomology.



                              Oh, and the illustrations, depending on how far you get with deciphering them, are pretty cool too:



                              enter image description here






                              share|cite|improve this answer











                              $endgroup$















                                26












                                26








                                26





                                $begingroup$

                                Elementary Applied Topology by Ghrist does a fantastic job surveying recent trends in the application of (co)homological methods to practical science and engineering. It goes all the way from Euler characteristic to sheaf cohomology.



                                Oh, and the illustrations, depending on how far you get with deciphering them, are pretty cool too:



                                enter image description here






                                share|cite|improve this answer











                                $endgroup$



                                Elementary Applied Topology by Ghrist does a fantastic job surveying recent trends in the application of (co)homological methods to practical science and engineering. It goes all the way from Euler characteristic to sheaf cohomology.



                                Oh, and the illustrations, depending on how far you get with deciphering them, are pretty cool too:



                                enter image description here







                                share|cite|improve this answer














                                share|cite|improve this answer



                                share|cite|improve this answer








                                answered Feb 11 at 22:09


























                                community wiki





                                Vidit Nanda






















                                    23












                                    $begingroup$

                                    For a casual and beautiful promenade, there is A Singular Mathematical Promenade by E. Ghys.



                                    For French readers, the collection
                                    Leçons de mathématiques d'aujourd'hui (there are now five volumes) presents a panorama of various subjects and research domains in (mostly pure) mathematics accessible to graduate or advanced undergraduate students.






                                    share|cite|improve this answer











                                    $endgroup$

















                                      23












                                      $begingroup$

                                      For a casual and beautiful promenade, there is A Singular Mathematical Promenade by E. Ghys.



                                      For French readers, the collection
                                      Leçons de mathématiques d'aujourd'hui (there are now five volumes) presents a panorama of various subjects and research domains in (mostly pure) mathematics accessible to graduate or advanced undergraduate students.






                                      share|cite|improve this answer











                                      $endgroup$















                                        23












                                        23








                                        23





                                        $begingroup$

                                        For a casual and beautiful promenade, there is A Singular Mathematical Promenade by E. Ghys.



                                        For French readers, the collection
                                        Leçons de mathématiques d'aujourd'hui (there are now five volumes) presents a panorama of various subjects and research domains in (mostly pure) mathematics accessible to graduate or advanced undergraduate students.






                                        share|cite|improve this answer











                                        $endgroup$



                                        For a casual and beautiful promenade, there is A Singular Mathematical Promenade by E. Ghys.



                                        For French readers, the collection
                                        Leçons de mathématiques d'aujourd'hui (there are now five volumes) presents a panorama of various subjects and research domains in (mostly pure) mathematics accessible to graduate or advanced undergraduate students.







                                        share|cite|improve this answer














                                        share|cite|improve this answer



                                        share|cite|improve this answer








                                        answered Feb 12 at 3:40


























                                        community wiki





                                        HYL






















                                            15












                                            $begingroup$

                                            Computing the Continuous Discretely by Beck and Robins.



                                            Good intro to the interplay of analysis (Fourier analysis and number theory), geometry, and combinatorics.



                                            Google books, pdf






                                            share|cite|improve this answer











                                            $endgroup$












                                            • $begingroup$
                                              A little more advanced, yet also presented with perceptive intuitions and illustrations, the books noted in mathoverflow.net/questions/31879/…
                                              $endgroup$
                                              – Tom Copeland
                                              Feb 18 at 11:17















                                            15












                                            $begingroup$

                                            Computing the Continuous Discretely by Beck and Robins.



                                            Good intro to the interplay of analysis (Fourier analysis and number theory), geometry, and combinatorics.



                                            Google books, pdf






                                            share|cite|improve this answer











                                            $endgroup$












                                            • $begingroup$
                                              A little more advanced, yet also presented with perceptive intuitions and illustrations, the books noted in mathoverflow.net/questions/31879/…
                                              $endgroup$
                                              – Tom Copeland
                                              Feb 18 at 11:17













                                            15












                                            15








                                            15





                                            $begingroup$

                                            Computing the Continuous Discretely by Beck and Robins.



                                            Good intro to the interplay of analysis (Fourier analysis and number theory), geometry, and combinatorics.



                                            Google books, pdf






                                            share|cite|improve this answer











                                            $endgroup$



                                            Computing the Continuous Discretely by Beck and Robins.



                                            Good intro to the interplay of analysis (Fourier analysis and number theory), geometry, and combinatorics.



                                            Google books, pdf







                                            share|cite|improve this answer














                                            share|cite|improve this answer



                                            share|cite|improve this answer








                                            answered Feb 12 at 21:19


























                                            community wiki





                                            Tom Copeland












                                            • $begingroup$
                                              A little more advanced, yet also presented with perceptive intuitions and illustrations, the books noted in mathoverflow.net/questions/31879/…
                                              $endgroup$
                                              – Tom Copeland
                                              Feb 18 at 11:17
















                                            • $begingroup$
                                              A little more advanced, yet also presented with perceptive intuitions and illustrations, the books noted in mathoverflow.net/questions/31879/…
                                              $endgroup$
                                              – Tom Copeland
                                              Feb 18 at 11:17















                                            $begingroup$
                                            A little more advanced, yet also presented with perceptive intuitions and illustrations, the books noted in mathoverflow.net/questions/31879/…
                                            $endgroup$
                                            – Tom Copeland
                                            Feb 18 at 11:17




                                            $begingroup$
                                            A little more advanced, yet also presented with perceptive intuitions and illustrations, the books noted in mathoverflow.net/questions/31879/…
                                            $endgroup$
                                            – Tom Copeland
                                            Feb 18 at 11:17











                                            12












                                            $begingroup$

                                            Regarding differential geometry and topology, there is the 3 volume "A Mathematical Gift - The Interplay Between Topology, Functions, Geometry, and Algebra" by K. Ueno, K. Shiga, S. Morita, T. Sunada. The level is "relaxed undergraduate mathematics". The book attempts to bring to the front the intuition behind some of the concepts encountered in geometry. (All the 4 authors are mathematicians working in Japanese universities.)






                                            share|cite|improve this answer











                                            $endgroup$

















                                              12












                                              $begingroup$

                                              Regarding differential geometry and topology, there is the 3 volume "A Mathematical Gift - The Interplay Between Topology, Functions, Geometry, and Algebra" by K. Ueno, K. Shiga, S. Morita, T. Sunada. The level is "relaxed undergraduate mathematics". The book attempts to bring to the front the intuition behind some of the concepts encountered in geometry. (All the 4 authors are mathematicians working in Japanese universities.)






                                              share|cite|improve this answer











                                              $endgroup$















                                                12












                                                12








                                                12





                                                $begingroup$

                                                Regarding differential geometry and topology, there is the 3 volume "A Mathematical Gift - The Interplay Between Topology, Functions, Geometry, and Algebra" by K. Ueno, K. Shiga, S. Morita, T. Sunada. The level is "relaxed undergraduate mathematics". The book attempts to bring to the front the intuition behind some of the concepts encountered in geometry. (All the 4 authors are mathematicians working in Japanese universities.)






                                                share|cite|improve this answer











                                                $endgroup$



                                                Regarding differential geometry and topology, there is the 3 volume "A Mathematical Gift - The Interplay Between Topology, Functions, Geometry, and Algebra" by K. Ueno, K. Shiga, S. Morita, T. Sunada. The level is "relaxed undergraduate mathematics". The book attempts to bring to the front the intuition behind some of the concepts encountered in geometry. (All the 4 authors are mathematicians working in Japanese universities.)







                                                share|cite|improve this answer














                                                share|cite|improve this answer



                                                share|cite|improve this answer








                                                answered Feb 11 at 22:03


























                                                community wiki





                                                Alex M.






















                                                    10












                                                    $begingroup$

                                                    I find the Carus Mathematical Monographs to be in this category.



                                                    Also, by Julian Havil - "The Irrationals" and "Exploring Gamma".






                                                    share|cite|improve this answer











                                                    $endgroup$








                                                    • 1




                                                      $begingroup$
                                                      Many Carus monographs are excellent. MY favorite one is P. Doyle and L. Snell, Random walks and electric networks.
                                                      $endgroup$
                                                      – Alexandre Eremenko
                                                      Feb 12 at 3:22






                                                    • 1




                                                      $begingroup$
                                                      The title is "Gamma. Exploring Euler's constant".
                                                      $endgroup$
                                                      – Alexandre Eremenko
                                                      Feb 13 at 18:19















                                                    10












                                                    $begingroup$

                                                    I find the Carus Mathematical Monographs to be in this category.



                                                    Also, by Julian Havil - "The Irrationals" and "Exploring Gamma".






                                                    share|cite|improve this answer











                                                    $endgroup$








                                                    • 1




                                                      $begingroup$
                                                      Many Carus monographs are excellent. MY favorite one is P. Doyle and L. Snell, Random walks and electric networks.
                                                      $endgroup$
                                                      – Alexandre Eremenko
                                                      Feb 12 at 3:22






                                                    • 1




                                                      $begingroup$
                                                      The title is "Gamma. Exploring Euler's constant".
                                                      $endgroup$
                                                      – Alexandre Eremenko
                                                      Feb 13 at 18:19













                                                    10












                                                    10








                                                    10





                                                    $begingroup$

                                                    I find the Carus Mathematical Monographs to be in this category.



                                                    Also, by Julian Havil - "The Irrationals" and "Exploring Gamma".






                                                    share|cite|improve this answer











                                                    $endgroup$



                                                    I find the Carus Mathematical Monographs to be in this category.



                                                    Also, by Julian Havil - "The Irrationals" and "Exploring Gamma".







                                                    share|cite|improve this answer














                                                    share|cite|improve this answer



                                                    share|cite|improve this answer








                                                    edited Feb 12 at 0:47


























                                                    community wiki





                                                    2 revs, 2 users 80%
                                                    David Campen








                                                    • 1




                                                      $begingroup$
                                                      Many Carus monographs are excellent. MY favorite one is P. Doyle and L. Snell, Random walks and electric networks.
                                                      $endgroup$
                                                      – Alexandre Eremenko
                                                      Feb 12 at 3:22






                                                    • 1




                                                      $begingroup$
                                                      The title is "Gamma. Exploring Euler's constant".
                                                      $endgroup$
                                                      – Alexandre Eremenko
                                                      Feb 13 at 18:19












                                                    • 1




                                                      $begingroup$
                                                      Many Carus monographs are excellent. MY favorite one is P. Doyle and L. Snell, Random walks and electric networks.
                                                      $endgroup$
                                                      – Alexandre Eremenko
                                                      Feb 12 at 3:22






                                                    • 1




                                                      $begingroup$
                                                      The title is "Gamma. Exploring Euler's constant".
                                                      $endgroup$
                                                      – Alexandre Eremenko
                                                      Feb 13 at 18:19







                                                    1




                                                    1




                                                    $begingroup$
                                                    Many Carus monographs are excellent. MY favorite one is P. Doyle and L. Snell, Random walks and electric networks.
                                                    $endgroup$
                                                    – Alexandre Eremenko
                                                    Feb 12 at 3:22




                                                    $begingroup$
                                                    Many Carus monographs are excellent. MY favorite one is P. Doyle and L. Snell, Random walks and electric networks.
                                                    $endgroup$
                                                    – Alexandre Eremenko
                                                    Feb 12 at 3:22




                                                    1




                                                    1




                                                    $begingroup$
                                                    The title is "Gamma. Exploring Euler's constant".
                                                    $endgroup$
                                                    – Alexandre Eremenko
                                                    Feb 13 at 18:19




                                                    $begingroup$
                                                    The title is "Gamma. Exploring Euler's constant".
                                                    $endgroup$
                                                    – Alexandre Eremenko
                                                    Feb 13 at 18:19











                                                    9












                                                    $begingroup$

                                                    I think that the "Number Theory" series by Kato, Kurokawa, Kurihara and Saito fits here. They are beautifully written and require only some undergradute algebra and analysis. They were published (translated) in the AMS Translations of Mathematical Monographs:



                                                    Number theory. 1. Fermat's dream



                                                    Number theory. 2. Introduction to class field theory



                                                    Number theory. 3. Iwasawa theory and modular forms






                                                    share|cite|improve this answer











                                                    $endgroup$

















                                                      9












                                                      $begingroup$

                                                      I think that the "Number Theory" series by Kato, Kurokawa, Kurihara and Saito fits here. They are beautifully written and require only some undergradute algebra and analysis. They were published (translated) in the AMS Translations of Mathematical Monographs:



                                                      Number theory. 1. Fermat's dream



                                                      Number theory. 2. Introduction to class field theory



                                                      Number theory. 3. Iwasawa theory and modular forms






                                                      share|cite|improve this answer











                                                      $endgroup$















                                                        9












                                                        9








                                                        9





                                                        $begingroup$

                                                        I think that the "Number Theory" series by Kato, Kurokawa, Kurihara and Saito fits here. They are beautifully written and require only some undergradute algebra and analysis. They were published (translated) in the AMS Translations of Mathematical Monographs:



                                                        Number theory. 1. Fermat's dream



                                                        Number theory. 2. Introduction to class field theory



                                                        Number theory. 3. Iwasawa theory and modular forms






                                                        share|cite|improve this answer











                                                        $endgroup$



                                                        I think that the "Number Theory" series by Kato, Kurokawa, Kurihara and Saito fits here. They are beautifully written and require only some undergradute algebra and analysis. They were published (translated) in the AMS Translations of Mathematical Monographs:



                                                        Number theory. 1. Fermat's dream



                                                        Number theory. 2. Introduction to class field theory



                                                        Number theory. 3. Iwasawa theory and modular forms







                                                        share|cite|improve this answer














                                                        share|cite|improve this answer



                                                        share|cite|improve this answer








                                                        edited Feb 12 at 1:42


























                                                        community wiki





                                                        2 revs, 2 users 95%
                                                        EFinat-S






















                                                            9












                                                            $begingroup$

                                                            Based on your first paragraph, I would highly recommend the series What's Happening in the Mathematical Sciences. These provide excellent summaries of a wide variety of cutting-edge mathematics topics. The authors are mathematicians and so the accuracy of the discussion is very high and there is enough detail to satisfy the casual interest of a mathematician, but they also don't get bogged down in too much detail.



                                                            There have even been times when I wanted to gain a thorough understanding of a new and unfamiliar toipc, and I found that the introduction in What's Happening was better than any other introduction I could find. Of course I then I had to turn to more technical texts for more detail, but the overall perspective provided by the What's Happening article was invaluable.






                                                            share|cite|improve this answer











                                                            $endgroup$

















                                                              9












                                                              $begingroup$

                                                              Based on your first paragraph, I would highly recommend the series What's Happening in the Mathematical Sciences. These provide excellent summaries of a wide variety of cutting-edge mathematics topics. The authors are mathematicians and so the accuracy of the discussion is very high and there is enough detail to satisfy the casual interest of a mathematician, but they also don't get bogged down in too much detail.



                                                              There have even been times when I wanted to gain a thorough understanding of a new and unfamiliar toipc, and I found that the introduction in What's Happening was better than any other introduction I could find. Of course I then I had to turn to more technical texts for more detail, but the overall perspective provided by the What's Happening article was invaluable.






                                                              share|cite|improve this answer











                                                              $endgroup$















                                                                9












                                                                9








                                                                9





                                                                $begingroup$

                                                                Based on your first paragraph, I would highly recommend the series What's Happening in the Mathematical Sciences. These provide excellent summaries of a wide variety of cutting-edge mathematics topics. The authors are mathematicians and so the accuracy of the discussion is very high and there is enough detail to satisfy the casual interest of a mathematician, but they also don't get bogged down in too much detail.



                                                                There have even been times when I wanted to gain a thorough understanding of a new and unfamiliar toipc, and I found that the introduction in What's Happening was better than any other introduction I could find. Of course I then I had to turn to more technical texts for more detail, but the overall perspective provided by the What's Happening article was invaluable.






                                                                share|cite|improve this answer











                                                                $endgroup$



                                                                Based on your first paragraph, I would highly recommend the series What's Happening in the Mathematical Sciences. These provide excellent summaries of a wide variety of cutting-edge mathematics topics. The authors are mathematicians and so the accuracy of the discussion is very high and there is enough detail to satisfy the casual interest of a mathematician, but they also don't get bogged down in too much detail.



                                                                There have even been times when I wanted to gain a thorough understanding of a new and unfamiliar toipc, and I found that the introduction in What's Happening was better than any other introduction I could find. Of course I then I had to turn to more technical texts for more detail, but the overall perspective provided by the What's Happening article was invaluable.







                                                                share|cite|improve this answer














                                                                share|cite|improve this answer



                                                                share|cite|improve this answer








                                                                answered Feb 12 at 15:08


























                                                                community wiki





                                                                Timothy Chow






















                                                                    8












                                                                    $begingroup$

                                                                    James Gleick - Chaos: Making a New Science is a popular history I still remember from 25 years ago.






                                                                    share|cite|improve this answer











                                                                    $endgroup$

















                                                                      8












                                                                      $begingroup$

                                                                      James Gleick - Chaos: Making a New Science is a popular history I still remember from 25 years ago.






                                                                      share|cite|improve this answer











                                                                      $endgroup$















                                                                        8












                                                                        8








                                                                        8





                                                                        $begingroup$

                                                                        James Gleick - Chaos: Making a New Science is a popular history I still remember from 25 years ago.






                                                                        share|cite|improve this answer











                                                                        $endgroup$



                                                                        James Gleick - Chaos: Making a New Science is a popular history I still remember from 25 years ago.







                                                                        share|cite|improve this answer














                                                                        share|cite|improve this answer



                                                                        share|cite|improve this answer








                                                                        answered Feb 12 at 10:29


























                                                                        community wiki





                                                                        Mike M






















                                                                            8












                                                                            $begingroup$

                                                                            I think “Proofs from THE BOOK” of Aigner and Ziegler may be of some interest. Although the book is not primarily historical, it contains such aspects as well. The book is casual. I do not know if it is advanced or too elementary for you.






                                                                            share|cite|improve this answer











                                                                            $endgroup$

















                                                                              8












                                                                              $begingroup$

                                                                              I think “Proofs from THE BOOK” of Aigner and Ziegler may be of some interest. Although the book is not primarily historical, it contains such aspects as well. The book is casual. I do not know if it is advanced or too elementary for you.






                                                                              share|cite|improve this answer











                                                                              $endgroup$















                                                                                8












                                                                                8








                                                                                8





                                                                                $begingroup$

                                                                                I think “Proofs from THE BOOK” of Aigner and Ziegler may be of some interest. Although the book is not primarily historical, it contains such aspects as well. The book is casual. I do not know if it is advanced or too elementary for you.






                                                                                share|cite|improve this answer











                                                                                $endgroup$



                                                                                I think “Proofs from THE BOOK” of Aigner and Ziegler may be of some interest. Although the book is not primarily historical, it contains such aspects as well. The book is casual. I do not know if it is advanced or too elementary for you.







                                                                                share|cite|improve this answer














                                                                                share|cite|improve this answer



                                                                                share|cite|improve this answer








                                                                                edited Feb 12 at 18:37


























                                                                                community wiki





                                                                                2 revs, 2 users 50%
                                                                                Jörg Neunhäuserer






















                                                                                    7












                                                                                    $begingroup$

                                                                                    For a casual book on cryptography and the mathematics behind it, I'll recommend Simon Singh's 'The Code Book'. The concept of modular mathematics and public key cryptography are explained beautifully.






                                                                                    share|cite|improve this answer











                                                                                    $endgroup$

















                                                                                      7












                                                                                      $begingroup$

                                                                                      For a casual book on cryptography and the mathematics behind it, I'll recommend Simon Singh's 'The Code Book'. The concept of modular mathematics and public key cryptography are explained beautifully.






                                                                                      share|cite|improve this answer











                                                                                      $endgroup$















                                                                                        7












                                                                                        7








                                                                                        7





                                                                                        $begingroup$

                                                                                        For a casual book on cryptography and the mathematics behind it, I'll recommend Simon Singh's 'The Code Book'. The concept of modular mathematics and public key cryptography are explained beautifully.






                                                                                        share|cite|improve this answer











                                                                                        $endgroup$



                                                                                        For a casual book on cryptography and the mathematics behind it, I'll recommend Simon Singh's 'The Code Book'. The concept of modular mathematics and public key cryptography are explained beautifully.







                                                                                        share|cite|improve this answer














                                                                                        share|cite|improve this answer



                                                                                        share|cite|improve this answer








                                                                                        answered Feb 12 at 3:59


























                                                                                        community wiki





                                                                                        Jay






















                                                                                            7












                                                                                            $begingroup$

                                                                                            Birth of a Theorem: A Mathematical Adventure, by Cedric Villani. I've linked to a review.



                                                                                            Also: Diaconis & Skyrms, Ten Great Ideas About Chance. I quote from the Preface:



                                                                                            This is a history book, a probability book, and a philosophy book. We give the history of what we see as great ideas in the development of probability, but we also pursue the philosophical import of these ideas....



                                                                                            At the beginning of this book we are thinking along with the pioneers, and the tools involved are simple. By the end, we are up to the present, and some technicalities have to be at arms length. We try to ease the flow of exposition by putting some details in appendices, which you can consult as you wish. We also try to provide ample resources for the reader who finds something interesting enough to dig deeper.






                                                                                            share|cite|improve this answer











                                                                                            $endgroup$

















                                                                                              7












                                                                                              $begingroup$

                                                                                              Birth of a Theorem: A Mathematical Adventure, by Cedric Villani. I've linked to a review.



                                                                                              Also: Diaconis & Skyrms, Ten Great Ideas About Chance. I quote from the Preface:



                                                                                              This is a history book, a probability book, and a philosophy book. We give the history of what we see as great ideas in the development of probability, but we also pursue the philosophical import of these ideas....



                                                                                              At the beginning of this book we are thinking along with the pioneers, and the tools involved are simple. By the end, we are up to the present, and some technicalities have to be at arms length. We try to ease the flow of exposition by putting some details in appendices, which you can consult as you wish. We also try to provide ample resources for the reader who finds something interesting enough to dig deeper.






                                                                                              share|cite|improve this answer











                                                                                              $endgroup$















                                                                                                7












                                                                                                7








                                                                                                7





                                                                                                $begingroup$

                                                                                                Birth of a Theorem: A Mathematical Adventure, by Cedric Villani. I've linked to a review.



                                                                                                Also: Diaconis & Skyrms, Ten Great Ideas About Chance. I quote from the Preface:



                                                                                                This is a history book, a probability book, and a philosophy book. We give the history of what we see as great ideas in the development of probability, but we also pursue the philosophical import of these ideas....



                                                                                                At the beginning of this book we are thinking along with the pioneers, and the tools involved are simple. By the end, we are up to the present, and some technicalities have to be at arms length. We try to ease the flow of exposition by putting some details in appendices, which you can consult as you wish. We also try to provide ample resources for the reader who finds something interesting enough to dig deeper.






                                                                                                share|cite|improve this answer











                                                                                                $endgroup$



                                                                                                Birth of a Theorem: A Mathematical Adventure, by Cedric Villani. I've linked to a review.



                                                                                                Also: Diaconis & Skyrms, Ten Great Ideas About Chance. I quote from the Preface:



                                                                                                This is a history book, a probability book, and a philosophy book. We give the history of what we see as great ideas in the development of probability, but we also pursue the philosophical import of these ideas....



                                                                                                At the beginning of this book we are thinking along with the pioneers, and the tools involved are simple. By the end, we are up to the present, and some technicalities have to be at arms length. We try to ease the flow of exposition by putting some details in appendices, which you can consult as you wish. We also try to provide ample resources for the reader who finds something interesting enough to dig deeper.







                                                                                                share|cite|improve this answer














                                                                                                share|cite|improve this answer



                                                                                                share|cite|improve this answer








                                                                                                edited Feb 12 at 9:25


























                                                                                                community wiki





                                                                                                2 revs
                                                                                                Gerry Myerson






















                                                                                                    6












                                                                                                    $begingroup$

                                                                                                    I personally enjoyed The KAM Story by H. Scott Dumas. It gives an overview of the history of KAM theory. Very enjoyable and yet informative.






                                                                                                    share|cite|improve this answer











                                                                                                    $endgroup$

















                                                                                                      6












                                                                                                      $begingroup$

                                                                                                      I personally enjoyed The KAM Story by H. Scott Dumas. It gives an overview of the history of KAM theory. Very enjoyable and yet informative.






                                                                                                      share|cite|improve this answer











                                                                                                      $endgroup$















                                                                                                        6












                                                                                                        6








                                                                                                        6





                                                                                                        $begingroup$

                                                                                                        I personally enjoyed The KAM Story by H. Scott Dumas. It gives an overview of the history of KAM theory. Very enjoyable and yet informative.






                                                                                                        share|cite|improve this answer











                                                                                                        $endgroup$



                                                                                                        I personally enjoyed The KAM Story by H. Scott Dumas. It gives an overview of the history of KAM theory. Very enjoyable and yet informative.







                                                                                                        share|cite|improve this answer














                                                                                                        share|cite|improve this answer



                                                                                                        share|cite|improve this answer








                                                                                                        answered Feb 11 at 20:46


























                                                                                                        community wiki





                                                                                                        Severin Schraven






















                                                                                                            6












                                                                                                            $begingroup$

                                                                                                            In 'How to bake a π?', Eugenia Cheng provides a nice view of mathematics in general and category theory in particular that may fit your definition of 'casual'. Either way it is a very nice reading.






                                                                                                            share|cite|improve this answer











                                                                                                            $endgroup$

















                                                                                                              6












                                                                                                              $begingroup$

                                                                                                              In 'How to bake a π?', Eugenia Cheng provides a nice view of mathematics in general and category theory in particular that may fit your definition of 'casual'. Either way it is a very nice reading.






                                                                                                              share|cite|improve this answer











                                                                                                              $endgroup$















                                                                                                                6












                                                                                                                6








                                                                                                                6





                                                                                                                $begingroup$

                                                                                                                In 'How to bake a π?', Eugenia Cheng provides a nice view of mathematics in general and category theory in particular that may fit your definition of 'casual'. Either way it is a very nice reading.






                                                                                                                share|cite|improve this answer











                                                                                                                $endgroup$



                                                                                                                In 'How to bake a π?', Eugenia Cheng provides a nice view of mathematics in general and category theory in particular that may fit your definition of 'casual'. Either way it is a very nice reading.







                                                                                                                share|cite|improve this answer














                                                                                                                share|cite|improve this answer



                                                                                                                share|cite|improve this answer








                                                                                                                answered Feb 11 at 22:10


























                                                                                                                community wiki





                                                                                                                Melquíades Ochoa






















                                                                                                                    5












                                                                                                                    $begingroup$

                                                                                                                    I would add 3 favorites:



                                                                                                                    The first two (more or less continuing one another) are by W Narkiewicz and are quite comprehensive combining history, some proofs, excellent references and state of the art results up to ~2010:



                                                                                                                    The Development of Prime Number Theory : From Euclid to Hardy and Littlewood



                                                                                                                    Rational Number Theory in the 20th Century: From PNT to FLT



                                                                                                                    Finally a superb exposition with proofs, explanations, history, from D Choimet and H Queffelec:



                                                                                                                    Twelve Landmarks of Twentieth-Century Analysis






                                                                                                                    share|cite|improve this answer











                                                                                                                    $endgroup$












                                                                                                                    • $begingroup$
                                                                                                                      Narkiewicz has just published "The Story of Algebraic Numbers in the First Half of the 20th Century From Hilbert to Tate".
                                                                                                                      $endgroup$
                                                                                                                      – EFinat-S
                                                                                                                      Feb 12 at 15:38















                                                                                                                    5












                                                                                                                    $begingroup$

                                                                                                                    I would add 3 favorites:



                                                                                                                    The first two (more or less continuing one another) are by W Narkiewicz and are quite comprehensive combining history, some proofs, excellent references and state of the art results up to ~2010:



                                                                                                                    The Development of Prime Number Theory : From Euclid to Hardy and Littlewood



                                                                                                                    Rational Number Theory in the 20th Century: From PNT to FLT



                                                                                                                    Finally a superb exposition with proofs, explanations, history, from D Choimet and H Queffelec:



                                                                                                                    Twelve Landmarks of Twentieth-Century Analysis






                                                                                                                    share|cite|improve this answer











                                                                                                                    $endgroup$












                                                                                                                    • $begingroup$
                                                                                                                      Narkiewicz has just published "The Story of Algebraic Numbers in the First Half of the 20th Century From Hilbert to Tate".
                                                                                                                      $endgroup$
                                                                                                                      – EFinat-S
                                                                                                                      Feb 12 at 15:38













                                                                                                                    5












                                                                                                                    5








                                                                                                                    5





                                                                                                                    $begingroup$

                                                                                                                    I would add 3 favorites:



                                                                                                                    The first two (more or less continuing one another) are by W Narkiewicz and are quite comprehensive combining history, some proofs, excellent references and state of the art results up to ~2010:



                                                                                                                    The Development of Prime Number Theory : From Euclid to Hardy and Littlewood



                                                                                                                    Rational Number Theory in the 20th Century: From PNT to FLT



                                                                                                                    Finally a superb exposition with proofs, explanations, history, from D Choimet and H Queffelec:



                                                                                                                    Twelve Landmarks of Twentieth-Century Analysis






                                                                                                                    share|cite|improve this answer











                                                                                                                    $endgroup$



                                                                                                                    I would add 3 favorites:



                                                                                                                    The first two (more or less continuing one another) are by W Narkiewicz and are quite comprehensive combining history, some proofs, excellent references and state of the art results up to ~2010:



                                                                                                                    The Development of Prime Number Theory : From Euclid to Hardy and Littlewood



                                                                                                                    Rational Number Theory in the 20th Century: From PNT to FLT



                                                                                                                    Finally a superb exposition with proofs, explanations, history, from D Choimet and H Queffelec:



                                                                                                                    Twelve Landmarks of Twentieth-Century Analysis







                                                                                                                    share|cite|improve this answer














                                                                                                                    share|cite|improve this answer



                                                                                                                    share|cite|improve this answer








                                                                                                                    answered Feb 12 at 3:47


























                                                                                                                    community wiki





                                                                                                                    Conrad












                                                                                                                    • $begingroup$
                                                                                                                      Narkiewicz has just published "The Story of Algebraic Numbers in the First Half of the 20th Century From Hilbert to Tate".
                                                                                                                      $endgroup$
                                                                                                                      – EFinat-S
                                                                                                                      Feb 12 at 15:38
















                                                                                                                    • $begingroup$
                                                                                                                      Narkiewicz has just published "The Story of Algebraic Numbers in the First Half of the 20th Century From Hilbert to Tate".
                                                                                                                      $endgroup$
                                                                                                                      – EFinat-S
                                                                                                                      Feb 12 at 15:38















                                                                                                                    $begingroup$
                                                                                                                    Narkiewicz has just published "The Story of Algebraic Numbers in the First Half of the 20th Century From Hilbert to Tate".
                                                                                                                    $endgroup$
                                                                                                                    – EFinat-S
                                                                                                                    Feb 12 at 15:38




                                                                                                                    $begingroup$
                                                                                                                    Narkiewicz has just published "The Story of Algebraic Numbers in the First Half of the 20th Century From Hilbert to Tate".
                                                                                                                    $endgroup$
                                                                                                                    – EFinat-S
                                                                                                                    Feb 12 at 15:38











                                                                                                                    5












                                                                                                                    $begingroup$

                                                                                                                    I enjoyed Allen Hatcher's "Algebraic Topology" very much.
                                                                                                                    It's free. You can find it online as a PDF on his university's webspace: https://pi.math.cornell.edu/~hatcher/AT/AT.pdf
                                                                                                                    Hope it will stay up there for long. He's been retired for a while now.



                                                                                                                    That was maybe the most enjoyable math book I've read.






                                                                                                                    share|cite|improve this answer











                                                                                                                    $endgroup$








                                                                                                                    • 11




                                                                                                                      $begingroup$
                                                                                                                      I would not classify AT as a "casual" reading. I actually suffered reading that book when I took a course in AT. Also, he published in 2017, so I do not think he is retired.
                                                                                                                      $endgroup$
                                                                                                                      – EFinat-S
                                                                                                                      Feb 12 at 15:46






                                                                                                                    • 1




                                                                                                                      $begingroup$
                                                                                                                      Okay, "casual" is probably not the right word, at least for the middle and later parts of the book. But I think for the beginning chapters it might be applicable. And when one's interest has been captivated, the following suffering is ameliorated by a sense of purpose. Hatcher has retired from teaching I think many years ago, but is still active in research and writing. It states so on his uni web page: pi.math.cornell.edu/~hatcher But I guess it's not to be expected then that his public works would disappear from the web anyway.
                                                                                                                      $endgroup$
                                                                                                                      – Moritz Raguschat
                                                                                                                      Feb 16 at 15:05















                                                                                                                    5












                                                                                                                    $begingroup$

                                                                                                                    I enjoyed Allen Hatcher's "Algebraic Topology" very much.
                                                                                                                    It's free. You can find it online as a PDF on his university's webspace: https://pi.math.cornell.edu/~hatcher/AT/AT.pdf
                                                                                                                    Hope it will stay up there for long. He's been retired for a while now.



                                                                                                                    That was maybe the most enjoyable math book I've read.






                                                                                                                    share|cite|improve this answer











                                                                                                                    $endgroup$








                                                                                                                    • 11




                                                                                                                      $begingroup$
                                                                                                                      I would not classify AT as a "casual" reading. I actually suffered reading that book when I took a course in AT. Also, he published in 2017, so I do not think he is retired.
                                                                                                                      $endgroup$
                                                                                                                      – EFinat-S
                                                                                                                      Feb 12 at 15:46






                                                                                                                    • 1




                                                                                                                      $begingroup$
                                                                                                                      Okay, "casual" is probably not the right word, at least for the middle and later parts of the book. But I think for the beginning chapters it might be applicable. And when one's interest has been captivated, the following suffering is ameliorated by a sense of purpose. Hatcher has retired from teaching I think many years ago, but is still active in research and writing. It states so on his uni web page: pi.math.cornell.edu/~hatcher But I guess it's not to be expected then that his public works would disappear from the web anyway.
                                                                                                                      $endgroup$
                                                                                                                      – Moritz Raguschat
                                                                                                                      Feb 16 at 15:05













                                                                                                                    5












                                                                                                                    5








                                                                                                                    5





                                                                                                                    $begingroup$

                                                                                                                    I enjoyed Allen Hatcher's "Algebraic Topology" very much.
                                                                                                                    It's free. You can find it online as a PDF on his university's webspace: https://pi.math.cornell.edu/~hatcher/AT/AT.pdf
                                                                                                                    Hope it will stay up there for long. He's been retired for a while now.



                                                                                                                    That was maybe the most enjoyable math book I've read.






                                                                                                                    share|cite|improve this answer











                                                                                                                    $endgroup$



                                                                                                                    I enjoyed Allen Hatcher's "Algebraic Topology" very much.
                                                                                                                    It's free. You can find it online as a PDF on his university's webspace: https://pi.math.cornell.edu/~hatcher/AT/AT.pdf
                                                                                                                    Hope it will stay up there for long. He's been retired for a while now.



                                                                                                                    That was maybe the most enjoyable math book I've read.







                                                                                                                    share|cite|improve this answer














                                                                                                                    share|cite|improve this answer



                                                                                                                    share|cite|improve this answer








                                                                                                                    answered Feb 12 at 14:44


























                                                                                                                    community wiki





                                                                                                                    Moritz Raguschat








                                                                                                                    • 11




                                                                                                                      $begingroup$
                                                                                                                      I would not classify AT as a "casual" reading. I actually suffered reading that book when I took a course in AT. Also, he published in 2017, so I do not think he is retired.
                                                                                                                      $endgroup$
                                                                                                                      – EFinat-S
                                                                                                                      Feb 12 at 15:46






                                                                                                                    • 1




                                                                                                                      $begingroup$
                                                                                                                      Okay, "casual" is probably not the right word, at least for the middle and later parts of the book. But I think for the beginning chapters it might be applicable. And when one's interest has been captivated, the following suffering is ameliorated by a sense of purpose. Hatcher has retired from teaching I think many years ago, but is still active in research and writing. It states so on his uni web page: pi.math.cornell.edu/~hatcher But I guess it's not to be expected then that his public works would disappear from the web anyway.
                                                                                                                      $endgroup$
                                                                                                                      – Moritz Raguschat
                                                                                                                      Feb 16 at 15:05












                                                                                                                    • 11




                                                                                                                      $begingroup$
                                                                                                                      I would not classify AT as a "casual" reading. I actually suffered reading that book when I took a course in AT. Also, he published in 2017, so I do not think he is retired.
                                                                                                                      $endgroup$
                                                                                                                      – EFinat-S
                                                                                                                      Feb 12 at 15:46






                                                                                                                    • 1




                                                                                                                      $begingroup$
                                                                                                                      Okay, "casual" is probably not the right word, at least for the middle and later parts of the book. But I think for the beginning chapters it might be applicable. And when one's interest has been captivated, the following suffering is ameliorated by a sense of purpose. Hatcher has retired from teaching I think many years ago, but is still active in research and writing. It states so on his uni web page: pi.math.cornell.edu/~hatcher But I guess it's not to be expected then that his public works would disappear from the web anyway.
                                                                                                                      $endgroup$
                                                                                                                      – Moritz Raguschat
                                                                                                                      Feb 16 at 15:05







                                                                                                                    11




                                                                                                                    11




                                                                                                                    $begingroup$
                                                                                                                    I would not classify AT as a "casual" reading. I actually suffered reading that book when I took a course in AT. Also, he published in 2017, so I do not think he is retired.
                                                                                                                    $endgroup$
                                                                                                                    – EFinat-S
                                                                                                                    Feb 12 at 15:46




                                                                                                                    $begingroup$
                                                                                                                    I would not classify AT as a "casual" reading. I actually suffered reading that book when I took a course in AT. Also, he published in 2017, so I do not think he is retired.
                                                                                                                    $endgroup$
                                                                                                                    – EFinat-S
                                                                                                                    Feb 12 at 15:46




                                                                                                                    1




                                                                                                                    1




                                                                                                                    $begingroup$
                                                                                                                    Okay, "casual" is probably not the right word, at least for the middle and later parts of the book. But I think for the beginning chapters it might be applicable. And when one's interest has been captivated, the following suffering is ameliorated by a sense of purpose. Hatcher has retired from teaching I think many years ago, but is still active in research and writing. It states so on his uni web page: pi.math.cornell.edu/~hatcher But I guess it's not to be expected then that his public works would disappear from the web anyway.
                                                                                                                    $endgroup$
                                                                                                                    – Moritz Raguschat
                                                                                                                    Feb 16 at 15:05




                                                                                                                    $begingroup$
                                                                                                                    Okay, "casual" is probably not the right word, at least for the middle and later parts of the book. But I think for the beginning chapters it might be applicable. And when one's interest has been captivated, the following suffering is ameliorated by a sense of purpose. Hatcher has retired from teaching I think many years ago, but is still active in research and writing. It states so on his uni web page: pi.math.cornell.edu/~hatcher But I guess it's not to be expected then that his public works would disappear from the web anyway.
                                                                                                                    $endgroup$
                                                                                                                    – Moritz Raguschat
                                                                                                                    Feb 16 at 15:05











                                                                                                                    3












                                                                                                                    $begingroup$

                                                                                                                    Silvanus P. Thompson's Calculus Made Easy is a good read. Per the reviews, the paperback and Kindle versions should be avoided in favor of the hardcover, although my softcover copy predates this by 2-3 decades.






                                                                                                                    share|cite|improve this answer











                                                                                                                    $endgroup$

















                                                                                                                      3












                                                                                                                      $begingroup$

                                                                                                                      Silvanus P. Thompson's Calculus Made Easy is a good read. Per the reviews, the paperback and Kindle versions should be avoided in favor of the hardcover, although my softcover copy predates this by 2-3 decades.






                                                                                                                      share|cite|improve this answer











                                                                                                                      $endgroup$















                                                                                                                        3












                                                                                                                        3








                                                                                                                        3





                                                                                                                        $begingroup$

                                                                                                                        Silvanus P. Thompson's Calculus Made Easy is a good read. Per the reviews, the paperback and Kindle versions should be avoided in favor of the hardcover, although my softcover copy predates this by 2-3 decades.






                                                                                                                        share|cite|improve this answer











                                                                                                                        $endgroup$



                                                                                                                        Silvanus P. Thompson's Calculus Made Easy is a good read. Per the reviews, the paperback and Kindle versions should be avoided in favor of the hardcover, although my softcover copy predates this by 2-3 decades.







                                                                                                                        share|cite|improve this answer














                                                                                                                        share|cite|improve this answer



                                                                                                                        share|cite|improve this answer








                                                                                                                        answered Feb 12 at 3:21


























                                                                                                                        community wiki





                                                                                                                        Crayoneater






















                                                                                                                            3












                                                                                                                            $begingroup$

                                                                                                                            Manfred Schroeder's book entitled Number Theory in Science and Communication: With Applications in Cryptography, Physics, Digital Information, Computing, and Self-Similarity is a wonderful introduction to applied discrete mathematics, or concrete mathematics, to use Knuth's phrase.



                                                                                                                            https://www.springer.com/gp/book/9783540852971



                                                                                                                            "A light-hearted and readable volume with a wide range of applications to which the author has been a productive contributor – useful mathematics outside the formalities of theorem and proof." Martin Gardner






                                                                                                                            share|cite|improve this answer











                                                                                                                            $endgroup$

















                                                                                                                              3












                                                                                                                              $begingroup$

                                                                                                                              Manfred Schroeder's book entitled Number Theory in Science and Communication: With Applications in Cryptography, Physics, Digital Information, Computing, and Self-Similarity is a wonderful introduction to applied discrete mathematics, or concrete mathematics, to use Knuth's phrase.



                                                                                                                              https://www.springer.com/gp/book/9783540852971



                                                                                                                              "A light-hearted and readable volume with a wide range of applications to which the author has been a productive contributor – useful mathematics outside the formalities of theorem and proof." Martin Gardner






                                                                                                                              share|cite|improve this answer











                                                                                                                              $endgroup$















                                                                                                                                3












                                                                                                                                3








                                                                                                                                3





                                                                                                                                $begingroup$

                                                                                                                                Manfred Schroeder's book entitled Number Theory in Science and Communication: With Applications in Cryptography, Physics, Digital Information, Computing, and Self-Similarity is a wonderful introduction to applied discrete mathematics, or concrete mathematics, to use Knuth's phrase.



                                                                                                                                https://www.springer.com/gp/book/9783540852971



                                                                                                                                "A light-hearted and readable volume with a wide range of applications to which the author has been a productive contributor – useful mathematics outside the formalities of theorem and proof." Martin Gardner






                                                                                                                                share|cite|improve this answer











                                                                                                                                $endgroup$



                                                                                                                                Manfred Schroeder's book entitled Number Theory in Science and Communication: With Applications in Cryptography, Physics, Digital Information, Computing, and Self-Similarity is a wonderful introduction to applied discrete mathematics, or concrete mathematics, to use Knuth's phrase.



                                                                                                                                https://www.springer.com/gp/book/9783540852971



                                                                                                                                "A light-hearted and readable volume with a wide range of applications to which the author has been a productive contributor – useful mathematics outside the formalities of theorem and proof." Martin Gardner







                                                                                                                                share|cite|improve this answer














                                                                                                                                share|cite|improve this answer



                                                                                                                                share|cite|improve this answer








                                                                                                                                answered Feb 12 at 9:04


























                                                                                                                                community wiki





                                                                                                                                kodlu
















                                                                                                                                    protected by Community Feb 12 at 14:44



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