Complex numbers proof with modulus argument question
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I'm having trouble with this complex numbers proof.
Prove $|(z-a)/(overline az-1)|=1$ if $a,z$ are any complex numbers, where $zne a$ and $|z|=1$.
I tried substituing $z,a$ for general Cartesian and polar forms, but I couldn't get past the algebra or required simplifications.
complex-numbers
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up vote
1
down vote
favorite
I'm having trouble with this complex numbers proof.
Prove $|(z-a)/(overline az-1)|=1$ if $a,z$ are any complex numbers, where $zne a$ and $|z|=1$.
I tried substituing $z,a$ for general Cartesian and polar forms, but I couldn't get past the algebra or required simplifications.
complex-numbers
add a comment |Â
up vote
1
down vote
favorite
up vote
1
down vote
favorite
I'm having trouble with this complex numbers proof.
Prove $|(z-a)/(overline az-1)|=1$ if $a,z$ are any complex numbers, where $zne a$ and $|z|=1$.
I tried substituing $z,a$ for general Cartesian and polar forms, but I couldn't get past the algebra or required simplifications.
complex-numbers
I'm having trouble with this complex numbers proof.
Prove $|(z-a)/(overline az-1)|=1$ if $a,z$ are any complex numbers, where $zne a$ and $|z|=1$.
I tried substituing $z,a$ for general Cartesian and polar forms, but I couldn't get past the algebra or required simplifications.
complex-numbers
complex-numbers
edited 27 mins ago
Parcly Taxel
36.3k136994
36.3k136994
asked 29 mins ago
anonymous
1,5901036
1,5901036
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add a comment |Â
3 Answers
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up vote
2
down vote
accepted
$$|(z-a)/(overline az-1)|=1iff|z-a|=|overline az-1|$$
Now, since $|z|=1$,
$$|overline az-1|=|overline a-1/z|=|overline a-overline z|=|a-z|=|z-a|$$
add a comment |Â
up vote
2
down vote
That's equivalent to $|z-a|=|overline az-1|$. As $|z|=1$, $overline z=z^-1$ and
so
$$|overline az-1|=|z||overline a -z^-1|=|overline a -overline z|
=|overlinea-z|=|a-z|.$$
add a comment |Â
up vote
2
down vote
Another approach is using $z=e^it$ then
$$left|dfracz-aoverlineaz-1right|=left|dfrace^it-aoverlineae^-it-1right|=dfrac=dfrac=1$$
add a comment |Â
3 Answers
3
active
oldest
votes
3 Answers
3
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
2
down vote
accepted
$$|(z-a)/(overline az-1)|=1iff|z-a|=|overline az-1|$$
Now, since $|z|=1$,
$$|overline az-1|=|overline a-1/z|=|overline a-overline z|=|a-z|=|z-a|$$
add a comment |Â
up vote
2
down vote
accepted
$$|(z-a)/(overline az-1)|=1iff|z-a|=|overline az-1|$$
Now, since $|z|=1$,
$$|overline az-1|=|overline a-1/z|=|overline a-overline z|=|a-z|=|z-a|$$
add a comment |Â
up vote
2
down vote
accepted
up vote
2
down vote
accepted
$$|(z-a)/(overline az-1)|=1iff|z-a|=|overline az-1|$$
Now, since $|z|=1$,
$$|overline az-1|=|overline a-1/z|=|overline a-overline z|=|a-z|=|z-a|$$
$$|(z-a)/(overline az-1)|=1iff|z-a|=|overline az-1|$$
Now, since $|z|=1$,
$$|overline az-1|=|overline a-1/z|=|overline a-overline z|=|a-z|=|z-a|$$
answered 19 mins ago
Parcly Taxel
36.3k136994
36.3k136994
add a comment |Â
add a comment |Â
up vote
2
down vote
That's equivalent to $|z-a|=|overline az-1|$. As $|z|=1$, $overline z=z^-1$ and
so
$$|overline az-1|=|z||overline a -z^-1|=|overline a -overline z|
=|overlinea-z|=|a-z|.$$
add a comment |Â
up vote
2
down vote
That's equivalent to $|z-a|=|overline az-1|$. As $|z|=1$, $overline z=z^-1$ and
so
$$|overline az-1|=|z||overline a -z^-1|=|overline a -overline z|
=|overlinea-z|=|a-z|.$$
add a comment |Â
up vote
2
down vote
up vote
2
down vote
That's equivalent to $|z-a|=|overline az-1|$. As $|z|=1$, $overline z=z^-1$ and
so
$$|overline az-1|=|z||overline a -z^-1|=|overline a -overline z|
=|overlinea-z|=|a-z|.$$
That's equivalent to $|z-a|=|overline az-1|$. As $|z|=1$, $overline z=z^-1$ and
so
$$|overline az-1|=|z||overline a -z^-1|=|overline a -overline z|
=|overlinea-z|=|a-z|.$$
answered 22 mins ago
Lord Shark the Unknown
92.9k956122
92.9k956122
add a comment |Â
add a comment |Â
up vote
2
down vote
Another approach is using $z=e^it$ then
$$left|dfracz-aoverlineaz-1right|=left|dfrace^it-aoverlineae^-it-1right|=dfrac=dfrac=1$$
add a comment |Â
up vote
2
down vote
Another approach is using $z=e^it$ then
$$left|dfracz-aoverlineaz-1right|=left|dfrace^it-aoverlineae^-it-1right|=dfrac=dfrac=1$$
add a comment |Â
up vote
2
down vote
up vote
2
down vote
Another approach is using $z=e^it$ then
$$left|dfracz-aoverlineaz-1right|=left|dfrace^it-aoverlineae^-it-1right|=dfrac=dfrac=1$$
Another approach is using $z=e^it$ then
$$left|dfracz-aoverlineaz-1right|=left|dfrace^it-aoverlineae^-it-1right|=dfrac=dfrac=1$$
answered 12 mins ago
Nosrati
24.1k61952
24.1k61952
add a comment |Â
add a comment |Â
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