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Is there a mathematical reason (like a proof) of why this happens? You can do it with examples and it is 'intuitive.' But the proof of why this happens is never shown in pedagogy, we just warn students to remember to flip the inequality when

• multiply or divide by a negative number both sides

$$-2>-3 => 2 < 3$$

• take reciprocals of same sign fractions both sides

$$frac34 > frac12 => frac43 < 2$$

Depending on the context and the previous curriculum, the following might work:

1. "less than" means "to the left of" on the number line.


2. Multiplying by a negative number flips numbers around 0.


3. Thus, "left of" becomes "right of", or "greater than".

I'm slightly concerned that

Is there a mathematical reason (like a proof) of why this happens?

is a purely mathematical question, but since you write "we just warn students" I will assume that this question is purposefully asked here on Math Educators StackExchange.

As to a proof:

Given $a>b$, subtract $a$ from both sides: $0 > b-a$.

Next, subtract $b$ from both sides: $-b > -a$.

Note that this final inequality is equivalent to $-a < -b$.

And so we have proved: If $a > b$, then $-a < -b$.

For multiplying or dividing by -1...
$$beginalign
a&>b\
a-b&>0\
\-(a-b)&<0\
-a&<-b
endalign
$$
(You can then extended to arbitrary negative numbers by multiplying or dividing by the [positive] magnitude.)

For taking reciprocals... assuming $ab>0$

$$beginalign
a&>b\
left(frac1abright)a&>left(frac1abright)b\
frac1b&>frac1a\
frac1a&<frac1b\
endalign
$$