Dear Uncle Colin,

Is there a rigorous explanation of why the direction of the inequality changes when you flip the sign?

- Fuzzy Logic, Inequality Puzzle

Hi, FLIP, and thanks for your message!

Here are my best efforts at a rigorous explanation; I’d be interested to read anyone else’s go at it!

Version 1

• Define the relationship $a > b$ to mean “there exists a positive number $x$ such that $a = b+x$”.
• Define the relationship $c < d$ to mean “there exists a positive number $y$ such that $c + y = d$”.
• Suppose $a > b$.
• i.e., there is a positive number $x$ such that $a = b + x$.
• Multiplying by -1, $-a = -b - x$.
• Adding $x$ to each side gives $-a + x = -b$
• So $-a < -b$, as required.

Version 2

• Suppose $a > b$.
• Add $(-b)$ to each side: $a - b > 0$.
• Add $(-a)$ to each side: $-b > -a$.
• This is the same thing as $-a < -b$.

I hope at least one of those is rigorous and clear enough!

- Uncle Colin