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