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