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Positive and Negative Numbers Proof Challenge

If you are comfortable with the algebraic column on the Positive/Negative rules you may be ready for this challenge. Talk to me if you want to get credit for it.

Mathematics is a beautiful combination of wild intuition and pure logic. We use our intuition to figure out what ought to be true, and we use our logic to decide exactly what is true. Intuition is important or we'd never know what ideas to test out. Logic is important to test our intuition. Sometimes our intuition is wrong: our intuition may tell us that there are twice as many whole numbers as there are even integers (Because every other whole number is even). But we can prove mathematically that those sets have exactly the same size.

Want to try out a real mathematical proof?

A proof, at least in theory, is so convincing that it doesn't rely on intuition at all. All you need to believe it is a belief that logic works and a belief in your starting assumptions.

So what I am asking you to do is, assuming that the positive/negative rules are true, prove that a - b = -(b - a). You will probably have to consider a bunch of cases:

  1. |a| > |b|, a and b are both positive
  2. |a| > |b|, a is positive and b is negative
  3. |a| > |b|, a is negative and b is positive
  4. |a| > |b|, a and b are both negative

Cases 5-8 are like the above but |b| > |a|.

Proof of case 1

Assume |a| > |b| and both a and b are positive. Then a - b = a + -b by the subtraction rule, and a + -b = |a| - |b| by the "one positive, one negative" addition rule. Let's give the name z to |a| - |b|. So, putting those all together, a - b = z.

Also, b - a = b + -a by the subtraction rule and b + -a = -(|a| - |b|) by the "one positive, one negative" addition rule. Also, since |a| - |b| = z, we can take the opposite of each and get -(|a| - |b|) = -z. Putting all those together we get b - a = -z. Taking the negative of each, we get -(b - a) = -(-z) = z.

We have already shown that a - b = z and -(b - a) = z. Putting those together, we get a - b = -(b - a). Whew! Only 7 more cases to go!

I don't need to see all 7 of the other cases. Can you do one or two more? Or, can you come up with a simpler proof? There is one, I think, using integer multiplication and the distributive law, neither of which we've covered yet.