Happy April Fools’ Day

Happy April Fools’ to the zero people reading this blog. In honor of the day, I thought I’d make my first post something a little bizarre (and not about statistics).

I’m not ashamed to admit that negative numbers have never made complete sense to me. I think that’s because I (and most kids) first learned about the natural numbers, which we used to count things: our fingers, dollars, candy bars, and so on. When we’re eventually introduced to negative numbers, we try to fit them into this counting game we’re familiar with, but it just doesn’t work. For example, people tried to explain -5 to me as the absence of five dollars, but that felt frustratingly intangible compared to the simple concept of having five dollars.

As we get older, our abilities to think abstractly improve. We learn to think of negative and positive numbers as being symmetrically opposed about a number 0. With this schema in mind, negative numbers began to make sense to me: if +5 is five steps to the right, then -5 is five steps to the left.

While I’m proud to say that at the age of 23 I feel fully comfortable adding and subtracting numbers, I still never really understood why we’re allowed to multiply them the way we do. I’m mainly talking about why the product of two negative numbers equals a positive number. This fact is often explained with a statement along the following lines: positive and negative numbers are “opposites” of each other, and the opposite of an opposite is just the original value, so -(-5) equals 5. This explanation is unsatisfying to me, because the notion of an “opposite” is not defined anywhere. In fact, “opposite” just feels like a synonym for the word negative, so this explanation boils down to re-stating that a negative times a negative equals a positive.

I still don’t have a good intuition for this concept, so I’ve decided that if I can prove it then I can at least convince myself that it is true. Here is my proof that negative one times negative one equals one:

Instead of thinking of multiplication rules as intuitive properties of a number system, it helps me to view them as facts which follow from application of the number system’s axioms. Because the axioms are all things that make sense intuitively, the fact that they imply certain multiplication rules means that I understand why the rules are the way they are. Nevertheless, if someone can come up with an explanation for why that relies only on our human instincts or on counting dollars and candy bars, I would love to hear it.