| Following on from my other reply... When we start teaching math to students, we start with counting blocks: "You have 2 piles of blocks, one pile of 3 and another pile of 2. If you put them together, you get a pile of 5 blocks!" That stops working as well when you deal with fractions. You can get away with 2.5 blocks, but 2.5 blocks is really 3 blocks, but one is a little smaller than the others. And at some point you can't use blocks to represent 2.3456 blocks. So you need different kinds of "natural" problems to represent those numbers. But, as you point out. There are some things that aren't really representable as "natural problems". For a long time the idea of 0 wasn't natural. (People were actually killed for talking about the idea of 0) I mean, what does it mean to have 0 chickens? You either have some chickens, and you say "I have N chickens", or you don't have any chickens and you say nothing. Why would you need a number to represent nothing? Maybe n^2.1 doesn't have a natural explanation. At least, not one you can hold in your hand. Can you imagine a shape with 2.1 dimensions to relate it to geometry? Probably not. But you can use geometry to prove that n^(a+b) = n^a * n^b and then you can apply those rules to "unnatural values" with an understanding of what is happening. The natural explanation of n^2 can be applied to the unnatural idea of n^2.1 Everything in math can't be understood with geometry or "natural examples", lots of math (most of math?) describes things that are not representable within the constraints of our physical world. That's what makes it so powerful! Also, not everything in math can just be calculated (see: irrational numbers) |
Yes! A long time ago I read a wonderful little book on just this bit of history:
https://www.amazon.com/Zero-Biography-Dangerous-Charles-Seif...