It's amazing, isn't it? This was one of the best parts of math class. It's so natural, too. Numbers, you count, 1, 2, 3, 4... it makes so much sense. And of

It's amazing, isn't it? This was one of the best parts of math class. It's so natural, too. Numbers, you count, 1, 2, 3, 4... it makes so much sense. And of course, rational numbers aren't too hard to grasp within that context. They're expressible with the nice numbers we know. Even today, there are a lot of people who consider pi a strange, unusual number, an oddity. If they're aware of the square root of two, or euler's constant, they may consider them to be unusual as well. At least in the US, it isn't until you've studied math for a while that you really do get into the notion of limits. A professor in Calc might talk about this in more depth (maybe when Taylor series come up) or high school geometry might mention that pi isn't actually that unusual. But I honestly don't blame people that this is hard to grasp. It's a long way from the "numbers" you understand from an early age, when you learn to count. But yeah, it says something that numbers that occur "naturally" are almost never representable as the ratio of two integers. The ratio of a circumference to a radius. The "long" side of a right triangle with "legs" of a single unit. The number whose natural log is 1. Almost every number that took us by surprise, that we didn't invent but were instead required to measure and explain, remarkably, isn't expressible the way we thought numbers would be. Can I go on a bit here and say that math gets hard the moment almost everything else gets hard, when you get out of your own head and try to see things as they actually are? When I was a kid, I loved "mechanical drawing", because it all looked so good on the page. I drew human creations, cars, houses, everything with nice lines, it all looked so sharp and snappy. And then you try to draw nature... where are the lines? It turns out you can't represent nature very well with lines. You have to draw what isn't there, not what is. The first time this comes up is when you try to draw someone's face. Same for numbers. Draw a cube, measure the volume. Ok, easy. Draw a cylinder, measure the volume. A little harder, you have to make a good circle, but it can be done. Measure the volume - now you need this strange, "irrational" number that. Now go draw a tree trunk... Hard. There isn't even a nice circle there. And to actually take the volume or surface area accurately, you need some way to measure the space under the curve, which requires calculus, little squares above and below the line, which can be made smaller and smaller, but they never reach zero. And the limit of the upper and lower sequence isn't an approximation of the number. The limit is the number. And it turns out that it's exceptionally unusual for that limit to be expressible as a ratio of two integers. It's just unreal, isn't it? Er, I mean, real. ;)