On Pi and Pythagoras

Here's a reason why π is interesting: Everyone knows it's 3.14159265358...etc, and since that's cumbersome you might just want to write it as approximately 22/7, which comes out to 3.142857142857...etc, which is a little too high. So maybe it would be more accurate to write 22/7 as the exact equivalent 220/70, and make that a tiny bit lower, like say nudge it down to 219/70. Now that comes out to 3.12857142857...etc, which is a little too small. Well then use another exact equivalent like 2200/700 instead, and nudge THAT down a little to 2199/700, a much smaller change, now 3.14142857143...etc -- oops, still too small. Well you can get closer and closer approximations by doing that with smaller and smaller changes, but you'll never find ANY two numbers (well, integers) that give exactly π when you divide them. It's irrational -- literally "not a ratio" of other numbers. It can't be made from other numbers and can't "line up with" other numbers.

Like, say you have a circle and you measure its diameter to be 10 feet and its circumference to be 10π feet (approximately 31.415926 feet, but we mean EXACTLY 10π feet since the ratio of the circle's circumference to diameter is π, by definition). If you lay out a room that is 10 feet long by 10π feet wide, then you can never cover that floor with square tiles, because the tiles' length and width are in the same units (being literally the same). Look, if they were square tiles that were 1 foot on each side, you'd say, of course you can't -- you can fit 31 of them across the width of the room, but then you'd have .415926...etc feet left untiled along the whole edge. So use smaller square tiles, that will work won't it? Maybe use tiles that are just 1 inch across. That'll fill in a bunch of that uncovered space, but it won't fill it exactly and there will still be some untiled floor. Fine, just use smaller and smaller tiles, at some point it will be fine enough tiling to fill in all the space exactly, right? Nope, never ever. The tiles can be 1 mm squares, or 1 billionth of an inch, and they will never exactly cover that width, because the width (the original circle's circumference) can't be measured in units of the length (the original circle's diameter). If it could, that would be a ratio, which π is not. You can make the room 10 million miles long by 10π million miles wide, and make your tiles just one square atom in size, and they won't cover that floor exactly.

The fact that there are irrational numbers like this drove Pythagoras nuts because he had a whole philosophy, a religion practically, that depended on numbers being the basic building blocks of the universe, and then it turned out you couldn't even measure all the NUMBERS in terms of each other, never mind the rest of the universe. "Irrational" in the non-mathematical sense still means "outside of what is rational", or outside of the way we think of things fitting together.

Here's a reason why π is NOT interesting though: it turns out MOST numbers are irrational. They're not some weird subset that doesn't fit in with the integers, they're actually the norm. So, ho-hum, the nerd holiday "Pi Day" (March 14 is "3/14") is a day to celebrate yet another irrational number with a non-repeating decimal notation, of which there are infinitely many. Someone should try to memorize all those other digits for once.