π, the ratio of a circle’s circumference to its diameter, is an irrational number, which means it can’t be written as a fraction a/b, where a and b are integers. That means that, unlike decimals like 1/4, or 0.25, or repeating decimals, like 1/3 or 0.33333333, it neither terminates nor repeats. It goes on forever without repeating itself: 3.1415926. . . . ad infinitum.
Now you can prove that pi must be irrational; Wikipedia gives six explanations, and I’ll put one video proof below. All of these proofs depend not on the geometry of a circle, but on the fact that pi appears in certain trigonometric relationships.
What I think is weird is not pi’s irrationality itself, but simply the fact that a ratio that so important for geometry turns out to be an irrational number. Why couldn’t it be THREE? As Jason Rosenhouse pointed out a long time ago, the Bible implies that it is three, showing that God was ignorant. And you may be aware that at the end of the 19th century, the Indiana state legislature considered (but rejected) a bill including an assertion that the value of pi was 3.2.
The answer whyπ is irrational is, I suspect, simply “that’s the way it is.” But if there’s some proof out there that the ratio of a circle’s circumference to its diameter, based on the geometry alone, must be irrational, I’d like to know about it.
And I’m a bit surprised that nobody uses the irrationality of π as an inexplicable fact about mathematics that implies the existence of God. And then the astute theomathematician could bring up the square root of two. . .