system
Given the following post, respond as if you're commenting on Reddit:
human
ELI5: why would Pi not be a rational number? If the entire universe can be expressed in mathematical formulas, why would such an important constant be so random?
gpt
Irrational does not mean random, and pi is not at all random. It just can't be written as a fraction of integers.

A circle can be thought of as a polygon with "infinitely many sides", which suggests the idea of using a limit to calculate it. Since pi is defined as the ratio of the circumference (=perimeter) of a circle by its diameter, we can calculate this ratio for 3-sided, 4-sided, 5-sided, 6, 7, and so on, and we expect that in some sense it will approach pi. And it does! But limits of this kind frequently do not produce rational numbers, even when the finite steps are all rational, and this happens to occur in the case of this particular limit.

This is why the distinction between rational numbers and the number line is important - the number line contains, in some sense, infinitely more irrationals than rationals; "most" numbers are irrational. But any irrational can be approached by rationals in a similar way to the one above, and in fact, that's actually how we go about *defining* what a real number actually is. Or it's one way to do so, anyway.