Why is $\frac 13$ rational?

Question

Could we say we know how it is going to behave so it is rational.

But we do not know how $\pi$ is going to behave.

Answer 1

By definition, a rational number is a number $q$ that can be written as a fraction in the form $q = a/b$ where $a$ and $b$ are integers and $b \neq 0$. So, $1/3$ is rational because it is exactly what you get when you divide one integer by another.

It turns out that the rational numbers are exactly the numbers whose decimals eventually terminate (as in $0.123$) or repeat (as in $0.12333\overline{3}$). This is not the same as saying that we don't know "how the number will behave". For example, the number $$ 0.101001000100001000001\dots $$ is irrational even though there's a clear pattern.

An interesting question, then, is how we know that $\pi$ isn't rational.