Does a proportion have to be a rational number?

Question

Does a proportion have to be a rational number?

For example, Assume we have a square with side $2$ units. We are throwing a circle of radius $1$ unit over the square. Let $X$ be the area of the square covered by the circle relative to the area of the whole square. Hence, $X$ takes value in the interval $[0,{\pi \over 4}]$.

Is $X$ considered a proportion?

Thanks in advance

Answer 1

If the definition of proportion is that it is a ratio, then no, a proportion does not have to be a rational number. It depends on whether one of the quantities in the proportion is irrational. The product of a nonzero rational and irrational number is irrational (and the same for division)...

And yes $X$ is definitely a ratio, as you have defined it...in fact for any square encompassing a circle we have $\pi r^2/(2r)^2=\pi / 4$, which is the ratio of the area of the circle to the area of the square.

Answer 2

Proportions don't have to be rational - it's perfectly reasonable to call any quotient $\frac{a}{b}$ a proportion, for arbitrary real numbers $a,b$.

In fact, there's a special name for two $a,b$ whose quotient is not rational. Such a pair of numbers $a,b$ is called incommensurable.

Whether or not a pair $a,b$ is incommensurable or not is significant if you're looking for integral solutions of $$ a\cdot n = b\cdot m \text{.} $$ A pair $n,m \in \mathbb{Z}$ with $a\cdot n = b \cdot m$ exists exactly if $a,b$ are not incommensurable, i.e. if $\frac{a}{b}$ is rational. You then have $\frac{a}{b} = \frac{m}{n}$.