This is an ancient conundrum going back to the era of classical Greece.
The Greeks first believed that for any two geometric magnitudes there would exist a common unit which would divide evenly into both magnitudes.
So given, for example, the diagonal of a square and its side they thought there would be a tiny unit which would divide a whole number of times (let is say $m$ times) into the diagonal and a whole number of times (let us say $n$ times) into the side. Then if $d$ is the magnitude of the diagonal and $s$ the magnitude of the side, then
$$ d=\frac{m}{n}\times s $$
However it was learned that in the case of the diagonal and side of a square, there was no such common unit. The actual formula is
$$ d=\sqrt{2}\times s $$
and $\sqrt{2}$ is an irrational number, meaning it is not equal to the ratio of any two whole numbers $m$ and $n$.
However, they realized that if one made the units sufficiently small one could approximate, in this case, $\sqrt{2}$ or any such irrational magnitude, to approximate the magnitude with a rational number to any degree of precision required.
For example, $\dfrac{239}{169}\approx1.414201183$ whereas $\sqrt{2}\approx1.4142135562$. So the rational number $\dfrac{239}{169}$ is approximately equal to $\sqrt{2}$ to four decimal place precision. So if we divide the side of a square into $169$ equal units, that unit will go approximately, but not exactly, $239$ times into the diagonal of the square giving an approximation of the ratio of the diagonal to the side of the square.
