3 is an integer.
$\frac 13$ is a rational number.
$\sqrt 3$ is an irrational number.
But what about $\pi$? Or some other irrational infinitely repeating number that cannot be written as a ratio of square roots and rational numbers?
I know pi can be written compactly as $\frac c{2r}$ or something similar but that's not what I'm after. I'm interested in the numbers that are irrational but not square roots.
Is there a name for that set?
Best Answer
There is a set of numbers called the “constructible numbers” which are the numbers you can get starting from $1$ using addition, subtraction, multiplication, division by a nonzero number, and taking the square root of a nonnegative number. So you would be looking for the non-constructible numbers.
The term “constructible” comes from Ancient Greek geometry. In Greek geometry, you constructed figures by drawing lines (with a straight edge) and circles (with a compass), but no other operations were allowed. The constructible numbers are exactly the lengths that you can construct using a straight edge and compass and starting with a line segment of length $1$. Two of the classic problems of Greek geometry are: is $\pi$ constructible (a problem known as “squaring the circle” - if $\pi$ were constructible, then given a circle, you could make a square with the same area), and is $\sqrt[3]{2}$ constructible (known as “doubling the cube” - if you could construct $\sqrt[3]{2}$, then you could take one cube and make another with twice the volume)? It turns out that both of these numbers are non-constructible.