# Is there a name for the set of irrational numbers that cannot be written with square roots?

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?

• The numbers you describe could be transcendental numbers.
– fwd
Commented Jan 22, 2022 at 18:17
• How is this a set-theory question? Commented Jan 22, 2022 at 18:20
• What exactly does it mean to be "written with square roots"? For instance, is $\frac{\sqrt{1 + \sqrt{3}} - \sqrt{5}}{\sqrt{7}-\sqrt{11}}$ "written with square roots"? Commented Jan 22, 2022 at 18:24
• from the tradition of compass and straightedge constructions, real numbers that can be expressed using just rational numbers, field operations, and square roots (of positive elements) , are sometimes called the constructible numbers. Commented Jan 22, 2022 at 18:25
• @fwd Not true, algebraic numbers cannot be written using radicals in general either. Commented Jan 22, 2022 at 18:39

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.