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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?

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    $\begingroup$ The numbers you describe could be transcendental numbers. $\endgroup$
    – fwd
    Commented Jan 22, 2022 at 18:17
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    $\begingroup$ How is this a set-theory question? $\endgroup$ Commented Jan 22, 2022 at 18:20
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    $\begingroup$ 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"? $\endgroup$
    – Lee Mosher
    Commented Jan 22, 2022 at 18:24
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    $\begingroup$ 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. $\endgroup$
    – Will Jagy
    Commented Jan 22, 2022 at 18:25
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    $\begingroup$ @fwd Not true, algebraic numbers cannot be written using radicals in general either. $\endgroup$
    – Peter
    Commented Jan 22, 2022 at 18:39

1 Answer 1

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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.

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    $\begingroup$ Exactly what I was looking for, thank you! $\endgroup$ Commented Jan 23, 2022 at 15:41

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