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Transcendence

In addition to being irrational, π is also a transcendental number, which means that it is not the solution of any non-constant polynomial equation with rational coefficients, such as x 5 120 − x 3 6 + x = 0 {\textstyle {\frac {x^{5}}{120}}-{\frac {x^{3}}{6}}+x=0}.

The transcendence of π has two important consequences: First, π cannot be expressed using any finite combination of rational numbers and square roots or n -th roots (such as 31 3 {\displaystyle {\sqrt{31}}} or 10 {\displaystyle {\sqrt {10}}}). Second, since no transcendental number can be constructed with compass and straightedge, it is not possible to " square the circle ". In other words, it is impossible to construct, using compass and straightedge alone, a square whose area is exactly equal to the area of a given circle. Squaring a circle was one of the important geometry problems of the classical antiquity. Amateur mathematicians in modern times have sometimes attempted to square the circle and claim success—despite the fact that it is mathematically impossible.