In mathematics, a square root of a number x is a number r such that r2
= x, or in words, a number r whose square (the result of multiplying the number by itself) is x. Every non-negative real number x has a unique non-negative square root, called the principal square root and denoted with a radical symbol as √x. For example, the principal square root of 9 is 3, denoted √9 = 3, because 32 = 3 × 3 = 9. If otherwise unqualified, "the square root" of a number refers to the principal square root: the square root of 2 is approximately 1.4142.
Square roots often arise when solving quadratic equations, or equations of the form ax2
+ bx + c = 0, due to the variable x being squared.
Every positive number x has two square roots. One of them is √x, which is positive, and the other −√x, which is negative. Together, these two roots are denoted ±√x. Square roots of negative numbers can be discussed within the framework of complex numbers. Square roots of objects other than numbers can also be defined.
Square roots of integers that are not perfect squares are always irrational numbers: numbers not expressible as a ratio of two integers. For example, √2 cannot be written exactly as m/n, where n and m are integers. Nonetheless, it is exactly the length of the diagonal of a square with side length 1. This has been known since ancient times, with the discovery that √2 is irrational attributed to Hipparchus, a disciple of Pythagoras.