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Abstract Photography Downtown DecaturIn 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 \sqrt x. For example, the principal square root of 9 is 3, denoted \sqrt 9 = 3, because 3^2 = 3\times3 = 9. The other square root of 9 is -3.
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Square roots often arise when solving quadratic equations, or equations of the form ax2 + bx + c = 0, due to the variable x being squared.
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Every positive number x has two square roots. One of them is \sqrt{x}, which is positive, and the other is -\sqrt{x}, which is negative. Together, these two roots are denoted \pm\sqrt{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.
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Square roots of integers that are not perfect squares are always irrational numbers: numbers not expressible as a ratio of two integers. For example, \sqrt 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 \sqrt 2 is irrational attributed to Hipparchus, a disciple of Pythagoras. (See square root of 2 for proofs of the irrationality of this number.)
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