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( Square root) In mathematics, a square root (often abbreviated as sqrt, especially as function names in computer-related contexts) of a number x is a number r such that r² = x, or, in other 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, which is denoted with a radical symbol as , or, using exponent notation, as x^1/2. For example, the principal square root of 9 is 3, denoted = 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.

Every positive number x has two square roots. One of them is , which is positive, and the other , which is negative. Together, these two roots are denoted . Square roots of negative numbers can be discussed within the framework of complex numbers. More generally, square roots can obviously be considered in any context in which a notion of "squaring" of some mathematical objects is defined (including algebras of matrices, endomorphism rings, etc).

Square roots of integers that are not perfect squares are always irrational numbers numbers not expressible as a ratio of two integers. For example, 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 is irrational attributed to Hippasus, a disciple of Pythagoras. (See square root of 2 for proofs of the irrationality of this number and quadratic irrational for a proof for all non-square natural numbers)

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