In mathematics, a square root of a number x is a number n such that n2 = x, or, in other words, a number n 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 x1/2. For example, the principal square root of 9 is 3, denoted , because 32 = 3 × 3 = 9 and 3 is non-negative.

The principal square root of a positive number, however, is only one of its two square roots. 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 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)

Similar to 0 things you like: