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In mathematics, a square root (√) of a number x is a number r such that r^{2} = x, or, in other words, a number r whose square (the result of multiplying the number by itself, or r × r) is x.
Every nonnegative real number x has a unique nonnegative square root, called the principal square root, denoted by a radical sign as . For positive x, the principal square root can also be written in exponent notation, as x^{1/2}. For example, the principal square root of 9 is 3, denoted , because 3^{2} = 3 × 3 = 9 and 3 is nonnegative. Although the principal square root of a positive number is only one of its two square roots, the designation "the square root" is often used to refer to the principal square root.
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 (see ± shorthand). 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 (that is to say they cannot be written exactly as m/n, where n and m are integers). This is the theorem Euclid X, 9 almost certainly due to Theaetetus dating back to circa 380 BC.^{[1]} The particular case is assumed to date back earlier to the Pythagoreans and is traditionally attributed to Hippasus. It is exactly the length of the diagonal of a square with side length 1.
The term whose root is being considered is known as the radicand. For example, in the expression , ab + 2 is the radicand. The radicand is the number or expression underneath the radical sign.
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