Square root

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In mathematics, a square root (√) 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, or r × r) is x.

Every non-negative real number x has a unique non-negative square root, called the principal square root, denoted by a radical sign as \scriptstyle \sqrt{x}. For positive x, the principal square root can also be written in exponent notation, as x1/2. For example, the principal square root of 9 is 3, denoted \scriptstyle \sqrt{9} \ = \ 3, because 32 = 3 × 3 = 9 and 3 is non-negative. 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 \scriptstyle \sqrt{x}, which is positive, and the other \scriptstyle -\sqrt{x}, which is negative. Together, these two roots are denoted \scriptstyle \pm\sqrt{x} (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 \scriptstyle \sqrt{2} 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 \scriptstyle \sqrt[n]{ab+2}, ab + 2 is the radicand. The radicand is the number or expression underneath the radical sign.


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