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) 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 x 1/2 . For example, the principal square root of 9 is 3, denoted \scriptstyle \sqrt{9} \ = \ 3, because and 3 is non-negative. Although the principal square root of a positive number is only one of its two square roots, the designation "
... see moreIn 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) 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 x 1/2 . For example, the principal square root of 9 is 3, denoted \scriptstyle \sqrt{9} \ = \ 3, because 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. For example, \scriptstyle \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 \scriptstyle \sqrt{2} is irrational attributed to Hippasus, a disciple of Pythagoras.
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.
Properties
The principal square root function \scriptstyle f(x) = \sqrt{x} (usually just referred to as the "square root function") is a function that maps the set of non-negative real numbers onto itself. In geometrical terms, the square root function maps the area of a square to its side length.
The square root of x is rational if and only if x is a rational number that can be represented as a ratio of two perfect squares. (See square root of 2 for proofs that this is an irrational number, and quadratic irrational for a proof for all non-square natural numbers.) The square root function maps rational numbers into algebraic numbers (a superset of the rational numbers).
For all real numbers x
\sqrt{x^2} = \left|x\right| = \begin{cases} x, & \mbox{if }x \ge 0 \\ -x, & \mbox{if }x < 0. \end{cases} (see absolute value) ...
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