In mathematics, a square root of a number x is a number y such that y2 = x; in other words, a number y whose square (the result of multiplying the number by itself, or y ⋅ y) is x. For example, 4 and −4 are square roots of 16, because 42 = (−4)2 = 16.
Every nonnegative real number x has a unique nonnegative square root, called the principal square root, which is denoted by
x
,
{\displaystyle {\sqrt {x}},}
where the symbol
{\displaystyle {\sqrt {~^{~}}}}
is called the radical sign or radix. For example, the principal square root of 9 is 3, which is denoted by
9
=
3
,
{\displaystyle {\sqrt {9}}=3,}
because 32 = 3 ⋅ 3 = 9 and 3 is nonnegative. The term (or number) whose square root is being considered is known as the radicand. The radicand is the number or expression underneath the radical sign, in this case 9.
Every positive number x has two square roots:
x
,
{\displaystyle {\sqrt {x}},}
which is positive, and
−
x
,
{\displaystyle -{\sqrt {x}},}
which is negative. Together, these two roots are denoted as
±
x
{\displaystyle \pm {\sqrt {x}}}
(see ± shorthand). 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. For positive x, the principal square root can also be written in exponent notation, as x1/2.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 the "square" of a mathematical object is defined. These include function spaces and square matrices, among other mathematical structures.