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In mathematics, a **square root** of a number x is a number y such that ; in other words, a number y whose *square* (the result of multiplying the number by itself, or ) is x.^{[1]} For example, 4 and −4 are square roots of 16 because .

Every nonnegative real number x has a unique nonnegative square root, called the *principal square root* or simply *the square root* (with a definite article, see below), which is denoted by where the symbol "" is called the *radical sign*^{[2]} or *radix*. For example, to express the fact that the principal square root of 9 is 3, we write . 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. For non-negative x, the principal square root can also be written in exponent notation, as .

Every positive number x has two square roots: (which is positive) and (which is negative). The two roots can be written more concisely using the ± sign as . 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.^{[3]}^{[4]}

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.

**^**Gel'fand, p. 120 Archived 2016-09-02 at the Wayback Machine**^**"Squares and Square Roots".*www.mathsisfun.com*. Retrieved 2020-08-28.**^**Zill, Dennis G.; Shanahan, Patrick (2008).*A First Course in Complex Analysis With Applications*(2nd ed.). Jones & Bartlett Learning. p. 78. ISBN 978-0-7637-5772-4. Archived from the original on 2016-09-01. Extract of page 78 Archived 2016-09-01 at the Wayback Machine**^**Weisstein, Eric W. "Square Root".*mathworld.wolfram.com*. Retrieved 2020-08-28.