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Square Root (√x) ○◂｜Definition｜1st｜20260107023950-00-⌔

Square root - Wikipedia

Square root

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In mathematics, a square root of a number x is a number y such that $y^2=x$; in other words, a number y whose square (the result of multiplying the number by itself, or $y\cdot y$) is x.¹ For example, 4 and −4 are square roots of 16 because $4^2=(-4{)}^2=16$.

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 $\sqrt{x},$ where the symbol √ is called the radical, among other names,² and the line over the number is called the vinculum³ (sometimes considered a part of the radical symbol itself⁴). For example, to express the fact that the principal square root of 9 is 3, we write $\sqrt{9}=3$. 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 $x^{1/2}$.

Every positive number x has two square roots: $\sqrt{x}$ (which is positive) and $-\sqrt{x}$ (which is negative). The two roots can be written more concisely using the ± sign as $\pm \sqrt{x}$. 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.⁵⁶

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.

Printed 2026-06-28.

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Footnotes

1. Gel’fand, p. 120 Archived 2016-09-02 at the Wayback Machine ↩

2. “Squares and Square Roots”. www.mathsisfun.com. Retrieved 2020-08-28. ↩

3. Cajori, Florian (2012) [1928]. A History of Mathematical Notations. Vol. I. Dover. p. 208. ISBN 978-0-486-67766-8. ↩

4. Weisstein, Eric W. “Radical”. mathworld.wolfram.com. Retrieved 2026-05-19. ↩

5. 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 ↩

6. Weisstein, Eric W. “Square Root”. mathworld.wolfram.com. Retrieved 2020-08-28. ↩

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