square root

The square root of a nonnegative real number x, written as x, is the unique nonnegative real number y such that y2=x. Thus, (x)2x. Or, x×xx.


9=3 because 30 and 32=3×3=9.


x2+2x+1=|x+1| (see absolute valueMathworldPlanetmathPlanetmathPlanetmath and even-even-odd rule) because (x+1)2=(x+1)(x+1)=x2+x+x+1=x2+2x+1.

In some situations it is better to allow two values for x. For example, 4=±2 because 22=4 and (-2)2=4.

Over nonnegative real numbers, the square root operationMathworldPlanetmath is left distributive over multiplicationPlanetmathPlanetmath and division, but not over addition or subtractionPlanetmathPlanetmath. That is, if x and y are nonnegative real numbers, then x×y=x×y and xy=xy.


x2y2=xy because (xy)2=xy×xy=x×x×y×y=x2×y2=x2y2.


925=35 because (35)2=3252=925.

On the other hand, in general, x+yx+y and x-yx-y. This error is an instance of the freshman’s dream error.

The square root notation is actually an alternative to exponentiation. That is, xx12. When it is defined, the square root operation is commutative with exponentiation. That is, xa=xa2=(x)a whenever both xa>0 and x>0. The restrictionsPlanetmathPlanetmath can be lifted if we extend the domain and codomain of the square root function to the complex numbersMathworldPlanetmathPlanetmath.

Negative real numbers do not have real square roots. For example, -4 is not a real number. This fact can be proven by contradictionMathworldPlanetmathPlanetmath (http://planetmath.org/ProofByContradiction) as follows: Suppose -4=x. If x is negative, then x2 is positive, and if x is positive, then x2 is also positive. Therefore, x cannot be positive or negative. Moreover, x cannot be zero either, because 02=0. Hence, -4.

For additional discussion of the square root and negative numbers, see the discussion of complex numbers.

The square root function generally maps rational numbers to algebraic numbersMathworldPlanetmath; x is rational if and only if x is a rational number which, after cancelling, is a fraction of two squares. In particular, 2 is irrational.

The function is continuousMathworldPlanetmathPlanetmath for all nonnegative x, and differentiableMathworldPlanetmathPlanetmath for all positive x (it is not differentiable for x=0). Its derivativePlanetmathPlanetmath is given by:


It is possible to consider square roots in rings other than the integers or the rationals. For any ring R, with x,yR, we say that y is a square root of x if y2=x.

When working in the ring of integersMathworldPlanetmath modulo n (http://planetmath.org/MathbbZ_n), we give a special name to members of the ring that have a square root. We say x is a quadratic residueMathworldPlanetmath modulo n if there exists y coprimeMathworldPlanetmathPlanetmath to x such that y2x(modn). Rabin’s cryptosystem is based on the difficulty of finding square roots modulo an integer n.

Title square root
Canonical name SquareRoot
Date of creation 2013-03-22 11:57:19
Last modified on 2013-03-22 11:57:19
Owner Wkbj79 (1863)
Last modified by Wkbj79 (1863)
Numerical id 27
Author Wkbj79 (1863)
Entry type Definition
Classification msc 11A25
Related topic CubeRoot
Related topic NthRoot
Related topic RationalNumber
Related topic IrrationalNumber
Related topic RealNumber
Related topic ComplexNumber
Related topic Complex
Related topic DerivativeOfInverseFunction
Related topic EvenEvenOddRule