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)2≡x. Or, √x×√x≡x.
Example.
√9=3 because 3≥0 and 32=3×3=9.
Example.
√x2+2x+1=|x+1| (see absolute value 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 operation is left distributive over multiplication
and division, but not over addition or subtraction
. That is, if x and y are nonnegative real numbers, then √x×y=√x×√y and √xy=√x√y.
Example.
√x2y2=xy because (xy)2=xy×xy=x×x×y×y=x2×y2=x2y2.
Example.
√925=35 because (35)2=3252=925.
On the other hand, in general, √x+y≠√x+√y and √x-y≠√x-√y. This error is an instance of the freshman’s dream error.
The square root notation is actually an alternative to exponentiation. That is, √x≡x12. 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 restrictions can be lifted if we extend the domain and codomain of the square root function to the complex numbers
.
Negative real numbers do not have real square roots. For example, √-4 is not a real number. This fact can be proven by contradiction (http://planetmath.org/ProofByContradiction) as follows: Suppose √-4=x∈ℝ. 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∉ℝ.
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 numbers; √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 continuous for all nonnegative x, and differentiable
for all positive x (it is not differentiable for x=0). Its derivative
is given by:
ddx(√x)=12√x |
It is possible to consider square roots in rings other than the integers or the rationals. For any ring R, with x,y∈R, we say that y is a square root of x if y2=x.
When working in the ring of integers 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 residue
modulo n if there exists y coprime
to x such that . Rabin’s cryptosystem is based on the difficulty of finding square roots modulo an integer .
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 |