square root
The square root of a nonnegative real number , written as , is the unique nonnegative real number such that . Thus, . Or, .
Example.
because and .
Example.
(see absolute value and even-even-odd rule) because .
In some situations it is better to allow two values for . For example, because and .
Over nonnegative real numbers, the square root operation is left distributive over multiplication
and division, but not over addition or subtraction
. That is, if and are nonnegative real numbers, then and .
Example.
because .
Example.
because .
On the other hand, in general, and . This error is an instance of the freshman’s dream error.
The square root notation is actually an alternative to exponentiation. That is, . When it is defined, the square root operation is commutative with exponentiation. That is, whenever both and . 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, is not a real number. This fact can be proven by contradiction (http://planetmath.org/ProofByContradiction) as follows: Suppose . If is negative, then is positive, and if is positive, then is also positive. Therefore, cannot be positive or negative. Moreover, cannot be zero either, because . Hence, .
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; is rational if and only if is a rational number which, after cancelling, is a fraction of two squares. In particular, is irrational.
The function is continuous for all nonnegative , and differentiable
for all positive (it is not differentiable for ). Its derivative
is given by:
It is possible to consider square roots in rings other than the integers or the rationals. For any ring , with , we say that is a square root of if .
When working in the ring of integers modulo (http://planetmath.org/MathbbZ_n), we give a special name to members of the ring that have a square root. We say is a quadratic residue
modulo if there exists coprime
to 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 |