# square root

The square root of a nonnegative real number $x$, written as $\sqrt{x}$, is the unique nonnegative real number $y$ such that ${y}^{2}=x$. Thus, ${(\sqrt{x})}^{2}\equiv x$. Or, $\sqrt{x}\times \sqrt{x}\equiv x$.

###### Example.

$\sqrt{9}=3$ because $3\ge 0$ and ${3}^{2}=3\times 3=9$.

###### Example.

$\sqrt{{x}^{2}+2x+1}=|x+1|$ (see absolute value^{} and even-even-odd rule) because ${(x+1)}^{2}=(x+1)(x+1)={x}^{2}+x+x+1={x}^{2}+2x+1$.

In some situations it is better to allow two values for $\sqrt{x}$. For example, $\sqrt{4}=\pm 2$ because ${2}^{2}=4$ 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 $\sqrt{x\times y}=\sqrt{x}\times \sqrt{y}$ and $\sqrt{{\displaystyle \frac{x}{y}}}={\displaystyle \frac{\sqrt{x}}{\sqrt{y}}}$.

###### Example.

$\sqrt{{x}^{2}{y}^{2}}=xy$ because ${(xy)}^{2}=xy\times xy=x\times x\times y\times y={x}^{2}\times {y}^{2}={x}^{2}{y}^{2}$.

###### Example.

$\sqrt{{\displaystyle \frac{9}{25}}}={\displaystyle \frac{3}{5}}$ because ${\left({\displaystyle \frac{3}{5}}\right)}^{2}={\displaystyle \frac{{3}^{2}}{{5}^{2}}}={\displaystyle \frac{9}{25}}$.

On the other hand, in general, $\sqrt{x+y}\ne \sqrt{x}+\sqrt{y}$ and $\sqrt{x-y}\ne \sqrt{x}-\sqrt{y}$. This error is an instance of the freshman’s dream error.

The square root notation is actually an alternative to exponentiation. That is, $\sqrt{x}\equiv {x}^{\frac{1}{2}}$. When it is defined, the square root operation is commutative with exponentiation. That is, $\sqrt{{x}^{a}}={x}^{\frac{a}{2}}={(\sqrt{x})}^{a}$ whenever both ${x}^{a}>0$ 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, $\sqrt{-4}$ is not a real number. This fact can be proven by contradiction^{} (http://planetmath.org/ProofByContradiction) as follows: Suppose $\sqrt{-4}=x\in \mathbb{R}$. If $x$ is negative, then ${x}^{2}$ is positive, and if $x$ is positive, then ${x}^{2}$ is also positive. Therefore, $x$ cannot be positive or negative. Moreover, $x$ cannot be zero either, because ${0}^{2}=0$. Hence, $\sqrt{-4}\notin \mathbb{R}$.

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^{}; $\sqrt{x}$ is rational if and only if $x$ is a rational number which, after cancelling, is a fraction of two squares. In particular, $\sqrt{2}$ 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:

$$\frac{d}{dx}\left(\sqrt{x}\right)=\frac{1}{2\sqrt{x}}$$ |

It is possible to consider square roots in rings other than the integers or the rationals. For any ring $R$, with $x,y\in R$, we say that $y$ is a square root of $x$ if ${y}^{2}=x$.

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 ${y}^{2}\equiv x\phantom{\rule{veryverythickmathspace}{0ex}}(modn)$. 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 |