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单词 NthRoot
释义

nth root


The phrase “the n-th root of a number” is a somewhat misleading concept that requires a fair amount of thought to make rigorous.

For n a positive integer, we define an n-th root of a number x to be a number y such that yn=x. The number n is said to be the index of the root. Note that the term “number” here is ambiguous, as the discussion can apply in a varietyMathworldPlanetmath of contexts (groups, rings, monoids, etc.) The purpose of this entry is specifically to deal with n-th roots of real and complex numbersMathworldPlanetmathPlanetmath.

In an effort to give meaning to the term the n-th root of a real number x, we define it to be the unique real number that y is an nth root of x and such that sign(x)=sign(y), if such a number exists. We denote this number by xn, or by x1n if x is positive. This specific nth root is also called the principal nth root.

Example: 814=3 because 34=3×3×3×3=81, and 3 is the unique positive real number with this property.

Example: If x+1 is a positive real number, then we can write x5+5x4+10x3+10x2+5x+15=x+1 because(x+1)5=(x2+2x+1)2(x+1)=x5+5x4+10x3+10x2+5x+1. (See the Binomial Theoremand .)

The nth root operationMathworldPlanetmath is distributive for multiplicationPlanetmathPlanetmath and division, but not for addition andsubtractionPlanetmathPlanetmath. That is, x×yn=xn×yn, andxyn=xnyn. However, except in special cases,x+ynxn+yn and x-ynxn-yn.

Example: 816254=35 because(35)4=3454=81625.

Note that when we restrict our attention to real numbers, expressions like -3 are undefined. Thus, for a more full definition of nth roots, we will have to incorporate the notion of complex numbers: The nth roots of a complex number t=x+yi are all the complex numbers z1,z2,,zn that satisfy the condition zkn=t. Applying the fundamental theorem of algebra (complex version) to the function xn-t tells us that n such complex numbers always exist (counting multiplicityMathworldPlanetmath).

One of the more popular methods of finding these roots is through trigonometryMathworldPlanetmath and the geometryMathworldPlanetmathPlanetmath of complex numbers. For a complex number z=x+iy, recall that we can put z in polar form: z=(r,θ), where r=x2+y22, and θ=π2 if x=0, and θ=arctanyx if x0. (See the Pythagorean TheoremMathworldPlanetmathPlanetmath.) For the specific procedures involved, see calculating the nth roots of a complex number.

Titlenth root
Canonical nameNthRoot
Date of creation2013-03-22 11:57:27
Last modified on2013-03-22 11:57:27
Ownermathcam (2727)
Last modified bymathcam (2727)
Numerical id25
Authormathcam (2727)
Entry typeDefinition
Classificationmsc 30-00
Classificationmsc 12D99
Synonymcomplex root
Synonymprincipal root
Related topicSquareRoot
Related topicCubeRoot
Related topicRealNumber
Related topicRationalNumber
Related topicComplex
Related topicIrrationalNumber
Related topicEvenEvenOddRule
Related topicExtensionOfValuationFromCompleteBaseField
Related topicRadical5
Related topicRadical6
Related topicExampleOfConvergingIncreasingSequence
Related topicNthRootByNewtonsMethod
Definesindex
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更新时间:2025/5/4 2:29:34