For a positive integer, we define an -th root of a number to be a number such that . The number is said to be the index of the root. Note that the term “number” here is ambiguous, as the discussion can apply in a variety of contexts (groups, rings, monoids, etc.) The purpose of this entry is specifically to deal with -th roots of real and complex numbers.
In an effort to give meaning to the term the -th root of a real number , we define it to be the unique real number that is an th root of and such that , if such a number exists. We denote this number by , or by if is positive. This specific th root is also called the principal th root.
Example: because , and is the unique positive real number with this property.
Example: If is a positive real number, then we can write because . (See the Binomial Theorem and .)
Example: because .
Note that when we restrict our attention to real numbers, expressions like are undefined. Thus, for a more full definition of th roots, we will have to incorporate the notion of complex numbers: The nth roots of a complex number are all the complex numbers that satisfy the condition . Applying the fundamental theorem of algebra (complex version) to the function tells us that such complex numbers always exist (counting multiplicity).
One of the more popular methods of finding these roots is through trigonometry and the geometry of complex numbers. For a complex number , recall that we can put in polar form: , where , and if , and if . (See the Pythagorean Theorem.) For the specific procedures involved, see calculating the nth roots of a complex number.
|Date of creation||2013-03-22 11:57:27|
|Last modified on||2013-03-22 11:57:27|
|Last modified by||mathcam (2727)|