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Principle nth root

The root of a complex number z in the interval [0, 2π) is called the principle value. The interval (−π, π] can also be taken.

 


Explanation

Roots of complex numbers must be reduced to arithmetic roots to enable calculations. Complex numbers can be converted by using the definition of the imaginairy unit i 2 = – 1. You then eliminate the minus sign. In a multiplication you get

where a and b are positive numbers.

 


Example 1

You can calculate the root of the imaginary unit if you write i as a square, so

in which we calculate a and b. Squaring both sides gives

what we dissolve in

As of course applies i = i, we get two equations with two unknowns

A calculation shows

By substitution you get

The principle value is then

so that

 


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