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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.



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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