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