Find four distinct complex numbers (which are neither purely imaginary nor purely real) such that each has an absolute value of 3.

Explanation:

Complex numbers have the general form a + bi, where a is the real part and b is the imaginary part.

Since, the numbers are neither purely imaginary nor purely real a ≠ 0 and b ≠ 0.

The absolute value of a complex number is its distance to the origin (0,0), so you use Pythagorean theorem to calculate the absolute value. Calling it |C|, that is:

Then, the work consists in finding pairs (a,b) for which:

You can do it by setting any arbitrary value less than 3 to a or b and solving for the other:

I will use b =0.5, b = 1, b = 1.5, b = 2

Then, four distinct complex numbers that have an absolute value of 3 are:

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