The equality relation “=” among the complex numbers is determined as consequence of the definition of the complex numbers as elements of the quotient ring , which enables the interpretation of the complex numbers as the ordered pairs of real numbers and also as the sums where .
This condition may as well be derived by using the field properties of and the properties of the real numbers:
The implication chain in the reverse direction is apparent.
If , then at least one of the real numbers and differs from 0. We can set
where is a uniquely determined positive number and is an angle which is uniquely determined up to an integer multiple of . In fact, the equations (2) yield
Thus (2) implies
The equations (4) are compatible, since the sum of the squares of their right sides is 1. So these equations determine the angle up to a multiple of . We can write the
Theorem. Every complex number may be represented in the polar form
where is the modulus and the argument of the number. Two complex numbers are equal if and only if they have equal moduli and, if the numbers do not vanish, their arguments differ by a multiple of .