equality of complex numbers
The equality relation “=” among the is determined as consequence of the definition of the complex numbers as elements of the quotient ring , which enables the 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 in the reverse direction is apparent.
If , then at least one of the real numbers and differs from 0. We can set
Thus (2) implies
The equations (4) are , since the sum of the squares of their 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 .
|Title||equality of complex numbers|
|Date of creation||2015-05-09 17:11:34|
|Last modified on||2015-05-09 17:11:34|
|Last modified by||pahio (2872)|