equality of complex numbers

The equality relation “=” among the is determined as consequence of the definition of the complex numbersMathworldPlanetmathPlanetmath as elements of the quotient ring /(X2+1), which enables the of the complex numbers as the ordered pairs  (a,b)  of real numbers and also as the sums a+ib where  i2=-1.

a1+ib1=a2+ib2a1=a2b1=b2 (1)

This condition may as well be derived by using the field properties of and the properties of the real numbers:

a1+ib1=a2+ib2 a2-a1=-i(b2-b1)

The implication in the reverse direction is apparent.

If  a+ib0,  then at least one of the real numbers a and b differs from 0.  We can set

a=rcosφ,b=rsinφ, (2)

where r is a uniquely determined positive number and φ is an angle which is uniquely determined up to an integer multiple of 2π.  In fact, the equations (2) yield



r=a2+b2. (3)

Thus (2) implies

cosφ=aa2+b2,sinφ=ba2+b2. (4)

The equations (4) are , since the sum of the squares of their is 1.  So these equations determine the angle φ up to a multiple of 2π.  We can write the

Theorem.  Every complex number may be represented in the polar form


where r is the modulus and φ 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 2π.

Title equality of complex numbers
Canonical name EqualityOfComplexNumbers
Date of creation 2015-05-09 17:11:34
Last modified on 2015-05-09 17:11:34
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 7
Author pahio (2872)
Entry type Topic
Classification msc 30-00
Related topic ModulusOfAComplexNumber
Related topic ArgumentOfProductAndQuotient
Related topic ComplexLogarithm