argument of product and quotient

Using the distributive law, we perform the multiplication


Using the addition formulasPlanetmathPlanetmath of cosine ( and sine ( we still obtain the formula

(cosφ1+isinφ1)(cosφ2+isinφ2)=cos(φ1+φ2)+isin(φ1+φ2). (1)

The inverse number of  cosφ2+isinφ2  is calculated as follows:


This equals  cosφ2-isinφ2,  and since the cosine is an even ( and the sine an odd function, we have

1cosφ2+isinφ2=cos(-φ2)+isin(-φ2). (2)

The equations (1) and (2) imply



cosφ1+isinφ1cosφ2+isinφ2=cos(φ1-φ2)+isin(φ1-φ2). (3)

According to the formulae (1) and (3), for the complex numbersMathworldPlanetmathPlanetmath


we have


Thus we have the

Theorem.  The modulusPlanetmathPlanetmath of the product of two complex numbers equals the product of the moduli of the factors and the argumentMathworldPlanetmath equals the sum of the arguments of the factors (  The modulus of the quotient of two complex numbers equals the quotient of the moduli of the dividend and the divisor and the argument equals the difference of the arguments of the dividend and the divisor.

Remark.  The equation (1) may be by induction generalised for more than two factors of the left hand ; then the special case where all factors are equal gives de Moivre identityMathworldPlanetmath.

Example.  Since

(2+i)(3+i)=5+5i= 5eπ4,

one has

Title argument of product and quotient
Canonical name ArgumentOfProductAndQuotient
Date of creation 2013-03-22 17:45:20
Last modified on 2013-03-22 17:45:20
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 8
Author pahio (2872)
Entry type Theorem
Classification msc 30-00
Classification msc 26A09
Synonym product and quotient of complex numbers
Related topic Argument
Related topic PolarCoordinates
Related topic ModulusOfComplexNumber
Related topic Complex
Related topic EqualityOfComplexNumbers