argument of product and quotient
Using the distributive law, we perform the multiplication
Using the addition formulas of cosine (http://planetmath.org/GoniometricFormulae) and sine (http://planetmath.org/GoniometricFormulae) we still obtain the formula
(1) |
The inverse number of is calculated as follows:
This equals , and since the cosine is an even (http://planetmath.org/EvenFunction) and the sine an odd function, we have
(2) |
The equations (1) and (2) imply
i.e.
(3) |
According to the formulae (1) and (3), for the complex numbers
we have
Thus we have the
Theorem. The modulus of the product of two complex numbers equals the product of the moduli of the factors and the argument equals the sum of the arguments of the factors (http://planetmath.org/Product). 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 identity.
Example. Since
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 |