Given two complex numbers, $a+bi$ and $c+di,$ we can ask about the quotient:

• $\frac{a+bi}{c+di}$
There are two ways to proceed. One is to remember that when you multiply a complex number by its complex conjugate the result is a real number. So we get this:

$\frac{a+bi}{c+di}~=~\frac{a+bi}{c+di}\times\frac{c-di}{c-di}$

The right-hand side multiples through, using the rules for multiplying rational numbers, to:

$\frac{a+bi}{c+di}\times\frac{c-di}{c-di}~=~\frac{(a+bi)(c-di)}{(c+di)(c-di)}$

Using the rules of multiplying complex numbers we get:

$\frac{(a+bi)(c-di)}{(c+di)(c-di)}~=~\frac{(ac+bd)+(bc-ad)i}{c^2+d^2}$

Alternatively, we can express the numbers using the polar representation of a complex number:

$z_1=re^{i\theta},\quad\quad~z_2=se^{i\phi}$

Then the quotient is:

$\frac{z_1}{z_2}~=~\frac{r}{s}e^{i(\theta-\phi)}$

This uses the index laws that $\frac{x^a}{x^b}~=~x^{a-b}.$

AbsoluteValue
AddingComplexNumbers
ArgandDiagram
CategoryMetaTopic
CauchySequence
CommonCoreDomains
CommonFactor
ComplexPlane
DifferenceOfTwoSquares
Divisor
DomainOfAFunction
FundamentalTheoremOfAlgebra
HolomorphicFunction
ImaginaryNumber
IsaacNewton
Logarithm
MagnitudeOfAComplexNumber
MathematicsTaxonomy
MultiplyingComplexNumbers
NewtonsMethod
PolarRepresentationOfAComplexNumber
Polynomial
PythagorasTheorem
Quaternion
RootsOfPolynomials
SiteNavigation
SquareRoot
TypesOfNumber
(none) Surd
ComplexConjugate
ComplexNumber
RationalisingTheDenominator
WhatIsATopic

## You are here

DividingComplexNumbers
IndexLaws
MultiplyingComplexNumbers
MultiplyingRationalNumbers
PolarRepresentationOfAComplexNumber
RealNumber
AddingComplexNumbers
ArgandDiagram
CategoryMetaTopic
ComplexPlane
ImaginaryNumber
Euler AlgebraicNumber
CauchySequence
CommonFactor
ContinuedFraction
DedekindCut
Denominator
IrrationalNumber
MagnitudeOfAComplexNumber
Numerator
PythagorasTheorem
RationalNumber
TranscendentalNumber

# Local neighbourhood - D3

 Last change to this page Full Page history Links to this page Edit this page   (with sufficient authority) Change password Recent changes All pages Search