[center][u][b]Preface: Polar and Exponential Representation of [/b][/u][math]z=a+bi[/math][/center]Using trigonometry we have the identification: [br][center][math]a=|z|cos(θ),b=|z|sin(θ)[/math][b].[/b][/center][b]The angle [math]θ[/math][/b][b] determined by [/b][math]z[/math][b] can be measured in degrees or radians [br]and restricted to be in a specific interval. [br]For example, [math]θ∈[0,2\pi)[/math][/b][b] or [math]θ∈(−\pi,\pi][/math][/b][b].[br]THUS:[br][center][u][math]z=|z|cos(θ)+|z|sin(θ)i=|z|[cos(θ)+sin(θ)i]=|z|cis(θ)[/math][/u][/center][u]The angle can be considered a function of [math]z[/math][/u][/b][u][b], called [u]the argument of [math]z[/math][/u][/b][b]: [math]Arg(z)=θ[/math][/b][b].[/b][left][b][br][br][/b]Now consider the Taylor Series for the real functions:[/left][center][br][math]e^x=1+x+\frac{x^2}{2}+\frac{x^3}{3!}+\frac{x^4}{4!}+\frac{x^5}{5!}+\frac{x^6}{6!}+\frac{x^7}{7!}+...[/math][br][math]cos(x)=1−\frac{x^2}{2}+\frac{x^4}{4!}−\frac{x^6}{6!}+...[/math][br][math]sin(x)=x−\frac{x^3}{3!}+\frac{x^5}{5!}−\frac{x^7}{7!}+...[/math][/center]Then using [math]θ[/math][u] in radian measure for[/u][/u] [math]x[/math][center][br][math]e^{iθ}=cos(θ)+sin(θ)i\equiv cis(θ)[/math] and  [math]z=|z|e^{iθ}[/math].[/center]Note:  When [math]θ=\pi[/math] this equation demonstrates that [math]e^{\pi i}=cos(\pi)+sin(\pi)i=−1[/math].

[center][u][b]Complex Multiplication: [/b][/u][math]z_1⋅z_2[/math][/center][b]Algebraically:[/b] If [math]z_1=a_1+b_1i[/math] and [math]z_2=a_2+b_2i[/math] then [math]z_1z_2=a_1a_2-b_1b_2+(a_1b_2+b_1a_2)i[/math].[br]Example: If [math]z_1=2+3i;z_2=1−i[/math][br]then [math]z_1⋅z_2=(2+3i)⋅(1−i)=2⋅1−3⋅i^2+3i⋅1+2⋅(−i)=(2+3)+(3−2)i=5+i[/math].[br][b][br]Geometrically:[/b] Use the polar or the exponential representation ([u]in radian measure[/u]) and the addition formulae for trigonometric functions: [br][br][center][math]z_1=|z_1|[cos(θ_1)+sin(θ_1)i]=|z1|e^{iθ_1}[/math][br][math]z_2=⋅|z_2|[cos(θ_2)+sin(θ_2)i]=|z_2|e^{iθ_2}[/math][br][/center][math]z_1⋅z_2=|z_1|[cos(θ_1)+sin(θ_1)i]⋅|z_2|[cos(θ_2)+sin(θ_2)i][/math][b]  [/b][math]=|z_1|⋅|z_2|[cos(θ_1)cos(θ_2)−sin(θ_1)sin(θ_2)+(sin(θ_1)cos(θ_2)+sin(θ_2)cos(θ_1))i][/math][math]=|z_1|⋅|z_2|[cos(θ_1+θ_2)+sin(θ_1+θ_2)i][/math][b][br]        [/b][math]=|z_1|⋅|z_2|cis(θ_1+θ_2)[/math][b][br][br]or more simply [u]using [math]θ[/math][/u][/b][b][u] in radian (or degree) measure[/u][/b][b][b]:[br][center][br][math]z_1⋅z_2=|z_1|e^{iθ_1}⋅|z_2|e^{iθ_2}[/math][b]  [/b][math]=|z_1|⋅|z_2|e^{\left(θ_1+θ_2\right)i}[/math][/center][br]Example: If [/b][math]z_1=\sqrt{2}cis(45°);z_2=2cis(30°)[/math] then[/b]  [math]z_1⋅z_2=2\sqrt{2}cis(75°)[/math]