Introduction

Division of Complex numbers is an undefined process by itself. There is no way to properly 'divide' a Complex number by another Complex number.

This means that if there is a Complex number that is a fraction that has something other than a pure Real number in the denominator, i.e. an Imaginary number or a Complex number, then we must convert that number into an equivalent fraction that we will be able to Mathematically manipulate.

When a Complex number appears in the denominator, we must rationalize the denominator to form an equivalent number that has a Complex number in the numerator only and a Real number in the denominator only.

We rationalize the denominator by multiplying both the top and the bottom of the fraction by the conjugate of the Complex number in the denominator, which is just equivalent to multiplication by 1

z=\frac{a+bi}{c+di} we must multiply it by \frac{c-di}{c-di}=1

then

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

Example

If z=\frac{1-2i}{2+2i} then the conjugate of the denominator is (2-2i) and

z=(\frac{1-2i}{2+2i})(\frac{2-2i}{2-2i})=\frac{(2-4)+(2-4)i}{(2)^{2}+(-2)^{2}}=

\frac{-2-2i}{8}=\frac{-1-i}{4}

We turned an undefined operation of division problem into an equivalent multiplication problem by using the conjugate. Multiplying both the numerator and the denominator by the conjugate of the denominator makes the denominator the square of its modulus and the numerator a new Complex number. The new Complex number with just a Real number in the denominator is equivalent to the old Complex number because we only multiplied by an expression that is equivalent to 1.

Example

z=\frac{3-2i}{3+4i} then

z=(\frac{3-2i}{3+4i})(\frac{3-4i}{3-4i})=\frac{(9-8)+(12-6)i}{9+16}=\frac{1+6i}{25}

Example

z=\frac{5-3i}{i} then

z=(\frac{5-3i}{i})(\frac{i}{i})=\frac{5i+3}{-1}=-3-5i

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The Reciprocal of a Complex Number

If we have a certain Complex number z=x+iy, we can find the value of its reciprocal z^{-1}=\frac{1}{x+iy} by rationalizing the denominator in the same way.

Example

If z=1-\sqrt{3}i with z^{-1}=\frac{1}{1-\sqrt{3}i} and z^{*}=1+\sqrt{3}i

(\frac{1}{1-\sqrt{3}})(\frac{1+\sqrt{3}}{1+\sqrt{3}})=\frac{1+\sqrt{3}}{1+3}=\frac{1+\sqrt{3}}{4}

This shows that z^{-1}=\frac{z^{*}}{zz^{*}}=\frac{z^{*}}{|z|^{2}}

meaning the reciprocal of a Complex number is equal to the conjugate of that Complex number divided by the square of the modulus. The inverse is a scaled version of the conjugate.

Visualizing the Division of Complex Numbers

The product of the division of  a Complex numbers z_1=x_1+iy_1 by another z_2=x_2+iy_2 is a new, scaled Complex number that has a new angle \theta_1-\theta_2. The numerator z_1 is multiplied by z_2^{*} and the reciprocal of the squared modulus of the denominator z_2, z_2z_2^{*}=\frac{1}{x_2^{2}+y_2^{2}}.

We subtract the angle during division because we are actually multiplying by the reflection of the denominator, which has an angle -\theta_2.

Example

z_1=3+4i and z_2=1-i then

\frac{z_1}{z_2}=\frac{(3+4i)}{(1-i)}\frac{(1+i)}{(1+i)}=\frac{(3-4)+(3+4)i}{2}=\frac{-1+7i}{2}

\frac{-1+7i}{2} is a scaled by \frac{1}{2} version of the regular product of

z_1z_2=(3+4i)(1+i)=(3-4)+(3+4)i=-1+7i

Complex Division

Modulus and Argument

Example

z_1=2+2i and z_2=1-i then

\frac{z_1}{z_2}=\frac{(2+2i)}{(1-i)}\frac{(1+i)}{(1+i)}=\frac{(2+2)+(2-2)i}{2}=\frac{4+oi}{2}=2

Dividing by 1-i subtracts an angle of \frac{\pi}{4} because 1+i has an angle of \frac{\pi}{4}.

This shows that arg(\frac{z_1}{z_2})=arg(z_1)-arg(z_2)

The new Complex number is located on the positive x-axis a distance of \frac{|z_1|}{|z_2|}=\frac{\sqrt{8}}{\sqrt{2}}=2 away from the origin.

This shows that |\frac{z_1}{z_2}|=\frac{|z_1|}{|z_2|}

Example

z_1=2+2i and z_2=1+i then

\frac{z_1}{z_2}=\frac{(2+2i)}{(1+i)}\frac{(1-i)}{(1-i)}=\frac{(2-2)+(2+2)i}{2}=\frac{0+4i}{2}=2i

Dividing by 1+i adds an angle of \frac{\pi}{4} because 1-i has an angle of -\frac{\pi}{4}. The new Complex number is located on the positive y-axis a distance of \frac{|z_1|}{|z_2|}=\frac{\sqrt{8}}{\sqrt{2}}=2 away from the origin.

 

 

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