How to Divide Complex Numbers

Co-authored by wikiHow Staff

Last Updated: October 11, 2022

A complex number is a number that can be written in the form $z=a+bi,$ where $a$ is the real component, $b$ is the imaginary component, and $i$ is a number satisfying $i^{{2}}=-1.$

Complex numbers satisfy many of the properties that real numbers have, such as commutativity and associativity. However, when an expression is written as the ratio of two complex numbers, it is not immediately obvious that the number is complex. But given that the complex number field must contain a multiplicative inverse, the expression ends up simply being a product of two complex numbers and therefore has to be complex. We show how to write such ratios in the standard form $a+bi$ in both Cartesian and polar coordinates.

Part 1

Part 1 of 2:

Cartesian Coordinates

1
Begin with the ratio. In this section, we will show how to divide two complex numbers and show why it works. ^{[1]
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Research source}
- ${\frac {2+3i}{3+6i}}$
2

Find the complex conjugate of the denominator. The complex conjugate ${\bar {z}},$ pronounced "z-bar," is simply the complex number with the sign of the imaginary part reversed. ^{[2]
X
Research source} For example, the conjugate of the number $3+6i$ is $3-6i.$

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3
Multiply the numerator and denominator by this complex conjugate. The reason why this works is that for any complex number $z,$ multiplying it by its conjugate yields a real number $z{\bar {z}}=(a+bi)(a-bi)=a^{{2}}+b^{{2}},$ because $a$ and $b$ are both real. The i's are then removed from the denominator. Remember that we are really multiplying by 1, so both the top and bottom must be multiplied by the same number. ^{[3]
X
Research source} The process and reasoning is similar to that of rationalizing the denominator.
- ${\frac {2+3i}{3+6i}}\cdot {\frac {3-6i}{3-6i}}={\frac {6+9i-12i+18}{9+36}}={\frac {24-3i}{45}}$
- Remember that $i^{{2}}=-1.$ This means that you need to pay close attention to the signs.
4
Simplify and separate the result into real and imaginary components. We now have a fraction with a real number in the denominator, so all that's left is to put it in the form $a+bi.$ ^{[4]
X
Research source}
- ${\frac {24-3i}{45}}={\frac {8-i}{15}}={\frac {8}{15}}-{\frac {1}{15}}i$
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Part 2

Part 2 of 2:

Polar Coordinates

1

Review rectangular graphs of complex numbers. You may have already learned how to graph a complex number on the complex plane. The horizontal axis denotes the real axis, while the vertical axis is the imaginary axis. Above is a graph of an arbitrary complex number on the complex plane. It is important to understand what these graphs mean, because the nature of complex numbers means that we can draw tight connections between their algebraic properties (as shown in part 1) and their geometrical properties.
2
Understand polar coordinates. In polar coordinates, we denote points with two variables $r$ and $\theta ,$ where $r$ is the distance from the pole and $\theta$ is the angle from the polar axis. ^{[5]
X
Research source} In the context of complex numbers, the magnitude of the number is called the modulus, while its angle from the polar axis is called its argument.
- Recall the coordinate conversions from Cartesian to polar.
  - $x=r\cos \theta$
  - $y=r\sin \theta$
- We can therefore write any complex number on the complex plane as $z=r(\cos \theta +i\sin \theta ).$ Using Euler's formula, we can compact this to $z=re^{{i\theta }}.$
- In this section, we will show that dealing with complex numbers in polar form is vastly simpler than dealing with them in Cartesian form.
3
Begin with the ratio. Let's do a different example.
- ${\frac {3+3i}{2+2{\sqrt {3}}i}}$
4
Convert your complex numbers to polar form. If your numbers are already in polar form, skip this step. Otherwise, use the relations below.
- $r={\sqrt {a^{{2}}+b^{{2}}}}$
- $\theta =\tan ^{{-1}}\left({\frac {b}{a}}\right)$
- In our example, we have two complex numbers to convert to polar. Let's label them as ${\frac {z_{{1}}}{z_{{2}}}}$ and use the notation $z=re^{{i\theta }}.$
  - $z_{{1}}=3+3i=3{\sqrt {2}}e^{{i\pi /4}}$
  - $z_{{2}}=2+2{\sqrt {3}}i=4e^{{i\pi /3}}$
5
Divide the two complex numbers. When we write out the numbers in polar form, we find that all we need to do is to divide the magnitudes and subtract the angles. Likewise, when we multiply two complex numbers in polar form, we multiply the magnitudes and add the angles.
- ${\begin{aligned}{\frac {z_{{1}}}{z_{{2}}}}={\frac {3{\sqrt {2}}e^{{i\pi /4}}}{4e^{{i\pi /3}}}}&={\frac {3{\sqrt {2}}}{4}}e^{{i\left({\frac {\pi }{4}}-{\frac {\pi }{3}}\right)}}\\&={\frac {3{\sqrt {2}}}{4}}e^{{-i\pi /12}}\end{aligned}}$
6
Convert back to Cartesian coordinates. Use the following relations. Our angle is $\theta =-{\frac {\pi }{12}},$ which is not typically shown on the unit circle. You can either use a calculator or find the exact values of $\cos \left(-{\frac {\pi }{12}}\right)$ and $\sin \left(-{\frac {\pi }{12}}\right)$ using trigonometric summation identities. Below, we use the identities to write the components in exact form. Such identities are available online or in textbooks.
- $a=r\cos {\theta }$
- $b=r\sin {\theta }$
- ${\begin{aligned}a&={\frac {3{\sqrt {2}}}{4}}\cos \left(-{\frac {\pi }{12}}\right)\\&={\frac {3{\sqrt {2}}}{4}}\cos \left({\frac {\pi }{4}}-{\frac {\pi }{3}}\right)\\&={\frac {3{\sqrt {2}}}{4}}\left(\cos {\frac {\pi }{4}}\cos {\frac {\pi }{3}}+\sin {\frac {\pi }{4}}\sin {\frac {\pi }{3}}\right)\\&={\frac {3{\sqrt {2}}}{4}}\left({\frac {{\sqrt {2}}}{2}}\cdot {\frac {1}{2}}+{\frac {{\sqrt {2}}}{2}}\cdot {\frac {{\sqrt {3}}}{2}}\right)\\&={\frac {3{\sqrt {2}}}{4}}\cdot {\frac {{\sqrt {2}}+{\sqrt {6}}}{4}}\\&={\frac {3+3{\sqrt {3}}}{8}}\end{aligned}}$
- ${\begin{aligned}b&={\frac {3{\sqrt {2}}}{4}}\sin \left(-{\frac {\pi }{12}}\right)\\&={\frac {3{\sqrt {2}}}{4}}\cdot {\frac {{\sqrt {2}}-{\sqrt {6}}}{4}}\\&={\frac {3-3{\sqrt {3}}}{8}}\end{aligned}}$
7
Write out the complex number in standard form. While the above conversions took quite a few steps, it is important to recognize that the act of converting from one coordinate system to another is where the "difficult" part is. When dealing with complex numbers purely in polar, the operations of multiplication, division, and even exponentiation (cf. De Moivres' formula) are very easy to do.
- ${\frac {3+3{\sqrt {3}}}{8}}+{\frac {3-3{\sqrt {3}}}{8}}i$
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It is easy to show why multiplying two complex numbers in polar form is equivalent to multiplying the magnitudes and adding the angles. Write two complex numbers in polar form and multiply them out. Then we can use trig summation identities to bring the real and imaginary parts together.
- ${\begin{aligned}z_{{1}}z_{{2}}&=r_{{1}}(\cos \theta _{{1}}+i\sin \theta _{{1}})r_{{2}}(\cos \theta _{{2}}+i\sin \theta _{{2}})\\&=r_{{1}}r_{{2}}(\cos \theta _{{1}}\cos \theta _{{2}}+i\sin \theta _{{1}}\cos \theta _{{2}}+i\sin \theta _{{2}}\cos \theta _{{1}}-\sin \theta _{{1}}\sin \theta _{{2}})\\&=r_{{1}}r_{{2}}((\cos \theta _{{1}}\cos \theta _{{2}}-\sin \theta _{{1}}\sin \theta _{{2}})+i(\sin \theta _{{1}}\cos \theta _{{2}}+\sin \theta _{{2}}\cos \theta _{{1}}))\\&=r_{{1}}r_{{2}}(\cos(\theta _{{1}}+\theta _{{2}})+i\sin(\theta _{{1}}+\theta _{{2}}))\end{aligned}}$
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How to Divide Complex Numbers

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Cartesian Coordinates

Polar Coordinates

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