Polar form of complex numbers

The polar form of complex numbers is a different way of writing them that makes some math problems easier to solve. Understanding how to convert between forms and learning a new way to handle operations with complex numbers is helpful in many disciplines, like engineering and physics where you often deal with cycles or rotations. Use this resource to learn how to express complex numbers in their polar form and complete calculations using this form.

Rectangular and polar forms

In Introduction to complex numbers, you learned about complex numbers expressed in the form \(z=x+yi\). This is called the rectangular form. \(\DeclareMathOperator{\cis}{cis} \DeclareMathOperator{\Arg}{Arg}\)

You also learned about Argand diagrams, which graphically represent complex numbers. Cartesian coordinates in the form \((x,y)\) can be used to indicate a point \(P\) on a plane, but so can polar coordinates in the form \((r,\theta)\), where \(r\) is the distance of \(P\) from the origin and \(\theta\) is the angle that \(\overrightarrow{OP}\) makes with the positive \(x\)-axis.

The polar form gives us direct information about the magnitude (distance) and angle of the complex number.

We can convert the polar coordinate of a complex number to the polar form of the number using:
\[\begin{align*} z & = x+yi\\
& = r\cos(\theta) + r\sin(\theta)\\
& = r(\cos(\theta) + \sin(\theta))\quad\textrm{which we abbreviate to}\\ \DeclareMathOperator{\cis}{cis}
& = r\cis(\theta)
\end{align*}\]

Therefore, the polar form of the complex number \(z\) is:

\[z=r\cis(\theta)\]

where \(r=\sqrt{x^{2}+y^{2}}\) and \(\theta=\tan^{-1}\left(\dfrac{y}{x}\right)\).

Modulus, argument and Argument

To find \(r\), we need to find the modulus of \(z\) or \(\left| z \right|\).
\[\begin{align*} \mod(z) & = \left| z \right|\\
& = \left| x+yi \right|\\
& = \sqrt{x^{2}+y^{2}}\\
& = r
\end{align*}\]

To find \(\theta\), we need to find the argument (lower case "a") of \(z\), or \(\arg(z)\). If \(\arg(z)=\theta\), then:

• \(\sin(\theta)=\dfrac{y}{\left| z \right|}\)
• \(\cos(\theta)=\dfrac{x}{\left| z \right|}\)
• \(\tan(\theta)=\dfrac{y}{x}\).

An infinite number of arguments of \(z\) exist. For example, if \(z=i\), then \(\arg(z)=\dfrac{\pi}{2}+2\pi n\), where \(n\in Z\) (i.e. \(n\) can be any integer).

We can also use a value called the Argument (capital "A"), where \(\Arg(z)=\theta\) for \(-\pi\leq\theta\leq\pi\). Because the Argument is restricted to \(-\pi\leq\theta\leq\pi\), \(\Arg(z)\) has only one value, compared to \(\arg(z)\) which has many values.

Example 1 – expressing complex numbers in polar form

Express \(z=1-i\) in polar form.

We start by finding the coordinates on the Cartesian plane. For \(z=1-i\), \(x=1\) and \(y=-1\). The point is in the fourth quadrant.

Remember that the polar form of \(z=r\cis(\theta)\). We can start by finding \(r\).
\[\begin{align*} r & = \sqrt{x^{2}+y^{2}}\\
& = \sqrt{1+1}\\
& = \sqrt{2}
\end{align*}\]

Next, let's find \(\theta\).
\[\begin{align*} \theta & = \tan^{-1}\left(\frac{y}{x}\right)\\
& = \tan^{-1}\left(\frac{-1}{1}\right)\\
& = \tan^{-1}(-1)\\
& = -\frac{\pi}{4}
\end{align*}\]

Putting it altogether:
\[\begin{align*} z & = r\cis(\theta)\\
& = \sqrt{2}\cis\left(-\frac{\pi}{4}\right)
\end{align*}\]

Exercise – expressing complex numbers in polar form

1. Find the polar form (in radians) of the following complex numbers.
1. \(z=-1+i\)
2. \(z=-\sqrt{3}+i\)
3. \(z=-3i\)
4. \(z=-2-4i\)
2. Express the following complex numbers in rectangular form.
1. \(3\cis\left(\dfrac{\pi}{4}\right)\)
2. \(\sqrt{7}\cis(\pi)\)
3. \(8\cis\left(\dfrac{\pi}{2}\right)\)
4. \(10\cis(0.41)\)
3. If \(z=2+i\) and \(w=1-4i\), find the polar form (in radians) of the following complex numbers.
1. \(\left| z \right|\)
2. \(\left| w \right|\)
3. \(\Arg(z)\)
4. \(\left| \overline{w} \right|\)
5. \(\Arg(zw)\)
6. \(zw\)

Operations on complex numbers in polar form

To add and subtract complex numbers, they must be converted from polar form to rectangular form.

For multiplication and division, we use a trigonometric identity.
To multiply:

\[z_{1}z_{2} = r_{1}r_{2}\cis(\theta_{1}+\theta_{2})\]

To divide:

\[\dfrac{z_{1}}{z_{2}} = \dfrac{r_{1}}{r_{2}}\cis(\theta_{1}-\theta_{2})\]

Example 1 – performing operations on complex numbers in polar form

If \(z_{1}=2\cis\left(\dfrac{\pi}{4}\right)\) and \(z_{2}=-3\cis\left(\dfrac{5\pi}{6}\right)\), find \(z_{1}z_{2}\) in polar form, where \(-\pi\leq\theta\leq\pi\).
\[\begin{align*} z_{1}z_{2} & = 2\cis\left(\frac{\pi}{4}\right) \times \left(-3\cis\left(\frac{5\pi}{6}\right)\right)\\
& = -6\cis\left(\frac{\pi}{4}+\frac{5\pi}{6}\right)\\
& = -6\cis\left(\frac{13\pi}{12}\right)\\
& = -6\cis\left(\frac{-11\pi}{12}\right)\quad\textrm{since }-\pi\leq\theta\leq\pi
\end{align*}\]

Exercise – performing operations on complex numbers in polar form

1. Simplify the following.
1. \(4\cis\left(\dfrac{\pi}{3}\right) \times 3\cis\left(\dfrac{\pi}{4}\right)\)
2. \(\dfrac{3\cis \left( \dfrac{5\pi}{6} \right)}{12 \cis \left( \dfrac{\pi}{6} \right)}\)
2. If \(u=6\cis\left(\dfrac{3\pi}{4}\right)\) and \(v=4\cis\left(-\dfrac{\pi}{4}\right)\), express \(\dfrac{u}{v}\) in polar form.
3. If \(z=1-\sqrt{3}i\), find \(\overline{z}\) and express both \(z\) and \(\overline{z}\) in polar form (in radians).