Polar form of complex numbers
Theory
Besides rectangular form \(a+bi\), a complex number can be written in polar (trigonometric) form using its distance and direction from the origin:
where \(r=|z|=\sqrt{a^2+b^2}\) is the modulus and \(\theta\) the argument (the angle from the positive real axis). This is often abbreviated \(z=r\,\text{cis}\,\theta\).
Conversions between the two forms:
How to convert to polar form
- Compute \(r\) from \(\sqrt{a^2+b^2}\).
- Find the reference angle \(\arctan\left|\dfrac{b}{a}\right|\).
- Adjust for the quadrant of \((a,b)\) to get \(\theta\).
- Write \(z=r(\cos\theta+i\sin\theta)\).
Find the modulus and argument.
| \(r\) | \(=\) | \(\sqrt{1^2+1^2}=\sqrt2\) |
| \(\theta\) | \(=\) | \(\arctan\dfrac{1}{1}=\dfrac{\pi}{4}\) |
| \(z\) | \(=\) | \(\sqrt2\left(\cos\dfrac{\pi}{4}+i\sin\dfrac{\pi}{4}\right)\) |
Evaluate the sine and cosine.
| \(z\) | \(=\) | \(2\left(\dfrac12+i\dfrac{\sqrt3}{2}\right)\) |
| \(=\) | \(1+\sqrt3\,i\) |
\(z\) is in Quadrant II, so add \(\pi\) to the reference angle.
| \(\text{ref}\) | \(=\) | \(\arctan\dfrac{1}{1}=\dfrac{\pi}{4}\) |
| \(\theta\) | \(=\) | \(\pi-\dfrac{\pi}{4}=\dfrac{3\pi}{4}\) |
\(z=0+3i\) lies straight up the imaginary axis.
| \(r\) | \(=\) | \(3\) |
| \(\theta\) | \(=\) | \(\dfrac{\pi}{2}\) |
Common pitfalls
Frequently asked questions
What is the polar form of a complex number?
\(z=r(\cos\theta+i\sin\theta)\), where \(r\) is the modulus (distance from the origin) and \(\theta\) is the argument (angle).
How do you find the modulus and argument?
Modulus \(r=\sqrt{a^2+b^2}\); argument \(\theta=\arctan\dfrac{b}{a}\), adjusted for the quadrant of the point.
Why adjust the argument for the quadrant?
Because \(\arctan\) returns only values in two quadrants; the actual point may lie elsewhere, so you add or subtract \(\pi\) as needed.
How do you convert polar back to rectangular?
Use \(a=r\cos\theta\) and \(b=r\sin\theta\), then write \(a+bi\).