Complex Numbers in Polar and Euler Forms
Modulus-Argument (Polar) Form
The modulus-argument form, also known as polar form, is an alternative method of representing complex numbers. For a complex number $z = a + bi$, its polar form is given by:
$$z = r(\cos \theta + i \sin \theta) = r \text{ cis } \theta$$
Where:
- $r$ is the modulus (magnitude) of the complex number, calculated as $r = \sqrt{a^2 + b^2}$
- $\theta$ is the argument (angle) of the complex number, calculated as $\theta = \arctan(\frac{b}{a})$
- $\text{cis } \theta$ is shorthand notation for $(\cos \theta + i \sin \theta)$
Consider the complex number $z = 1 + \sqrt{3}i$. To convert it to polar form:
- Calculate the modulus: $r = \sqrt{1^2 + \sqrt{3}^2} = 2$
- Calculate the argument: $\theta = \arctan(\sqrt3) = \frac{\pi}{3}$ radians or $60°$
Therefore, $z = 2(\cos\frac{\pi}3 + i \sin\frac\pi3) = 2 \text{ cis } \frac\pi3$
NoteThe argument $\theta$ is not unique; it can be expressed as any angle that differs by multiples of $2\pi$ radians or $360°$. Always consider the quadrant of the complex number when determining $\theta$.
Euler Form
The Euler form of a complex number is closely related to the polar form and is based on Euler's formula:
$$e^{ix} = \cos x + i \sin x$$
Using this, we can express a complex number in Euler form as:
$$z = re^{i\theta}$$
Where $r$ and $\theta$ are the same as in the polar form.
ExampleFor the complex number $z = 1+\sqrt{3}i$, which we found earlier to have $r = 2$ and $\theta =\frac\pi3$, the Euler form would be:
$z = 2e^{\frac{\pi}3i}$
Converting Between Forms
Cartesian to Polar/Euler
- Calculate $r = \sqrt{a^2 + b^2}$
- Calculate $\theta = \tan^{-1}(\frac{b}{a})$, adjusting for the correct quadrant
- Express as $r(\cos \theta + i \sin \theta)$ or $re^{i\theta}$
Polar/Euler to Cartesian
- Expand $r(\cos \theta + i \sin \theta)$ or $re^{i\theta}$ to $r\cos\theta + ir\sin\theta$
- Simplify to $a + bi$ form by calculating $r\cos\theta$ and $\r\sin\theta$
Students often forget to adjust the argument $\theta$ for the correct quadrant when converting from Cartesian to polar form. Always check which quadrant the complex number lies in!
