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!
Operations in Polar and Euler Forms
Addition and Subtraction
These operations are typically performed in Cartesian form, then converted if necessary.
Multiplication
In polar or Euler form, multiplication becomes: $$(r_1(\cos \theta_1 + i \sin \theta_1))(r_2(\cos \theta_2 + i \sin \theta_2)) = r_1r_2(\cos(\theta_1 + \theta_2) + i \sin(\theta_1 + \theta_2))$$
Or in Euler form: $$(r_1e^{i\theta_1})(r_2e^{i\theta_2}) = r_1r_2e^{i(\theta_1 + \theta_2)}$$
ExampleMultiply $z_1 = 2(\cos \frac{\pi}{4} + i \sin \frac{\pi}{4})$ and $z_2 = 3(\cos \frac{\pi}{3} + i \sin \frac{\pi}{3})$:
$z_1z_2 = 2 \cdot 3 (\cos(\frac{\pi}{4} + \frac{\pi}{3}) + i \sin(\frac{\pi}{4} + \frac{\pi}{3}))$ $= 6(\cos \frac{7\pi}{12} + i \sin \frac{7\pi}{12})$
Division
Division in polar or Euler form is: $$\frac{r_1(\cos \theta_1 + i \sin \theta_1)}{r_2(\cos \theta_2 + i \sin \theta_2)} = \frac{r_1}{r_2}(\cos(\theta_1 - \theta_2) + i \sin(\theta_1 - \theta_2))$$
Or in Euler form: $$\frac{r_1e^{i\theta_1}}{r_2e^{i\theta_2}} = \frac{r_1}{r_2}e^{i(\theta_1 - \theta_2)}$$
TipMultiplication and division in polar or Euler form are often simpler than in Cartesian form, especially when dealing with roots of unity or trigonometric expressions.
Geometric Interpretation
The polar and Euler forms provide a geometric interpretation of complex numbers and operations:
- The modulus $r$ represents the distance from the origin to the point in the complex plane.
- The argument $\theta$ represents the angle from the positive real axis to the line from the origin to the point.
Multiplication
Geometrically, multiplying complex numbers in polar form:
- Multiplies their moduli
- Adds their arguments
This results in a rotation and scaling in the complex plane.
Division
Division in polar form:
- Divides their moduli
- Subtracts their arguments
This results in a rotation in the opposite direction and inverse scaling.
