Number and Algebra

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Complex numbers can be represented in various forms, including Cartesian, modulus-argument (also known as polar), and Euler form. Converting between these forms is a fundamental skill in complex number analysis.

Cartesian Form: A complex number in Cartesian form is represented as \( z = a + bi \), where \( a \) and \( b \) are real numbers, and \( i \) is the imaginary unit.

Modulus-Argument (Polar) Form: In this form, a complex number is represented as \( z = r(\cos(\theta) + i\sin(\theta)) \), where \( r \) is the modulus (or magnitude) of the complex number and \( \theta \) is the argument (or angle) in radians. The modulus is given by \( r = \sqrt{a^2 + b^2} \) and the argument is \( \theta = \arctan\left(\frac{b}{a}\right) \).

Euler Form: Euler's formula relates the exponential function and the trigonometric functions. A complex number in Euler form is given by \( z = re^{i\theta} \). Using Euler's formula, this can be expanded as \( z = r(\cos(\theta) + i\sin(\theta)) \).

Key Formulae:

$$ \text{Cartesian to Polar:} \\ r = \sqrt{a^2 + b^2} \\ \theta = \arctan\left(\frac{b}{a}\right) $$

$$ \text{Polar to Cartesian:} \\ a = r\cos(\theta) \\ b = r\sin(\theta) $$

$$ \text{Euler Form:} \\ z = re^{i\theta} = r(\cos(\theta) + i\sin(\theta)) $$

Example: Convert the complex number \( z = 3 + 4i \) to its modulus-argument and Euler forms.

Using the formulae:

$$ r = \sqrt{3^2 + 4^2} = 5 \\ \theta = \arctan\left(\frac{4}{3}\right) $$

The modulus-argument form is \( z = 5(\cos(\theta) + i\sin(\theta)) \) and the Euler form is \( z = 5e^{i\theta} \).

Key Properties:

$$\text{If } z=|z|cis \theta, \text{ then } z^* = |z| cis(- \theta)$$ $$cis \theta \times cis \phi = cis(\theta + \phi)$$ $$\dfrac{cis \theta}{cis \phi} = cis(\theta - \phi)$$ $$cis(\theta + k2\pi)=cis \theta \text{ for all } k \in \mathbb{R}$$ $$\sqrt{2} cis \frac{\pi}{4} \text{ is the polar form of } 1 + i$$ $$\sqrt{2} e^{i \frac{\pi}{4}} \text{ is the Euler form of } 1 + i$$