Complex Numbers

1. Representation of Complex Numbers

A complex number is of the form $z = x + iy$, where $x$ is the real part and $y$ is the imaginary part. The real part is denoted by Re($z$) and the imaginary part by Im($z$).

• Standard Form: $z = x + iy$
• Rectangular Form: $z = a + bi$
• Geometric Representation: The complex number $z = x + iy$ can be represented as a point $(x, y)$ in the complex plane, where the x-axis represents the real axis and the y-axis represents the imaginary axis.

2. Arithmetic Operations

Complex numbers can be added, subtracted, multiplied, and divided using similar rules to real numbers, with the added consideration of $i^2 = -1$.

Addition and Subtraction

$(x + iy) + (a + bi) = (x + a) + i(y + b)$

$(x + iy) - (a + bi) = (x - a) + i(y - b)$

Multiplication

$(x + iy)(a + bi) = xa + xbi + iay + i^2 yb = xa + xbi + iay - yb = (xa - yb) + i(xb + ya)$

Division

To divide by a complex number, multiply the numerator and denominator by the conjugate of the denominator.

If $z_1 = a + bi$ and $z_2 = c + di$, then $\frac{z_1}{z_2} = \frac{(a + bi)(c - di)}{(c + di)(c - di)} = \frac{(ac + bd) + i(bc - ad)}{c^2 + d^2} = \frac{ac + bd}{c^2 + d^2} + i\frac{bc - ad}{c^2 + d^2}$

3. Modulus-Argument Form

A complex number $z = x + iy$ can be expressed in polar form as $z = r(\cos \theta + i \sin \theta)$, where:

• $r = |z| = \sqrt{x^2 + y^2}$ is the modulus (or magnitude) of $z$.
• $\theta = \arg(z)$ is the argument of $z$, which is the angle between the positive real axis and the line segment connecting the origin to the point $(x, y)$ in the complex plane. $-\pi < \theta \le \pi$.

The modulus and argument are related by the following equations:

$r = |z| = \sqrt{x^2 + y^2}$

$\cos \theta = \frac{x}{r}$

$\sin \theta = \frac{y}{r}$

$\theta = \arctan(\frac{y}{x})$ (Note: be careful with the quadrant of $\theta$ when using arctan)

4. Loci

A locus is the set of all points in the complex plane that satisfy a given condition.

• Constant Modulus: $|z| = r$ represents a circle with radius $r$ centered at the origin.
• Constant Argument: $\arg(z) = \theta$ represents a ray starting from the origin at an angle $\theta$ with the positive real axis.
• Equation involving Real and Imaginary Parts: Equations of the form $Re(z) = a$ or $Im(z) = b$ represent horizontal and vertical lines, respectively.

5. De Moivre’s Theorem

De Moivre’s Theorem provides a way to find powers and roots of complex numbers in polar form.

If $z = r(\cos \theta + i \sin \theta)$, then for any integer $n$:

$z^n = r^n(\cos(n\theta) + i \sin(n\theta))$

Similarly, the $n$th roots of a complex number $z$ are given by:

$z^{1/n} = r^{1/n} \left( \cos\left(\frac{\theta + 2k\pi}{n}\right) + i \sin\left(\frac{\theta + 2k\pi}{n}\right) \right)$, where $k = 0, 1, 2, ..., n-1$

Topic	Key Concepts
Representation	Standard Form, Rectangular Form, Complex Plane
Arithmetic	Addition, Subtraction, Multiplication, Division, Conjugate
Modulus-Argument Form	Modulus, Argument, Polar Form, De Moivre's Theorem
Loci	Constant Modulus, Constant Argument, Equations involving Real and Imaginary Parts