Complex numbers

 Quick reference for complex numbers: 

The formula for dividing two complex numbers is: $$\frac{a + bi}{c + di} = \frac{(a + bi)(c - di)}{c^2 + d^2} = \frac{ac + bd}{c^2 + d^2} + i\frac{bc - ad}{c^2 + d^2}$$ Rotation and scaling.

Complex Numbers: Cartesian Representation vs. The Gaussian Plane

Complex numbers are defined as numbers having both a real and an imaginary component. They can be expressed through standard algebraic notation or visualized through geometric mapping.

1. The Cartesian (Rectangular) Representation

The Cartesian form represents a complex number z as the sum of two distinct components.

• Algebraic Formula: $$z = x + iy$$
• Real Component ($\Re$): Represented by the value x
• Imaginary Component ($\Im$): Represented by the value y, where i is the imaginary unit defined as $\sqrt{-1}$
• Usage: This form is standard for algebraic manipulation, particularly addition and subtraction.

2. The Gaussian Plane (The Complex Plane)

The Gaussian plane is a two-dimensional geometric coordinate system where complex numbers are mapped as points or vectors.

• Horizontal (Real) Axis: Corresponds to the real component x
• Vertical (Imaginary) Axis: Corresponds to the imaginary component y
• Vector Interpretation: A complex number is visualized as a vector extending from the origin to the point (x, y)
• Geometric Metrics:
  • Modulus ($r$): The length of the vector, calculated as $r = \sqrt{x^2 + y^2}$
  • Argument ($\theta$): The angle the vector makes with the positive real axis, calculated as $\theta = \tan^{-1}(y/x)$

The Polar Form of a Complex Number

While the Cartesian form ($x + iy$) describes a complex number as a point on a grid, the Polar Form describes it by its distance from the origin and its rotation angle.

1. The Polar Formula

Any complex number z can be written in terms of its length (modulus) and its phase (angle):

• Formula:$$z = r e^{i\theta}$$
• Modulus ($r$): The distance from the origin (length of the vector).
• Phase ($\theta$): The angle measured counter-clockwise from the positive real axis.

2. Conversion Formulas

To move between the Cartesian (rectangular) and Polar representations, use the following trigonometric relationships:

• From Cartesian to Polar: $$r = \sqrt{x^2 + y^2}$$ $$\theta = \tan^{-1}\left(\frac{y}{x}\right)$$
• From Polar to Cartesian: $$x = r \cos\theta$$ $$y = r \sin\theta$$

3. Euler’s Formula

The link between these two forms is established by Euler’s Formula, which defines the complex exponential:

$$e^{i\theta} = \cos\theta + i\sin\theta$$ $$e^{i\pi} + 1 = 0$$   Mathematical beauty

4. Complex Conjugate in Polar Form

Finding the complex conjugate ($z^*$) in polar form is remarkably simple—you just negate the angle:

$$z^* = r e^{-i\theta}$$

This operation reflects the vector across the real axis in the Gaussian plane.