Complex Numbers PM3

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Complex Numbers
Summary

This guide covers the essential theory and formulas for Complex Numbers (Pure Mathematics 3), including imaginary numbers, Argand diagrams, polar forms, and Loci.

1. Imaginary Numbers

The set of complex numbers combines real numbers and imaginary numbers. We define the imaginary unit i as:

i = √-1
i² = -1

Calculating with roots:

• √-a = i√a (for a > 0)
• Example: √-9 = √(9 × -1) = 3i

2. Cartesian Form

A complex number z is written as:

z = x + iy

• x: Real part (Re z)
• y: Imaginary part (Im z)

Complex Conjugate

The conjugate of z = x + iy is denoted by z* (or z̄). It reflects the number across the Real axis.

If z = x + iy, then z* = x - iy

Property: The product of a complex number and its conjugate is always real:

zz* = (x + iy)(x - iy) = x² + y²

3. The Complex Plane (Argand Diagram)

Complex numbers are represented on a 2D plane where the x-axis is Real and the y-axis is Imaginary.

4. Modulus & Argument

Modulus (|z|)

The distance from the origin to the point.

|z| = r = √(x² + y²)

Argument (arg z)

The angle θ made with the positive real axis. The Principal Argument is usually in radians such that:

-π < arg z ≤ π

Calculated using trigonometry (always draw a diagram to check the quadrant!):

tan α = |y / x|

5. Polar Forms

Using modulus (r) and argument (θ), we can write complex numbers in two other useful forms.

Modulus-Argument Form:
z = r(cos θ + i sin θ)

Exponential Form (Euler):
z = re^iθ

Operations in Polar Form

When multiplying or dividing, exponential form is easiest:

• Multiplication: Multiply moduli, ADD arguments.
  → |z₁z₂| = r₁r₂
  → arg(z₁z₂) = θ₁ + θ₂
• Division: Divide moduli, SUBTRACT arguments.
  → |z₁ / z₂| = r₁ / r₂
  → arg(z₁ / z₂) = θ₁ - θ₂

6. Roots & Equations

Quadratic Equations

If b² - 4ac < 0, the equation has 2 complex roots which are conjugate pairs (e.g., a + bi and a - bi).

Polynomials

Any polynomial with real coefficients: If z = a + bi is a root, then z* = a - bi is also a root.

7. Loci (Paths of z)

Loci describe shapes on the Argand diagram based on algebraic rules. Here are the three most common standard forms.

A. Circle

Distance from a fixed point z₁ is constant r.

|z - z₁| = r

Note: If equation is |z + 3| = 2, center is at -3.

B. Perpendicular Bisector

Distance from point z₁ equals distance from point z₂.

|z - z₁| = |z - z₂|

C. Half-Line (Ray)

Angle from a fixed point z₁ is constant α.

arg(z - z₁) = α

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