Complex Numbers

1.1 Complex numbers and the imaginary unit

A complex number is written as:

z = a + bi

where:

• a is the real part
• b is the imaginary part
• i is the imaginary unit with i² = -1

1.2 Equality, real part, and imaginary part

Two complex numbers are equal if their real parts are equal and their imaginary parts are equal.

If  a + bi = c + di  then  a = c  and  b = d

1.3 Addition, subtraction, and multiplication

Let z1 = a + bi and z2 = c + di.

z1 + z2 = (a + c) + (b + d)i
z1 - z2 = (a - c) + (b - d)i
z1 z2   = (ac - bd) + (ad + bc)i

Quick examples

(3 + 2i) + (1 - 5i) = 4 - 3i
(2 + i)(4 - 3i)     = 11 - 2i

1.4 Conjugate and division

The conjugate of z = a + bi is:

z̄ = a - bi

Useful identity:

z z̄ = (a + bi)(a - bi) = a² + b²

Division (multiply top and bottom by the conjugate):

(a + bi) / (c + di) = (a + bi)(c - di) / (c² + d²)

1.5 Modulus (absolute value)

The modulus of z = a + bi is:

|z| = √(a² + b²)

Properties to remember:

• |z| ≥ 0
• |z| = 0 only when z = 0
• |z1 z2| = |z1| |z2|

1.6 Polar form (basic idea)

Any non-zero complex number can also be written using magnitude and direction:

z = r (cos θ + i sin θ)

where r = |z|.

Exercise-style practice (mixed)

1. Write the real and imaginary parts of:

z = -5 + 7i

2. Simplify:

(4 - 3i) + (2 + 9i)

3. Multiply:

(1 - 2i)(3 + i)

4. Find the conjugate and modulus of:

z = 6 - 8i

5. Simplify:

(2 + i) / (1 - i)