Where:
- a is the real part
- b is the imaginary part
- i is the square root of -1
You're simply combining two numbers:
• One on the real axis
• One on the imaginary axis
Examples:
3 + 4i → real = 3, imaginary = 4
-2 - 5i → real = -2, imaginary = -5
7 → real only (imaginary part is 0)
6i → imaginary only (real part is 0)
Why complex numbers?
They help describe 2D quantities — like angle & magnitude, or oscillations —
common in physics, engineering, and signal processing as we will learn later.
+
i
+
i
Geometric Transformations
Addition: Translation - Z₁ is translated by vector Z₂
Subtraction: Translation by inverse - Z₁ moved by -Z₂
Multiplication: Rotation and dilation - Z₁ rotated by arg(Z₂) and scaled by |Z₂|
Division: Inverse rotation and dilation - Z₁ rotated by -arg(Z₂) and scaled by 1/|Z₂|
Conjugate, Argument, and Modulus of a Complex Number
Given a complex number: z = a + bi
Conjugate: The conjugate of z is z̄ = a - bi. It flips the sign of the imaginary part. Example: If z = 3 + 4i, then z̄ = 3 - 4i.
Modulus (Magnitude): The modulus is the distance from the origin: |z| = √(a² + b²). Example: If z = 3 + 4i, then |z| = √(3² + 4²) = 5.
Argument (Angle): The argument is the angle θ the vector makes with the real axis: arg(z) = arctan(b/a) (in degrees or radians). Example: If z = 3 + 4i, then arg(z) = arctan(4/3) ≈ 53.13°.
Summary Table:
z
Conjugate
Modulus
Argument
3 + 4i
3 - 4i
5
53.13°
-2 - 5i
-2 + 5i
√29 ≈ 5.39
-111.80°
7
7
7
0°
6i
-6i
6
90°
Why are these useful?
- The conjugate helps with division and finding real/imaginary parts.
- The modulus tells you the size (distance from origin).
- The argument tells you the direction (angle) in the complex plane.
These three together let you describe any complex number in both rectangular (a + bi) and polar (r∠θ) forms.