U3 Complex Numbers(KA/PRE/U3)

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Unit 3: Complex Numbers

The Complex Plane

Complex numbers are represented as points (or vectors) on a plane where the x-axis is the real part and the y-axis is the imaginary part.

Modulus and Argument

The modulus of a complex number z = a + bi is √(a² + b²). The argument (angle) is the angle with the real axis.

Polar Form of Complex Numbers

z = r (cos θ + i sin θ) = r·e^(iθ)

Graphically Multiplying Complex Numbers

Multiplying complex numbers adds their angles and multiplies their moduli.

Complex Conjugates

The conjugate of z = a + bi is a - bi. It reflects the point across the real axis.

Distance and Midpoint of Complex Numbers

Distance between z₁ and z₂ is √[(a₂ - a₁)² + (b₂ - b₁)²]. Midpoint is ((a₁ + a₂)/2, (b₁ + b₂)/2).

Fundamental Theorem of Algebra

Every non-zero polynomial of degree n has exactly n complex roots (counting multiplicity).

Quiz: Test Your Understanding

What is the conjugate of z = 3 + 4i?

  1. 3 + 4i
  2. 3 - 4i
  3. -3 + 4i

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