Complex Number Arithmetic
Add, Subtract, Multiply, and Divide numbers in the Complex Plane.
Number Z1 (Blue)
Number Z2 (Red)
Result Z (Green)
Argand Diagram
Visual: Vector Addition uses the “Tip-to-Tail” or Parallelogram method.
The Geometry of Complex Arithmetic
Complex numbers aren’t just abstract symbols; they have beautiful geometric meanings when plotted on the 2D Complex Plane (Argand Diagram).
Addition & Subtraction
Rule: Combine Real parts with Real, Imaginary with Imaginary.
(a+bi) + (c+di) = (a+c) + (b+d)i
Geometry: This is exactly like Vector Addition. If you place the tail of the second vector at the tip of the first, the result points to the final location.
Multiplication & Division
Rule: Use FOIL for rectangular, or multiply magnitudes and add angles for polar.
r₁∠θ₁ × r₂∠θ₂ = (r₁r₂) ∠ (θ₁+θ₂)
Geometry: Multiplication is a Rotation (add angles) and a Scaling (multiply lengths). Division is the opposite (subtract angles, divide lengths).
Why is this useful?
In electrical engineering (AC circuits), voltage and current are represented as rotating vectors (phasors). A capacitor or inductor essentially “rotates” the current by 90 degrees relative to the voltage. Using complex multiplication (multiplying by i or -i) handles this rotation automatically, turning difficult calculus problems into simple algebra.
Frequently Asked Questions (FAQ)
Q: What is the conjugate?
A: The conjugate of a + bi is a – bi. Geometrically, it’s a reflection over the Real (x) axis. We use it in division to remove ‘i’ from the denominator.
Q: How do I divide manually?
A: Multiply the top and bottom by the conjugate of the bottom number.
(2+i)/(3-i) × (3+i)/(3+i)
Q: What if I multiply by i?
A: Multiplying any complex number by i rotates it exactly 90° counter-clockwise. Multiplying by -1 (which is i²) rotates it 180°.