Complex Numbers Algebra: Advanced Operations Guide

A complex number is z = a + bi with real a, b and i² = −1; a is the real part, b the imaginary part. Advanced algebra covers multiplication and division, the conjugate z̄ = a − bi, the modulus |z| = √(a² + b²), powers, and De Moivre's formula, which converts repeated multiplication into angle arithmetic.

These operations are the working language of AC circuits, signal processing, control theory, and quantum mechanics, where phasors, impedances, and probability amplitudes are manipulated with exactly these rules. Fluency here is a genuine professional skill, not just exam material.

Key Concepts

• Conjugate: z̄ = a − bi; reflects across the real axis, with z·z̄ = a² + b² = |z|².
• Modulus: |z| = √(a² + b²), distance from the origin; multiplicative: |zw| = |z|·|w|.
• Division: 1/z = z̄/|z|² for z ≠ 0, so (a + bi)⁻¹ = (a − bi)/(a² + b²).
• De Moivre: (cos θ + i sin θ)^n = cos(nθ) + i sin(nθ) for integer n.
• Triangle inequality: |z + w| ≤ |z| + |w|, equality only when z, w point the same way from the origin.

Formulas

• (a + bi) ± (c + di) = (a ± c) + (b ± d)i.
• (a + bi)(c + di) = (ac − bd) + (ad + bc)i, the −bd coming from i² = −1.
• (a + bi)/(c + di) = ((a + bi)(c − di))/(c² + d²) for c + di ≠ 0.
• |z/w| = |z|/|w| (w ≠ 0); |z^n| = |z|^n.
• For z = r(cos θ + i sin θ): z^n = r^n(cos(nθ) + i sin(nθ)).

Worked Example

Compute (1 + i)^8. Square first: (1 + i)² = 1 + 2i + i² = 2i. Then (1 + i)^8 = (2i)^4. Since (2i)² = 4i² = −4 and (−4)² = 16, the answer is 16. For division, compute (3 + 4i)/(1 + 2i) by multiplying with the conjugate 1 − 2i: numerator (3 + 4i)(1 − 2i) = 3 − 6i + 4i − 8i² = 11 − 2i; denominator (1 + 2i)(1 − 2i) = 1 + 4 = 5. Result: (11 − 2i)/5 = 2.2 − 0.4i. Verify: (2.2 − 0.4i)(1 + 2i) = 2.2 + 4.4i − 0.4i − 0.8i² = 3 + 4i.

Common Misconceptions

• "√(−1)·√(−1) = √((−1)(−1)) = 1." The rule √(ab) = √a√b fails for negatives; correctly, i·i = i² = −1.
• "Complex numbers can be ordered." No ordering respects the field operations; 2i < 3i is meaningless.
• "|z + w| = |z| + |w| always." It is an inequality: for z = 1, w = i, |z + w| = √2 ≈ 1.414 but |z| + |w| = 2.

Real-World Applications

• AC phasors: multiplying and dividing complex impedances computes phase shifts and resonance.
• Signal processing: filtering is multiplication in the complex frequency domain of the Fourier transform.
• Quantum mechanics: probability amplitudes are complex, with conjugation and modulus giving measurable probabilities.

Advanced complex algebra is a small rule set: distribute with i² = −1, divide via the conjugate, and let De Moivre handle powers. Once automatic, complex numbers become a fast, reliable computational tool.