Algebra of Complex Numbers
The algebra of complex numbers extends the familiar arithmetic of real numbers into a richer system where every polynomial equation has a solution. A complex number has the form z = a + bi, where a and b are real numbers and i is the imaginary unit satisfying i² = −1. Just as we add, subtract, multiply, and divide real numbers, we can perform all four arithmetic operations on complex numbers, following the same field axioms with the extra rule that i² = −1 simplifies expressions. Addition and subtraction are straightforward — real parts combine with real parts, and imaginary parts combine with imaginary parts, much like collecting like terms in algebra. Multiplication uses the distributive law and the identity i² = −1 to collapse results back into the form a + bi. Division is the most subtle operation: it requires multiplying numerator and denominator by the complex conjugate of the denominator, a technique that eliminates the imaginary part from the denominator and yields a real divisor. Mastering the algebra of complex numbers is essential for further study in polar form, De Moivre's theorem, signal processing, and quantum mechanics. Every operation on complex numbers has both an algebraic interpretation (manipulating symbols) and a geometric interpretation on the Argand plane — understanding both deepens mathematical fluency and prepares students for advanced topics in pure and applied mathematics.
- Addition and Subtraction
- Multiplication
- The Complex Conjugate: Key to Division
- Division
- Powers of i and Integer Exponents
Addition and Subtraction
Given z₁ = a + bi and z₂ = c + di, addition is defined by adding corresponding parts: z₁ + z₂ = (a + c) + (b + d)i. Subtraction works the same way: z₁ − z₂ = (a − c) + (b − d)i. This is analogous to adding vectors component-by-component and corresponds geometrically to the parallelogram law on the Argand plane. Example: (3 + 4i) + (1 − 2i) = (3+1) + (4−2)i = 4 + 2i.
Multiplication
Multiplication uses the distributive (FOIL) method combined with i² = −1. For z₁ = a + bi and z₂ = c + di: z₁ · z₂ = (ac − bd) + (ad + bc)i. Derivation: (a + bi)(c + di) = ac + adi + bci + bdi² = ac + adi + bci − bd = (ac − bd) + (ad + bc)i. Example: (2 + 3i)(1 − 4i) = (2·1 − 3·(−4)) + (2·(−4) + 3·1)i = (2 + 12) + (−8 + 3)i = 14 − 5i. Note: multiplication by i rotates a complex number by 90° counterclockwise on the Argand plane.
The Complex Conjugate: Key to Division
The complex conjugate of z = a + bi is z̄ = a − bi. Crucially, z · z̄ = a² + b², which is always a non-negative real number. This property is exploited in division. Geometrically, z̄ is the reflection of z across the real axis. If z = 3 + 4i, then z̄ = 3 − 4i and z · z̄ = 9 + 16 = 25. The modulus |z| = √(a² + b²) satisfies |z|² = z · z̄.
Division
To divide z₁ by z₂ (where z₂ ≠ 0), multiply numerator and denominator by z̄₂: z₁/z₂ = (z₁ · z̄₂)/(z₂ · z̄₂) = (z₁ · z̄₂)/|z₂|². The denominator becomes the real number c² + d², and the result is in a + bi form. Example: (1 + 2i)/(3 + 4i) = [(1 + 2i)(3 − 4i)] / [(3)² + (4)²] = (3 − 4i + 6i − 8i²)/25 = (3 + 2i + 8)/25 = (11 + 2i)/25 = 11/25 + (2/25)i.
Powers of i and Integer Exponents
Powers of i cycle with period 4: i¹ = i, i² = −1, i³ = −i, i⁴ = 1, i⁵ = i, … To find iⁿ, compute n mod 4: if the remainder is 0 → 1; 1 → i; 2 → −1; 3 → −i. Example: i²⁷ = i^(4·6+3) = i³ = −i. For general integer powers of z, use the binomial theorem or polar form. The set of complex numbers with these operations forms an algebraically closed field — every non-constant polynomial with complex coefficients has at least one complex root (Fundamental Theorem of Algebra).
The Key Equation
- z_1 = a + bi — Dividend (dimensionless): The complex number being divided; a is its real part and b is its imaginary part.
- z_2 = c + di — Divisor (dimensionless): The complex number dividing z₁; must be non-zero, i.e., c² + d² ≠ 0.
- \bar{z}_2 = c - di — Complex conjugate of z₂ (dimensionless): Obtained by negating the imaginary part of z₂; used to rationalise the denominator.
- c^2 + d^2 — Squared modulus of z₂ (dimensionless): Always a positive real number (when z₂ ≠ 0); equals |z₂|² and serves as the real denominator after rationalisation.
Worked Example
Problem. Let z₁ = 5 + 3i and z₂ = 2 − i. Compute: (a) z₁ + z₂, (b) z₁ · z₂, and (c) z₁ ÷ z₂. Express all answers in the form a + bi.
Answer. z₁ + z₂ = 7 + 2i; z₁ · z₂ = 13 + i; z₁ ÷ z₂ = 7/5 + (11/5)i
Origins · 1 minCarl Friedrich Gauss and how this was discovered
Gauss provided the first rigorous proof of the Fundamental Theorem of Algebra in his 1799 doctoral dissertation, establishing that every polynomial of degree n with complex coefficients has exactly n complex roots (counted with multiplicity). He also championed the geometric interpretation of complex numbers as points in a plane, which is now called the Gaussian plane or Argand plane, and developed the arithmetic of what he named 'complex numbers' — a term he coined. His work in number theory, including Gaussian integers (complex numbers a + bi where a, b ∈ ℤ), showed that the algebraic operations on complex numbers obey deep arithmetic laws. Gauss reportedly calculated the sum 1 + 2 + 3 + ... + 100 = 5050 instantly as a young child by noticing the pairing trick, but he was equally remarkable in complex analysis — he discovered many results attributed to others because he refused to publish work he considered incomplete.
Related Concepts
- Complex Number Definition — Algebra of complex numbers is built directly on the definition z = a + bi and the foundational rule i² = −1. You must understand what a complex number is before you can add, multiply, or divide them.
- Modulus and Conjugate — The complex conjugate z̄ = a − bi is the essential tool for division of complex numbers, and the modulus |z| = √(a² + b²) arises directly from the product z · z̄. Both concepts are inseparable from algebraic operations.
- Argand Plane — Every algebraic operation on complex numbers has a geometric meaning on the Argand plane: addition corresponds to vector addition, and multiplication scales and rotates vectors. The Argand plane gives visual intuition to abstract algebra.
- Polar Form of Complex Numbers — Polar form r(cosθ + i sinθ) makes multiplication and division of complex numbers elegantly simple: multiply moduli and add arguments. It is a direct extension of the Cartesian algebraic operations studied here.
- De Moivre's Theorem and Roots — De Moivre's theorem (cosθ + i sinθ)ⁿ = cos(nθ) + i sin(nθ) generalises complex multiplication to integer and fractional powers, requiring solid mastery of the algebraic operations on complex numbers as a prerequisite.