Modulus and Conjugate
Every complex number z = a + bi carries two fundamental pieces of geometric information: how far it sits from the origin in the complex plane, and the mirror image it produces when its imaginary part is flipped. These two ideas — the modulus and the conjugate — are central to nearly every calculation involving complex numbers, from solving polynomial equations to analysing electrical circuits and signal processing. The modulus |z| of a complex number is its distance from the origin, computed using the Pythagorean theorem: |z| = √(a² + b²). Because it measures length, the modulus is always a non-negative real number. The conjugate z̄ (read 'z-bar') is formed by negating the imaginary part: if z = a + bi, then z̄ = a − bi. Geometrically, conjugation reflects the point across the real axis. Together, the modulus and conjugate unlock powerful algebraic tools. Their product z · z̄ = |z|² is always real, which allows division of complex numbers without any guesswork. The modulus satisfies a multiplicative property |z₁z₂| = |z₁||z₂|, mirroring the absolute value of real numbers. Mastering these two concepts is the gateway to polar form, Euler's formula, and the deeper geometry of the complex plane.
- Defining the Conjugate
- Defining the Modulus
- The Key Identity: z · z̄ = |z|²
- Properties of the Modulus
- Dividing Complex Numbers Using the Conjugate
- Conjugate Properties
Defining the Conjugate
Given z = a + bi where a, b ∈ ℝ, the complex conjugate is z̄ = a − bi. Only the sign of the imaginary part changes; the real part stays fixed. For example, if z = 3 + 4i, then z̄ = 3 − 4i. Conjugation is an involution: taking the conjugate twice returns the original number, (z̄)̄ = z.
Defining the Modulus
The modulus of z = a + bi is |z| = √(a² + b²). This is exactly the Euclidean distance from the origin (0, 0) to the point (a, b) in the complex plane. For z = 3 + 4i, |z| = √(9 + 16) = √25 = 5. The modulus is always ≥ 0, and |z| = 0 if and only if z = 0.
The Key Identity: z · z̄ = |z|²
Multiplying z = a + bi by its conjugate z̄ = a − bi gives: z · z̄ = (a + bi)(a − bi) = a² − (bi)² = a² + b² = |z|². The result is a real, non-negative number. This identity is fundamental: it turns the problem of dividing by a complex number into dividing by a real number, which is straightforward.
Properties of the Modulus
The modulus obeys four important rules: (i) |z| ≥ 0, with equality only when z = 0; (ii) |z̄| = |z|, since negating the imaginary part does not change the distance; (iii) |z₁z₂| = |z₁||z₂|, the product rule; (iv) |z₁ + z₂| ≤ |z₁| + |z₂|, the triangle inequality. These mirror the properties of absolute value on the real line.
Dividing Complex Numbers Using the Conjugate
To compute z₁/z₂, multiply numerator and denominator by z̄₂: z₁/z₂ = (z₁ · z̄₂)/(z₂ · z̄₂) = (z₁ · z̄₂)/|z₂|². The denominator is now a real number, so the division is simple. Example: (1 + 2i)/(3 + 4i) = (1 + 2i)(3 − 4i)/25 = (3 − 4i + 6i − 8i²)/25 = (3 + 2i + 8)/25 = (11 + 2i)/25.
Conjugate Properties
Conjugation distributes over addition and multiplication: (z₁ + z₂)̄ = z̄₁ + z̄₂ and (z₁z₂)̄ = z̄₁ · z̄₂. Also, z + z̄ = 2 Re(z) = 2a (a real number), and z − z̄ = 2i Im(z) = 2bi (a purely imaginary number). These identities allow real and imaginary parts to be extracted cleanly from any complex expression.
The Key Equation
- z — Complex number (dimensionless): A number of the form a + bi where a and b are real numbers and i = √(−1)
- a — Real part (dimensionless): Re(z); the horizontal coordinate of z in the complex plane
- b — Imaginary part (dimensionless): Im(z); the vertical coordinate of z in the complex plane
- |z| — Modulus (dimensionless): The distance from the origin to z in the complex plane; always a non-negative real number
- \bar{z} — Complex conjugate (dimensionless): The reflection of z across the real axis; formed by replacing i with −i
Worked Example
Problem. Let z = 5 − 12i. Find (a) the conjugate z̄, (b) the modulus |z|, (c) the product z · z̄, and (d) the reciprocal 1/z in the form a + bi.
Answer. z̄ = 5 + 12i, |z| = 13, z · z̄ = 169, and 1/z = 5/169 + (12/169)i
Origins · 1 minJean-Robert Argand and how this was discovered
In 1806, Argand published a landmark pamphlet introducing the geometric representation of complex numbers as points in a plane — now called the Argand diagram or complex plane. His work made the modulus interpretable as a geometric distance, giving visual meaning to |z| = √(a² + b²) and turning conjugation into a visible reflection. Although Gauss and Wessel had similar ideas, Argand's publication was the first widely circulated geometric treatment of complex numbers. Argand was an amateur mathematician and Parisian bookseller with no formal university training. He published his geometric theory of complex numbers anonymously in a self-printed pamphlet, yet it became one of the most influential short works in the history of mathematics.
Related Concepts
- Introduction to Complex Numbers — The modulus and conjugate are defined in terms of the real and imaginary parts a and b of z = a + bi, so a solid understanding of what complex numbers are is essential before working with these operations.
- Polar Form of Complex Numbers — The modulus |z| is exactly the r in the polar representation z = r(cos θ + i sin θ), making it the bridge between rectangular and polar forms of complex numbers.
- The Argand Diagram — The Argand diagram is the visual setting where modulus becomes distance from the origin and conjugation becomes geometric reflection, giving both concepts their clearest interpretation.
- Dividing Complex Numbers — Dividing complex numbers relies entirely on multiplying by the conjugate to create a real denominator via z · z̄ = |z|², making conjugates indispensable for this operation.