The Argand Diagram and Modulus

The Complex Plane — Giving Geometry to Numbers

The Argand diagram (named after Jean-Robert Argand, 1806) is a coordinate plane where each complex number z = a + bi is plotted as the point (a, b). The horizontal axis represents real numbers; the vertical axis represents purely imaginary numbers. This geometric representation transforms algebraic operations into geometric transformations — a perspective that becomes essential in polar form and De Moivre’s theorem.

On the Argand diagram, the conjugate z̅ = a − bi is the reflection of z across the real axis. Adding two complex numbers corresponds to vector addition (parallelogram law). These geometric interpretations give deep insight into complex arithmetic.

Modulus as Distance

The modulus |z| = √(a² + b²) is simply the distance from the origin O to the point z, by Pythagoras’ theorem. This is why |z| ≥ 0 always, with |z| = 0 only when z = 0.

Key modulus properties (worth memorising):

• |zw| = |z||w|: the modulus of a product equals the product of the moduli
• |z/w| = |z|/|w|: the modulus of a quotient equals the quotient of the moduli
• |z²| = |z|², and more generally |zⁿ| = |z|ⁿ
• |z̅| = |z|: the conjugate has the same distance from the origin
• |z + w| ≤ |z| + |w| (triangle inequality): distance from O to z + w cannot exceed the sum of distances to z and w

Argument — The Angle

The argument arg(z) is the angle θ that the line from the origin to z makes with the positive real axis, measured anticlockwise. The principal argument is restricted to (−π, π], meaning −π < θ ≤ π.

To find arg(z) for z = a + bi:

1. Compute the reference angle α = tan⁻¹(|b|/|a|) (always positive, in first quadrant)
2. Adjust based on the quadrant of (a, b):
   • Quadrant 1 (a > 0, b > 0): arg(z) = +α
   • Quadrant 2 (a < 0, b > 0): arg(z) = π − α
   • Quadrant 3 (a < 0, b < 0): arg(z) = −(π − α) = α − π
   • Quadrant 4 (a > 0, b < 0): arg(z) = −α

Special cases: arg(positive real) = 0, arg(negative real) = π, arg(positive imaginary) = π/2, arg(negative imaginary) = −π/2.

Geometric Sets in the Complex Plane

Conditions on |z| and arg(z) describe geometric regions. For example:

• |z| = r is a circle of radius r centred at the origin
• |z − z⊂0;| = r is a circle of radius r centred at z⊂0;
• arg(z) = θ is a ray (half-line) from the origin at angle θ
• |z − z⊂1;| = |z − z⊂2;| is the perpendicular bisector of the segment from z⊂1; to z⊂2;

These geometric interpretations are tested in exam problem-solving questions and are excellent for developing spatial reasoning.