Argand Diagrams and Modulus-Argument Form

The Argand diagram

An Argand diagram represents complex numbers geometrically: the real part on the horizontal axis and the imaginary part on the vertical axis. So z = a + bi is the point (a, b).

Modulus and argument

• The modulus |z| is the distance from the origin:

|z| = √(a² + b²)

• The argument arg z is the angle from the positive real axis, measured in radians, usually in the range −π < θ ≤ π:

tan θ = b ÷ a

Take care to place the angle in the correct quadrant (use a sketch).

Modulus–argument (polar) form

A complex number can be written as:

z = r(cos θ + i sin θ)     where r = |z|, θ = arg z

This is useful for multiplication, division and powers.

Multiplying and dividing in mod-arg form

• Multiply: multiply the moduli, add the arguments.
• Divide: divide the moduli, subtract the arguments.

|z₁z₂| = |z₁||z₂|,   arg(z₁z₂) = arg z₁ + arg z₂

De Moivre's theorem

For raising to a power:

[r(cos θ + i sin θ)]ⁿ = rⁿ(cos nθ + i sin nθ)

Used to find powers and roots of complex numbers, and to derive trig identities.

Loci on the Argand diagram

Equations describe sets of points (loci):

• |z − a| = r → a circle of radius r centred at the point a.
• |z − a| = |z − b| → the perpendicular bisector of the line joining a and b.
• arg(z − a) = θ → a half-line from a at angle θ.

Worked example

Find the modulus and argument of z = 1 + i.

• |z| = √(1² + 1²) = √2.
• arg z = tan⁻¹(1/1) = π/4 (first quadrant). ✓

Common mistakes

• Putting the argument in the wrong quadrant (always sketch it).
• Forgetting the argument range (−π < θ ≤ π).
• Multiplying/dividing moduli but forgetting to add/subtract arguments.

Exam tips

• Sketch the point on an Argand diagram to get the correct quadrant for arg z.
• Use mod-arg form for products, quotients and powers (De Moivre).
• For loci, recognise the standard forms (circle, perpendicular bisector, half-line).

Key facts to remember

• Argand diagram: real (x) vs imaginary (y); |z| = √(a²+b²), arg z = angle (tan θ = b/a, correct quadrant).
• Mod-arg form z = r(cos θ + i sin θ); multiply → multiply moduli, add arguments; De Moivre: zⁿ = rⁿ(cos nθ + i sin nθ).
• Loci: |z − a| = r (circle), |z − a| = |z − b| (perpendicular bisector), arg(z − a) = θ (half-line).