Modulus, argument & the Argand diagram
The Argand diagram plots a + bi at the point (a, b), so |z| = √(a² + b²) is the distance from the origin and arg z is the angle from the positive real axis. The principal argument is conventionally taken in (−π, π], and computing it needs a quadrant check — arctan(b/a) alone cannot distinguish the second quadrant from the fourth.
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Method: how to approach it
The order below is what examiners expect to see, and each step carries its own marks.
- Plot the pointReal part on the horizontal axis, imaginary on the vertical. Sketching makes the quadrant obvious.
- Compute the modulus|z| = √(a² + b²), always non-negative.
- Compute the reference anglearctan|b/a| gives the acute angle to the real axis.
- Adjust for the quadrantQ1: θ. Q2: π − θ. Q3: θ − π (to stay in the principal range). Q4: −θ.
Worked example
Find the modulus and principal argument of z = −1 + i√3.
- |z| = √(1 + 3) = 2.
- The point (−1, √3) lies in the second quadrant.
- Reference angle: arctan(√3/1) = π/3.
- In Q2 the argument is π − π/3 = 2π/3.
Answer. |z| = 2 and arg z = 2π/3, so z = 2(cos 2π/3 + i sin 2π/3).
Where marks get dropped
These are the specific errors that cost credit on modulus, argument & the argand diagram questions — QED's rubric penalises each of them separately.
- Trusting arctan(b/a) without a quadrant check. For −1 + i√3 it gives −π/3, which is in the wrong half-plane entirely.
- Reporting an argument outside (−π, π] when the principal value is required. Add or subtract 2π to bring it into range.
- Forgetting that the modulus is always positive, even when both components are negative.
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Modulus, argument & the Argand diagram — frequently asked questions
What is the principal argument?
The unique value of arg z in (−π, π]. Arguments are only defined up to adding multiples of 2π, so a convention is needed.
What does |z − w| represent?
The distance between the points z and w on the Argand diagram. This is why |z − a| = r describes a circle of radius r centred at a.
Does arg 0 exist?
No. The origin has no direction, so its argument is undefined — though its modulus is 0.
The rest of Complex Numbers
Arithmetic, the Argand plane, polar form and De Moivre. Each subtopic below has its own method, worked example and mark-losing traps.
- 1The imaginary unit i & complex arithmetic
- 2Conjugates & division
- 3Modulus, argument & the Argand diagram
- 4Polar & exponential form re^{iθ}
- 5Multiplying & dividing in polar form
- 6De Moivre’s theorem & powers
- 7nth roots & roots of unity
- 8Solving polynomial equations over ℂ
- 9The fundamental theorem of algebra & conjugate roots
- 10Euler’s formula & trigonometric identities
- 11Loci & regions in the complex plane
- 12Complex multiplication as rotation & scaling
- 13Applications: phasors & AC circuits
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