Plot Argand points and find modulus and argument
Read a Cartesian complex number as a point, calculate its distance from the origin and choose its angle in the correct quadrant.
Exam scope and assessment notes
Pure Mathematics 3 (Paper 3, A Level) · Mathematics 9709, 2026–2027, section 3.9. Paper 1 algebra and radian trigonometry are assumed. Use Cartesian working for products, quotients and the two square roots of a non-zero complex number. Arguments usually use −π < θ ≤ π; an alternative interval 0 ≤ θ < 2π is acceptable unless specified. The argument of zero is undefined. General nth-root methods are beyond this mapped sequence. Written practice is self-checked; human subject-teacher review remains pending.
Before you begin
- Use Pythagoras and radian trigonometry.
- Locate points in Cartesian quadrants.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
Where is −3 + 4i on an Argand diagram?
What you will learn
- Represent complex numbers and displacements on an Argand diagram.
- Find modulus and a valid argument with explicit quadrant and zero checks.
The two parts become perpendicular coordinates
An Argand diagram plots Re z horizontally and Im z vertically. The number −3 + 4i is the point (−3, 4), and the origin represents zero. With equal axis scales, the distance to the origin is |z| = √(x² + y²); here it is 5. Modulus is a non-negative real quantity and is zero only for z = 0. The distance between points a and b is |b − a|, not generally the difference of their individual moduli.
The argument is the directed angle from the positive real axis to the origin-to-point vector, positive anticlockwise. For −3 + 4i it lies in quadrant II, so θ = π − arctan(4/3) ≈ 2.214 radians, or 126.9°. A calculator’s arctan(−4/3) alone gives a quadrant IV angle pointing in the opposite direction. Sketch or inspect signs before accepting an inverse-tangent value.
Angles are equivalent modulo a full turn
Arguments differ by 2π. The usual principal interval is −π < θ ≤ π; for −2 − 2i its principal argument is −3π/4, while 5π/4 is equivalent in 0 ≤ θ < 2π. On the negative real axis the usual principal value is π, not −π. Follow a question’s specified interval and use radians in algebraic expressions unless degrees are requested.
Zero has modulus zero but no argument: its vector has no direction. The conjugate reflects a point across the real axis; its argument is the negative of the original modulo 2π, with the negative real-axis endpoint represented consistently. A displacement from a to b has complex value b − a and its own modulus and argument. These describe the length and direction of the movement, rather than either point’s position from the origin.
Pause and explain
What is the argument of z = 0?
Worked example
Plot z = −3 + 4i and find its modulus and principal argument.
Show the worked solution
- Plot (−3, 4) in quadrant II with equal axis scales.
- Use |z| = √(9 + 16) = 5.
- Its quadrant II argument is π − arctan(4/3) ≈ 2.214 rad, or 126.9°.
Answer |z| = 5; arg z = π − arctan(4/3) ≈ 2.214 rad (126.9°).
Try a topic test
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Test this topic →From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
Find the modulus and principal argument of 1 + i√3.
Give me a hint
Its point is in quadrant I with tangent √3.
Compare my reasoning
- The modulus is √(1 + 3) = 2.
- The quadrant I angle with tangent √3 is π/3.
- The coordinates 2cos(π/3) = 1 and 2sin(π/3) = √3 check the direction.
Modulus 2; principal argument π/3.
Look for these in your work
- Calculate modulus from both squared components.
- Check the angle against both coordinate signs.
Find the modulus of −2 − 2i and give its argument in each interval −π < θ ≤ π and 0 ≤ θ < 2π.
Give me a hint
Its point lies in quadrant III; the reference angle is π/4.
Compare my reasoning
- The modulus is √8 = 2√2.
- In the principal interval choose −3π/4.
- Adding 2π gives 5π/4 in the non-negative interval; both point to the same quadrant.
Modulus 2√2; arguments −3π/4 and 5π/4 in the respective intervals.
Look for these in your work
- State which interval each angle belongs to.
- Use equivalent directions rather than the quadrant I arctangent alone.
A point moves from a = 1 + 2i to b = 4 − 2i. Find the complex displacement, its distance and principal direction angle.
Give me a hint
Use b − a before calculating distance or direction.
Compare my reasoning
- The displacement is 3 − 4i.
- Its modulus is √(9 + 16) = 5.
- It points to quadrant IV, so its argument is −arctan(4/3) ≈ −0.9273 rad, or −53.1°.
Displacement 3 − 4i; distance 5; direction −arctan(4/3) ≈ −0.9273 rad (−53.1°).
Look for these in your work
- Use the directed difference from start to finish.
- Check the displacement’s quadrant and angle convention.
Common mistakes
- Check the quadrant before choosing an inverse-tangent angle.
- Argand axes show the real and imaginary coefficients, with equal scales for metric geometry.
- The argument of zero is undefined; normalise the negative-axis endpoint consistently.
What can you explain now?
What is the principal argument of −4 using −π < θ ≤ π?
Compare with the explanation
π.
It points along the negative real axis; −π lies outside this specified interval.
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Make it stick
Plot a non-zero complex point or displacement and find its distance and directed angle, checking the quadrant and argument interval.
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.