Maths › Complex numbers › Modulus, argument and loci
Modulus, argument and loci
Length and direction take over from across and up: every complex number is a distance from the origin at an angle, multiplication becomes rotation, and equations in z draw circles and lines.
Builds on Complex arithmetic and the Argand diagram and Radians, arcs and small angles.
IN THIS TOPIC
- Find the modulus and argument of a complex number, handling every quadrant.
- Convert between x + yi and modulus-argument form, and multiply and divide in it.
- Sketch and describe the standard loci: circles, perpendicular bisectors, half-lines.
WHAT YOU PROBABLY THINK
The argument of z is tan⁻¹(y/x), whatever the quadrant z is in.
Length and angle
The modulus |z| is z's distance from the origin, √(x2 + y2), and the argument arg z is the angle from the positive real axis, measured in radians and reported in (−π, π]. Together they pin the point as surely as x and y do, and the conversion is the trigonometry of one right triangle:
with r = |z| and θ = arg z. The opening claim is the quadrant trap. tan⁻¹(y/x) only lands in the right half plane; for a number like −1 + i the calculator's −π/4 must be corrected to 3π/4 by a sketch. Always plot the point first and read which quadrant the angle belongs to.
Multiplying in modulus-argument form
Multiplication has a clean geometric reading: moduli multiply and arguments add. Dividing divides the moduli and subtracts the arguments. A product that looks messy in x + yi form can be almost mental arithmetic in modulus-argument form.
WORKED EXAMPLE
A product done both ways
Find (1 + √3 i)(√3 + i), using modulus-argument form, and check directly.
Both factors have modulus 2; the arguments are π/3 and π/6. So the product has modulus 4 and argument π/3 + π/6 = π/2: the product is 4i.
Directly: (1 + √3 i)(√3 + i) = √3 + i + 3i + √3 i2 = (√3 − √3) + 4i = 4i, as claimed.
Multiplying by a complex number scales by its modulus and rotates by its argument; here the rotation carried the product onto the imaginary axis.
Loci: equations that draw
An equation in z picks out a set of points, a locus, and three shapes cover the syllabus. |z − a| = r says 'distance from a is r': a circle, centre a, radius r. |z − a| = |z − b| says 'equidistant from a and b': the perpendicular bisector of the segment joining them. arg(z − a) = θ says 'the direction from a is θ': a half-line from a (excluding a itself) at angle θ.
YOUR TURN
Reading three loci
Describe the loci |z − 4| = 3, |z| = |z − 2i|, and arg(z − 1) = π/4.
Show the working
|z − 4| = 3: a circle, centre 4 (the point (4, 0)), radius 3.
|z| = |z − 2i|: points equidistant from 0 and 2i, the horizontal line through i, that is the line with equation y = 1 on the diagram.
arg(z − 1) = π/4: a half-line starting at 1 (excluded) heading up-right at π/4 to the real axis. Naming centre and radius, or the two fixed points, is what the marks attach to.
THE EXAM BIT
- Report arguments in (−π, π] in radians; plot the point before trusting any inverse tan.
- Convert to modulus-argument form before multiplying or dividing; convert back only if asked.
- For |z − a| = r, read the centre from what is subtracted: z − (2 + i) means centre 2 + i.
- A half-line locus excludes its endpoint; say so when describing arg(z − a) = θ.
CHECK YOURSELF
Sketch the locus |z − 3i| = 3, and state its centre and radius. Where does it meet the axes?
Show a hint
Read the centre from the subtraction, then think about how far the circle reaches.
Show the answer
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Modulus is distance from the origin; argument is the angle in (−π, π], read from a sketch.
Products multiply moduli and add arguments; |z − a| = r is a circle, centre a, radius r.
WORKBOOK
Printable practice for this topic: original exam-style questions with room to work, and a fully worked answer book. Free to use; please do not redistribute or sell.
CHECK YOUR PROGRESS
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- Find the modulus and argument of a complex number, handling every quadrant.
- Convert between x + yi and modulus-argument form, and multiply and divide in it.
- Sketch and describe the standard loci: circles, perpendicular bisectors, half-lines.
Open the full revision checklist to see every objective in the course in one place.
No animated video for this topic yet; these notes stand alone.