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 scales and rotates, 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 in every quadrant.
- Convert between x + yi and modulus-argument form, and multiply and divide in it.
- Sketch and describe circles, perpendicular bisectors and half-lines.
- Shade the region satisfying an inequality in z.
COMMON MISCONCEPTION
The argument of z is tan⁻¹(y/x), whatever the quadrant z is in.
Length and angle
The modulus |z| is the distance of z from the origin, √(x2 + y2). The argument arg z is the angle from the positive real axis, measured in radians and reported in the principal range (−π, π]. Every z except zero has one, since the origin points in no direction and so has no argument at all. 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 booklet never prints that conversion, so it has to come from you. Now the quadrant trap. tan⁻¹(y/x) only ever lands in the right half plane, so for a number like −1 + i the calculator returns −π/4 and you must correct it to 3π/4 from a sketch. Plot the point first. Every year candidates lose two marks on a number in the second or third quadrant because they trusted the calculator instead of the picture.
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 once both factors are in modulus-argument form.
Two conditions ride along with those statements. Neither number may be zero, because zero has a modulus but no argument at all, and the sum or difference of two arguments can land outside the principal range −π < θ ≤ π, in which case add or subtract 2π to bring it back before quoting it. Marks go on the second of those more often than on the arithmetic.
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, and the arguments are π/3 and π/6. So the product has modulus 4 and argument π/3 + π/6 = π/2, which makes it 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.
One caution. After adding two arguments you may leave the principal range, so subtract 2π to bring the answer back into (−π, π] before quoting it.
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 of centre a and radius r. |z − a| = |z − b| says 'equidistant from a and b', giving the perpendicular bisector of the segment joining them. arg(z − a) = θ says 'the direction from a is θ', a half-line from a at angle θ, with a itself excluded, since z = a would ask for the argument of zero.
GUIDED PRACTICE
Reading three loci
Describe the loci |z − 4| = 3, |z| = |z − 2i|, and arg(z − 1) = π/4.
Show the working
|z − 4| = 3 is a circle of centre 4, the point (4, 0), and radius 3.
|z| = |z − 2i| gives the points equidistant from 0 and 2i, the horizontal line through i, with equation y = 1 on the diagram.
arg(z − 1) = π/4 is a half-line starting at 1, which is excluded, heading up and right at π/4 to the real axis. Marks attach to naming the centre and radius, or the two fixed points, so name them.
Regions, not only edges
Swap the equals sign for an inequality and the locus fattens into a region. |z − a| < r is the inside of the circle, |z − a| > r the outside. |z − a| ≤ |z − b| is the half plane on a's side of the perpendicular bisector, boundary included. Questions often stack two conditions and ask you to shade what satisfies both, so draw each boundary first, decide inside or outside by testing one convenient point such as the origin, then shade the overlap. Dashed for strict, solid for inclusive.
ASSESSMENT FOCUS
- Report arguments in radians in (−π, π], and plot the point before trusting any inverse tan.
- Convert to modulus-argument form before multiplying or dividing, and convert back only if asked.
- Read the centre from what is subtracted. z − (2 + i) means centre 2 + i, not −2 − i.
- A half-line locus excludes its endpoint. Say so when describing arg(z − a) = θ.
- For regions, test the origin to decide which side you want, and mark strict boundaries dashed.
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
A circle of centre 3i, the point (0, 3), and radius 3. It meets the imaginary axis at 0 and 6i, and touches the real axis only at the origin, since the centre sits exactly one radius above it.
Modulus is distance from the origin; argument is the angle in (−π, π], read from a sketch.
Products multiply moduli and add arguments; quotients divide and subtract.
|z − a| = r is a circle, |z − a| = |z − b| a bisector, arg(z − a) = θ a half-line.
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.
Or read them with their worked answers on the modulus, argument and loci questions page.
CHECK YOUR PROGRESS
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- Find the modulus and argument of a complex number in every quadrant.
- Convert between x + yi and modulus-argument form, and multiply and divide in it.
- Sketch and describe circles, perpendicular bisectors and half-lines.
- Shade the region satisfying an inequality in z.
Open the full revision checklist to see every objective in the course in one place.