Modulus and argument

Modulus

The modulus of z = a + bi is |z| = sqrt(a² + b²), the distance from O to the point (a, b). It is never negative.

|3 + 4i| = 5 and |z*| = |z| for every z. Note zz* = |z|².

Argument

The argument arg z is the angle between the positive real axis and the line from O to z, measured anticlockwise as positive. In F1 it is given in radians.

The principal argument lies in the range -π < arg z ≤ π. Points below the real axis have a negative argument.

Use a sketch

tan^-1(b/a) only gives angles between -π/2 and π/2, so it is right only in the first and fourth quadrants.

• Second quadrant (a < 0, b > 0): arg z = π - tan^-1|b/a|.
• Third quadrant (a < 0, b < 0): arg z = -(π - tan^-1|b/a|).

On the axes: arg(positive real) = 0, arg(negative real) = π, arg(ki) = π/2 and arg(-ki) = -π/2 for k > 0.

Worked example

Find the modulus and the argument of z = -2 + 2sqrt(3)i.

1. |z| = sqrt(4 + 12) = 4.
2. z is in the second quadrant. The angle with the negative real axis is tan^-1(2sqrt(3)/2) = π/3.
3. arg z = π - π/3.

Answer: |z| = 4, arg z = 2π/3

Check yourself

1. Find |5 - 12i|.
   Show answer

   13. |5 - 12i| = sqrt(25 + 144) = sqrt(169) = 13.

2. Find arg(1 + sqrt(3)i).
   • π/6
   • π/3
   • 2π/3
   • -π/3
   Show answer

   π/3. First quadrant: tan^-1(sqrt(3)/1) = π/3.

3. Find arg(-3 - 4i), in radians to 3 decimal places.
   Show answer

   -2.214. Third quadrant. The angle with the negative real axis is tan^-1(4/3) = 0.927. So arg z = -(π - 0.927) = -2.214.

4. Find arg(4 - i), in radians to 3 decimal places.
   Show answer

   -0.245. Fourth quadrant, so arg z = -tan^-1(1/4) = -0.245.

5. Find arg(-2 + i), in radians to 3 decimal places.
   Show answer

   2.678. Second quadrant: arg z = π - tan^-1(1/2) = π - 0.464 = 2.678.

6. What is arg(-5)?
   • π
   • 0
   • -π/2
   • -π
   Show answer

   π. -5 lies on the negative real axis. The principal argument is π (the range is -π < θ ≤ π, so -π is excluded).

7. z has modulus 6 and argument π/5. Which describes z*?
   • Modulus -6, argument π/5
   • Modulus 6, argument π - π/5
   • Modulus 6, argument -π/5
   • Modulus 1/6, argument -π/5
   Show answer

   Modulus 6, argument -π/5. z* is the reflection of z in the real axis: the same distance from O, with the angle reflected.

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Sources

• Pearson Edexcel International Advanced Level in Mathematics, Further Mathematics and Pure Mathematics specification (issue 3), Unit FP1
• Pearson Edexcel IAL Further Pure Mathematics 1 scheme of work

Last updated 8 November 2026. Spotted a mistake? Tell us.