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The Argand diagram

Edexcel IAL Further Mathematics · WFM01 Further Pure Mathematics 1

An Argand diagram plots z = a + bi as the point (a, b), with a real axis and an imaginary axis. Sums, differences, products and quotients all have a geometric meaning on it.

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Plotting

Draw and label both axes (Re and Im) with an accurate scale. A complex number can be shown as a point or as a vector from O. Conjugate pairs are mirror images in the real axis, so plot them as a pair.

Sums and differences

Adding complex numbers adds their vectors: z_1 + z_2 is the fourth vertex of the parallelogram with sides Oz_1 and Oz_2.

z_1 - z_2 is the vector from z_2 to z_1, so |z_1 - z_2| is the distance between the points.

Products and quotients

Multiplying by a real number k > 0 enlarges the vector by scale factor k. Multiplying by i rotates it 90° anticlockwise about O; multiplying by -1 rotates it 180°.

In general |z_1 z_2| = |z_1||z_2| and |z_1/z_2| = |z_1|/|z_2|: the point z_1 z_2 is |z_2| times as far from O as z_1, and z_1/z_2 is 1/|z_2| times as far.

Worked example

z_1 = 1 + 2i and z_2 = 3 - i. Show z_1, z_2 and z_1 + z_2 on an Argand diagram and describe the shape O, z_1, z_1 + z_2, z_2.

  1. Plot (1, 2), (3, -1) and z_1 + z_2 = 4 + i at (4, 1).
  2. The vector from z_2 to z_1 + z_2 equals the vector Oz_1, and the vector from z_1 to z_1 + z_2 equals Oz_2.

Answer: A parallelogram, with z_1 + z_2 = 4 + i at the vertex opposite O

Key idea: Sum: parallelogram rule. |z_1 - z_2| is the distance between two points. Multiplying by i rotates by 90° anticlockwise.

Check yourself

  1. Find the distance between the points representing 2 + 3i and -1 - i on an Argand diagram.
    Show answer

    5. z_1 - z_2 = 3 + 4i and |3 + 4i| = 5.

  2. z = 3 + i. The point iz is plotted. Find its real part.
    Show answer

    -1. iz = 3i + i2 = -1 + 3i, so the real part is -1. The point (3, 1) has been rotated 90° anticlockwise to (-1, 3).

  3. On an Argand diagram, which transformation maps z to iz?
    • Rotation 90° clockwise about O
    • Reflection in the real axis
    • Rotation 90° anticlockwise about O
    • Enlargement scale factor 2 about O
    Show answer

    Rotation 90° anticlockwise about O. |i| = 1, and i(a + bi) = -b + ai: the point (a, b) moves to (-b, a).

  4. On an Argand diagram, how is z* related to z?
    • Reflection of z in the imaginary axis
    • Reflection of z in the real axis
    • Rotation of z through 180° about O
    • Reflection of z in the line y = x
    Show answer

    Reflection of z in the real axis. z* = a - bi: the point (a, b) goes to (a, -b).

  5. z_1 = 1 + 2i and z_2 = 3 - i. O, z_1, P and z_2 form a parallelogram with P opposite O. Find the imaginary part of the number at P.
    Show answer

    1. P represents z_1 + z_2 = 4 + i, so the imaginary part is 1.

  6. |z_1| = 3 and |z_2| = 4. How far from O is the point representing z_1 z_2?
    Show answer

    12. |z_1 z_2| = |z_1||z_2| = 3 × 4 = 12.

  7. Which vector on the Argand diagram represents z_1 - z_2?
    • From the point z_1 to the point z_2
    • From O to the midpoint of z_1 and z_2
    • From O to the point z_1 + z_2
    • From the point z_2 to the point z_1
    Show answer

    From the point z_2 to the point z_1. z_2 + (z_1 - z_2) = z_1, so z_1 - z_2 takes you from z_2 to z_1.

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