H2 Maths Argand Diagrams: Why Multiplying by i Turns a Point

Check the syllabus before following older advice

A JC student on SGExams asked whether an Argand diagram operation seen in prelim papers was relevant to the A-Levels. The discussion dates from 2022. SEAB’s H2 Mathematics 9758 syllabus for the 2026 examination explicitly includes the geometrical effect of multiplying by i. It excludes the polar and exponential forms of complex numbers from this section. For another examination year, consult that year’s syllabus.

You can explain the rotation entirely in Cartesian form. Plot the real part on the horizontal axis and the imaginary coefficient on the vertical axis. The point for 3 + 2i is therefore three units to the right and two up.

Make the change algebraically first

Multiply a + bi by i. The result is ai + bi², which equals −b + ai because i² = −1. The coordinate rule is therefore (a, b) → (−b, a). For 3 + 2i, that gives −2 + 3i, which is two units left and three up. Sketch arrows from the origin to see the turn.

A quick direction check is to start at 1, the point (1, 0). Multiplication by i takes it to (0, 1), so the turn is anticlockwise. The modulus stays the same because a² + b² is unchanged by swapping the coordinates and changing one sign. The origin itself stays fixed.

Separate rotation from reflection

Conjugating a + bi gives a − bi and changes only the vertical coordinate. For 3 + 2i this gives 3 − 2i, a reflection across the real axis. Negating the same number gives −3 − 2i, changing both coordinates. Check your sketch by comparing all three operations using the same starting point.

Apply multiplication by i twice and the rule gives (−a, −b), which is the same as multiplying by −1. Four applications return the original point. These are quick consistency checks when a sign feels uncertain.

Try a point in a different quadrant

Start with −2 + 4i. Predict the new quadrant before calculating. Multiplication by i gives −4 − 2i, so the point moves from the upper left to the lower left. Its modulus remains √20. Next reverse the operation: if iz = −4 − 2i, multiplying both sides by −i recovers z = −2 + 4i.

When practising with older papers, note the syllabus year before treating every question as a current requirement. Pair one diagram question with one algebra question to check that you can move comfortably between the two representations.

Explain the move without relying on a picture

Pick an arbitrary point above the real axis and explain why its new real coordinate must be negative after multiplication by i. Then choose a point on the imaginary axis, such as 5i. The new point is −5 on the real axis. This checks a boundary case that may be missed when all examples start in a quadrant. In a written answer, state the coordinate rule and use the diagram to support your reasoning.

Choose a suitable practice question from our H2 Maths papers.

Sources and further reading

• Complex numbers H2 Maths Argand questions (5 September 2022).
• Official syllabus reference. Checked on 5 November 2026.