H2 Maths Complex Numbers | Free Notes & Worked Examples

Q: What does H2 Maths Notes (JC 1-2): 4) Complex Numbers cover?
A: Complex numbers in Cartesian form, modulus, argument, conjugates, quadratic roots, operations, and Argand-diagram transformations for H2 Maths Topic 4.

Before you begin
Review your Additional Maths introduction to complex arithmetic so operations with $a + bi$ feel fluent. For the 2026 and 2027 syllabus, focus on Cartesian form and Argand representation. Complex loci, polar/exponential representations, and De Moivre workflows are not part of the current H2 Maths 9758 examinable scope.

• Complex numbers pair algebra with geometry: Separate real and imaginary parts.
• Conjugates make denominators real and reflect points: Use $\bar{z}$ when dividing.
• Argand transformations are visual: Track conjugation, negation, translations, and multiplication by $i$ as movements on the plane.

Concrete example: To simplify $\frac{3+4i}{1-2i}$, multiply top and bottom by $1+2i$. The denominator becomes real, so the result can be written as $a+bi$

Status: SEAB's current H2 Mathematics (9758) syllabus PDFs for 2026 and 2027 list quadratic roots, modulus/argument, conjugates, operations, Argand representation, and selected geometrical effects. They do not list complex loci.

4.1 | Core Concepts and Forms

Representations

• Cartesian form: $z = a + bi$ with real part $\Re(z) = a$ and imaginary part $\Im(z) = b$.

Modulus and argument

• Compute $|z| = \sqrt{a^2 + b^2}$

Argument quadrant checkpoint

Before writing $\arg z$, locate the point $(a,b)$ first. The calculator angle or inverse-tangent value is only safe after the quadrant is checked.

Misconception check: $\arctan\left(\frac{b}{a}\right)$ gives a reference calculation, not the full answer by itself. The point's quadrant decides the principal argument.

Conjugates

• Conjugate $\overline{z} = a - bi$.
• Useful identity: multiplying a complex number by its conjugate produces the squared modulus ($|z|^2$).
• Reciprocal form: for non-zero z, multiply numerator and denominator by the conjugate so the denominator becomes the real scalar $|z|^2$

4.2 | Operations and Quadratic Roots

• Addition/subtraction: combine real and imaginary parts directly.
• Multiplication: expand brackets and use $i^2 = -1$.
• Division: multiply numerator and denominator by the conjugate to make the denominator real.
• Quadratic equations: apply the quadratic formula in $\mathbb{C}$. For real coefficients, non-real roots occur in conjugate pairs.

Example 1 -- Quadratic with complex roots

Solve $z^2 - 4z + 13 = 0$.

$$z = \frac{4 \pm \sqrt{16 - 52}}{2} = \frac{4 \pm \sqrt{-36}}{2} = 2 \pm 3i.$$

Example 2 -- Simplify a division

Compute $\dfrac{3 + 4i}{1 - 2i}$.

$$\frac{3 + 4i}{1 - 2i} \times \frac{1 + 2i}{1 + 2i} = \frac{(3 + 4i)(1 + 2i)}{1 + 4} = \frac{3 + 6i + 4i + 8i^2}{5} = \frac{-5 + 10i}{5} = -1 + 2i.$$

4.3 | Argand Diagram Interpretation

• Representation: $z=a+bi$ is plotted as the point $(a,b)$.
• Conjugation: $\overline{z}$ reflects the point across the real axis.
• Negation: $-z$

Transformation checkpoint

Before sketching, name the operation first. The operation tells you the movement.

Worked check: if $z=2-i$, then $\overline{z}=2+i$, $-z=-2+i$

Misconception check: modulus and argument describe a point's distance and angle from the origin; they are still part of current 9758. Complex loci are different and are kept only in the appendix below for old-syllabus or enrichment use.

Example 3 -- Argand transformation

Let $z=-1+2i$. Find $iz$ and describe the effect on the Argand diagram.

$$iz=i(-1+2i)=-i+2i^2=-2-i.$$

So the point $(-1,2)$ rotates $90^\circ$ anticlockwise to $(-2,-1)$.

4.4 | Calculator Workflows

• Use complex mode to verify arithmetic; keep the calculator in radians for argument work.
• Use Abs / Arg (or equivalent) to check modulus and principal argument, but keep the written solution in Cartesian form.
• For quadratic equations, a polynomial solver can sanity-check roots-still show the analytic method and present answers as $a \pm bi$.

4.5 | Exam Watch Points

• State the branch (degrees vs radians) for arguments and stay consistent throughout a question.
• If your equation has real coefficients and one root is $a + bi$, state the conjugate root $a - bi$ explicitly.
• On Argand diagram questions, include a short sentence describing the geometry after the algebra (for example, reflection in the real axis or rotation by $90^\circ$ anticlockwise).
• For division, multiply numerator and denominator by the conjugate to show the simplification explicitly.

Practice Quiz

Check your fluency with Cartesian operations, quadratic roots, modulus, argument, and Argand transformations.

4.6 | Quick Revision Checklist

• Perform conjugation, modulus, and division confidently in Cartesian form.
• Solve quadratic equations with complex roots and state conjugate pairs.
• Interpret Argand diagram transformations with a labelled sketch.
• Explain the effect of conjugation and modulus in plain language after calculations.

Next step: stay in sync via the H2 Maths notes hub and revisit Sub-topic 4.1 - Argand diagrams for detailed drills.

Want weekly guided practice on Complex Numbers? Our H2 Maths tuition programme builds fluency in this topic through structured problem sets and exam-style drills.

Common exam mistakes

• Giving the argument in degrees instead of radians (or vice versa): Always check what the question requires and keep your GC in the correct mode. The principal argument is typically expressed in exact radian form (e.g. $\pi/4$) unless degrees are specifically requested.
• Stating the argument outside the principal range $(-\pi, \pi]$: The principal argument must satisfy $-\pi \lt \arg z \leq \pi$

Frequently asked questions

Is Topic 4 (Complex Numbers) in Paper 1 or Paper 2?
Topic 4 is Pure Mathematics and can appear in Paper 1 (100 marks) or Paper 2 Section A (40 marks). For the current 9758 syllabus, focus on Cartesian complex numbers, modulus, argument, conjugate, operations, Argand representation, and the listed geometrical effects.

Are polar and exponential forms of complex numbers examinable in 2026?
No. The 2026 H2 Maths (9758) syllabus explicitly excludes polar and exponential (Euler) form, and De Moivre's theorem is also excluded. All complex number work is in Cartesian form $a + bi$.

When do complex roots appear in conjugate pairs?
Conjugate pairs occur when the polynomial has real coefficients. If a polynomial has complex (non-real) coefficients, roots need not come in conjugate pairs. For H2 Maths, almost all polynomial questions involve real coefficients, so always state the conjugate pair unless the coefficients are explicitly complex.

Appendix: Old Syllabus / Enrichment - Complex Loci

Complex loci are useful when working through older TYS or extension questions, but they are not listed in the current SEAB H2 Mathematics 9758 syllabus for 2026 or 2027.

• Circle: $|z - (a + bi)| = r$ represents a circle centre $(a, b)$ with radius $r$.
• Half-plane: $\Re(z) \gt c$

Locus boundary checkpoint

For Argand region questions, draw the boundaries first, then decide which side or sector is included. Treat each condition as a geometry instruction before shading.

Worked check: $|z-(2+i)| \leq 3$ uses centre $(2,1)$, not $(2,-1)$. If it is combined with

Sources

• SEAB H2 Mathematics syllabus (9758), examinations from 2026 - Topic 4 Complex numbers (Cartesian form, quadratic roots, modulus/argument, conjugate, operations, Argand representation, and selected geometrical effects). The syllabus does not list complex loci and explicitly excludes complex numbers expressed in polar/exponential form: SEAB H2 Mathematics 9758 syllabus PDF
• SEAB H2 Mathematics syllabus (9758), examinations from 2027 - Topic 4 keeps the same complex-number scope checked for this page: SEAB H2 Mathematics 9758 syllabus PDF