Argand Diagrams Cheat Sheet | H2 Maths Modulus & Argument

Q: What does H2 Maths Notes (JC 1-2): 4.1) Complex Numbers and Argand Diagrams cover?
A: Cartesian-form complex numbers, Argand diagrams, modulus, argument, conjugates, transformations, and the algebra you need for Topic 4.1.

Before you revise
Stick to Cartesian $x + iy$: for the 2026 and 2027 syllabus, complex loci, polar/exponential form, and De Moivre workflows are not part of the current H2 Maths 9758 examinable scope. Focus on modulus/argument, conjugates, quadratic roots, and Argand-diagram transformations.

• $x+iy$ is a point on a plane: Plot $(x,y)$.
• Modulus is distance; argument is angle: Translate algebra into geometry.
• Transformations move points predictably: Track reflections, translations, negation, and multiplication by $i$ on the diagram.

Concrete example: if $z=2-i$, then $iz=1+2i$. On the Argand diagram, multiplication by $i$ rotates the point $(2,-1)$

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

Core Definitions

• Complex number $z = x + iy$ with real part $x$ and imaginary part $y$.
• Modulus $\lvert z \rvert = \sqrt{x^2 + y^2}$

Argument quadrant checkpoint

When finding $\arg z$, calculate the reference angle, then place it in the correct quadrant.

Worked check: for $z=-1-\sqrt{3}i$, the point is in the third quadrant and the reference angle is $\pi/3$. The principal argument is $-\pi+\pi/3=-2\pi/3$

Misconception check: $\tan^{-1}(y/x)$ gives a ratio angle, not the whole answer. The Argand diagram decides the sign and quadrant.

Algebra of Complex Numbers

• Addition/subtraction: combine real and imaginary parts.
• Multiplication: $(a + ib)(c + id) = (ac - bd) + i(ad + bc)$

Example -- Division

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

Complex Roots of Quadratic Equations

• The quadratic formula works in $\mathbb{C}$; complex roots occur when the discriminant is negative.
• If a polynomial has real coefficients and $a + bi$ is a root, then $a - bi$ is also a root (conjugate pair).

Example -- Complex roots

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

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

Argand Diagram Transformations

• Conjugation reflects across the real axis: $x + iy \mapsto x - iy$.
• Negation rotates by $180^\circ$ about the origin: $z \mapsto -z$

Example -- Multiplication by $i$

If $z = 2 - i$, then $iz = i(2 - i) = 1 + 2i$. On the Argand diagram, $(2, -1)$

Calculator Workflow

• Use GC complex mode to check arithmetic; keep the calculator in radians for argument work.
• Use Abs / Arg (or equivalent) to verify modulus and principal argument while showing the algebraic setup.
• For transformations, plot one input point and one output point to check the direction of the movement.

Exam Watch Points

• Always state principal value of argument in the interval requested (usually $-\pi \lt \arg z \leq \pi$).
• When sketching an Argand diagram, label the real axis, imaginary axis, original point, and transformed point.
• Provide exact trigonometric forms-avoid rounding angles unless final answer requires degrees.
• For quadratic roots, give answers in $a \pm bi$ form and show the discriminant step before introducing $i$

Practice Quiz

Drill Argand plotting, modulus/argument, and transformation effects before tackling calculus.

Quick Revision Checklist

• Perform conjugation, modulus, and division confidently.
• Solve quadratic equations with complex roots and state conjugate pairs where relevant.
• Plot complex numbers and transformations with accurate annotations.
• Describe geometric effects of conjugation, negation, and multiplication by $i$.

Next steps: Keep the H2 Maths notes hub bookmarked and move on to Topic 4 - Complex numbers deep dive before tackling calculus in Topic 5.

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

Common exam mistakes

• Giving argument outside the principal range: The principal argument must satisfy $-\pi \lt \arg z \leq \pi$. Writing $5\pi/4$ instead of $-3\pi/4$ for a third-quadrant number is a standard error that loses the accuracy mark.
• Forgetting to state the conjugate root: For a real-coefficient polynomial, if $a + bi$

Frequently asked questions

Is Topic 4.1 in Paper 1 or Paper 2?
Topic 4.1 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 or exponential forms of complex numbers tested 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 $a + bi$.

Are complex loci still tested in H2 Maths?
No. Complex loci are useful for older TYS or enrichment practice, but they are not listed in the current SEAB H2 Mathematics 9758 syllabus for 2026 or 2027.

Appendix: Old Syllabus / Enrichment - Complex Loci

Complex loci are kept here for students working through older papers or extension problems. Do not treat this appendix as current 9758 examinable core content.

• $\lvert z - a \rvert = r$ represents a circle centre $a$ radius $r$.
• $\arg(z - a) = \theta$

Locus translation checkpoint

Before expanding algebra, identify the geometry hidden inside the condition.

Example - perpendicular bisector

Locus of points equidistant from $2 + i$ and $-1 + 3i$:

$$\lvert z - (2 + i) \rvert = \lvert z - (-1 + 3i) \rvert.$$

Let $z = x + iy$. Squaring both sides yields $(x - 2)^2 + (y - 1)^2 = (x + 1)^2 + (y - 3)^2$

Inequality regions

For inequality loci, draw the equality boundary first, then decide which side or sector is included. Treat the final answer as a region with boundary information, not just an equation.

Worked check: $\lvert z-(1+i)\rvert \le 2$ and $0 \le \arg(z-(1+i)) \le \pi/2$

Sources

• SEAB H2 Mathematics syllabus (9758), examinations from 2026 - Topic 4.1 Complex numbers expressed in cartesian form and Argand diagrams (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 4.1 keeps the same complex-number scope checked for this page: SEAB H2 Mathematics 9758 syllabus PDF