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)
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Marcus Pang·Managing Director (Maths)
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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) to (1,2).
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 ∣z∣=x2+y2; argument argz is the angle from positive real axis.
Conjugate zˉ=x−iy; identity zzˉ=∣z∣2
Argument quadrant checkpoint
When finding argz, calculate the reference angle, then place it in the correct quadrant.
Position of z=x+iy
Reference angle
Principal argument to write
Common trap
x>0,y>0
tan−1(y/x)
Positive acute angle.
None if both coordinates are positive.
x<0,y>0
tan−1(∣y/x∣)
π−
x<0,y<0
tan−1(∣y/x∣)
−π+
x>0,y<0
tan−1(∣y/x∣)
Negative acute angle.
Losing the minus sign because the reference angle is positive.
Point on an axis
No triangle needed.
0, π, π/2, or −π/2.
Treating x=0
Worked check: for z=−1−3i, the point is in the third quadrant and the reference angle is π/3. The principal argument is −π+π/3=−2π/3, not 4π/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).
Division: multiply numerator and denominator by complex conjugate.
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 −π<argz≤π. Writing 5π/4 instead of −3π/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 is a root then a−bi must also be explicitly stated. Simply solving for one root and stopping loses the conjugate pair mark.
Writing the argument of a negative real number as 0: The argument of a negative real number −k (where k>0) is π, not 0. The argument of a positive real number is 0. Confusing these is a frequent error when z lies on the negative real axis.
Reversing the multiplication-by-i rotation: Multiplication by i rotates a point by 90∘ anticlockwise about the origin. Check this with 1↦i before applying it to a harder point.
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.
∣z−a∣=r represents a circle centre a radius r.
arg(z−a)=θ is a ray from a making angle θ.
Linear loci: express z=x+iy, substitute into the condition, separate real/imaginary parts, and simplify.
Locus translation checkpoint
Before expanding algebra, identify the geometry hidden inside the condition.
Condition type
Geometry to draw first
Algebra route
Misconception check
∣z−a∣=r
Circle centred at point a.
Let a=p+iq, then write (x−p)2+(y−q)2=r2.
The centre is a, not −a.
∣z−a∣=∣z−b∣
Perpendicular bisector of the segment from a to b.
Square both sides, expand, and simplify to a line.
Do not try to subtract the two moduli directly.
arg(z−a)=θ
Ray starting from point a.
Draw the ray at angle θ from the positive real direction, then state endpoint exclusion if needed.
The angle is measured from a
Inequality such as ∣z−a∣≤r
Region inside or outside the boundary.
Draw the equality boundary first, then test which side is included.
Boundary is solid for ≤ or ≥, dashed for strict inequalities.
Example - perpendicular bisector
Locus of points equidistant from 2+i and −1+3i:
∣z−(2+i)∣=∣z−(−1+3i)∣.
Let z=x+iy. Squaring both sides yields (x−2)2+(y−1)2=(x+1)2+(y−3)2, which simplifies to 6x−4y+5=0.
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.
Inequality cue
Boundary to draw first
How to choose the shaded region
Common trap
∣z−a∣<r
Circle centred at a with radius r, drawn dashed.
Shade inside the circle.
Drawing a solid boundary even though points exactly on the circle are excluded.
∣z−a∣≥r
Circle centred at a with radius r, drawn solid.
Shade outside the circle, including the boundary.
Shading the disk because the circle equation looked familiar.
arg(z−a)≤θ with a lower ray also given
Rays starting from a.
Shade the angle sector between the rays, respecting strict or inclusive boundary signs.
Measuring the rays from the origin instead of from a.
Two conditions together
Draw both boundaries before shading.
Keep only the overlap that satisfies both conditions.
Shading each condition separately and forgetting to take the intersection.
Worked check: ∣z−(1+i)∣≤2 and 0≤arg(z−(1+i))≤π/2 give the quarter-disk of radius 2 centred at (1,1) above and to the right of the centre. The two straight boundaries start at (1,1), not at the origin.
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
