MTH1020 · Analysis of Change
Complex Numbers: The Argand Plane & Arithmetic
Week 5 of Monash MTH1020 Analysis of Change introduces the complex numbers ℂ: the imaginary unit i with i² = −1, the number-system hierarchy ℕ ⊂ ℤ ⊂ ℚ ⊂ ℝ ⊂ ℂ, and the Argand plane on which z = x + yi is the point (x, y). It develops complex conjugates, the four arithmetic operations — with division realised by multiplying through by the conjugate — and the conjugate root theorem for real-coefficient polynomials. Note that for the S1-2026 cohort complex numbers were tested in the mid-semester tests but excluded from the final; confirm the scope for your cohort on Moodle.
What this chapter covers
- 01Number systems ℕ ⊂ ℤ ⊂ ℚ ⊂ ℝ ⊂ ℂ; motivation x² = −1 ⇒ invent i with i² = −1
- 02Complex number z = x + yi; real part Re(z) = x, imaginary part Im(z) = y (a real number)
- 03The Argand (complex) plane: z ↔ (x, y); real axis horizontal, imaginary axis vertical
- 04Complex conjugate z̄ = x − yi (reflection in the real axis); z + z̄ = 2Re(z); z·z̄ = x² + y² = |z|²
- 05Addition/subtraction (add real and imaginary parts); multiplication using i² = −1; powers of i cycle mod 4
- 06Division by realising the denominator: z₁/z₂ = z₁z̄₂/|z₂|²
- 07Equating complex numbers: a + bi = c + di ⇔ a = c and b = d (used to solve equations)
- 08Conjugate root theorem: real-coefficient polynomials have complex roots in conjugate pairs
Complex division into x + yi form, with conjugate and modulus
- +1Realise the denominator. Multiply numerator and denominator by the conjugate of the denominator, 1 + i: (3 + 2i)/(1 − i) × (1 + i)/(1 + i).
- +1Expand the numerator using i² = −1. (3 + 2i)(1 + i) = 3 + 3i + 2i + 2i² = 3 + 5i − 2 = 1 + 5i.
- +1Expand the denominator. (1 − i)(1 + i) = 1 − i² = 1 − (−1) = 2. So (3 + 2i)/(1 − i) = (1 + 5i)/2 = 1/2 + (5/2)i.
- +1Conjugate and modulus. The conjugate of z = 1/2 + (5/2)i is z̄ = 1/2 − (5/2)i. The modulus is |z| = √((1/2)² + (5/2)²) = √(1/4 + 25/4) = √(26/4) = √26/2 ≈ 2.55.
Key terms
- Imaginary unit (i)
- The number with i² = −1, invented so that equations like x² + 1 = 0 have solutions. Its powers cycle with period 4: i⁰ = 1, i¹ = i, i² = −1, i³ = −i.
- Complex number (z = x + yi)
- A number with real part Re(z) = x and imaginary part Im(z) = y, where x, y are real. If Im(z) = 0 it is real; if Re(z) = 0 it is pure imaginary.
- Argand plane
- The plane in which z = x + yi is plotted as the point (x, y): the horizontal axis is the real axis, the vertical axis the imaginary axis.
- Complex conjugate (z̄)
- The reflection of z = x + yi in the real axis: z̄ = x − yi. It satisfies z + z̄ = 2Re(z) and z·z̄ = x² + y² = |z|².
- Realising the denominator
- Dividing complex numbers by multiplying numerator and denominator by the conjugate of the denominator, so z₁/z₂ = z₁z̄₂/|z₂|² has a real denominator.
- Conjugate root theorem
- If a polynomial has real coefficients and z is a root, then z̄ is also a root — complex roots of real polynomials come in conjugate pairs.
Complex Numbers: The Argand Plane & Arithmetic FAQ
How do you divide two complex numbers?
Multiply the numerator and denominator by the conjugate of the denominator. That turns the denominator into z₂·z̄₂ = |z₂|², a positive real number, so the quotient becomes z₁z̄₂/|z₂|², which you can then split into real and imaginary parts. For example, (3 + 2i)/(1 − i) becomes (3 + 2i)(1 + i)/((1 − i)(1 + i)) = (1 + 5i)/2 = 1/2 + (5/2)i. The whole trick is 'realising' the denominator.
Is the imaginary part of z a real number or does it include the i?
The imaginary part Im(z) is the real coefficient of i, not the term with i attached. For z = 1/2 + (5/2)i, Re(z) = 1/2 and Im(z) = 5/2 — both real numbers. Writing Im(z) = (5/2)i is a common MTH1020 slip that costs marks, because it confuses the component with the full imaginary term.
What does the conjugate root theorem let me do?
If a polynomial has real coefficients and you know one complex root, the theorem hands you a second root for free — its conjugate. So a root of 1 + 2i forces 1 − 2i to be a root too, and their product gives the real quadratic factor (x − (1+2i))(x − (1−2i)) = x² − 2x + 5. That is how you factor real polynomials with complex roots into real quadratic factors.
Are complex numbers on the MTH1020 exam?
For the S1-2026 cohort a lecturer said complex numbers were not on the final exam but were still tested in the mid-semester tests. This is cohort-specific and not in the Monash Handbook, so treat it as a planning note and confirm the scope for your cohort on Moodle. Learn the material regardless — it is examined somewhere in the unit and Sia can walk you through conjugates, division and the Argand plane step by step, checking your reasoning without doing graded work for you.
Exam move
Anchor everything on i² = −1 and the conjugate. Drill the four operations until division is automatic — the reflex is 'multiply top and bottom by the conjugate of the denominator' — and always report answers in clean x + yi form. Keep a card distinguishing Re(z), Im(z) (both real) and the conjugate z̄, and practise the powers-of-i cycle for quick simplifications. For polynomials, rehearse using the conjugate root theorem to pair complex roots and build real quadratic factors. Because complex numbers were tested in the mid-semester tests for the recent cohort, make sure this is solid before the tests even if it may be off the final — confirm the exam scope for your cohort on Moodle. Ask Sia to generate fresh division and conjugate-root problems and check each step.
Working through Complex Numbers: The Argand Plane & Arithmetic in MTH1020? Sia is AskSia’s AI Mathematics tutor — ask any MTH1020 Complex Numbers: The Argand Plane & Arithmetic question and get a clear, step-by-step explanation grounded in how MTH1020 is taught and assessed. Read this chapter free, then take your hardest questions to Sia.
