A-level Further Maths Year 12 · Topic 1
Complex numbers
Complex numbers let us solve every quadratic, even those with no real roots.
We will add, multiply and divide complex numbers and draw them on an Argand diagram.
Then we use modulus and argument, describe loci, and find the roots of polynomials.
- 4 subtopics
- 65 questions
- Free: 1.1 notes and 10 questions
In this topic
- 1.1Complex numbers: arithmetic and solving quadratics with complex rootsFree below · 19 questions
- 1.2The Argand diagram: modulus, argument and modulus–argument formOpen in Full Marks · 18 questions
- 1.3Loci and regions in the Argand diagramOpen in Full Marks · 17 questions
- 1.4Roots of polynomials with real coefficients: conjugate pairsOpen in Full Marks · 11 questions
1.1 · Free notes
Complex numbers: arithmetic and solving quadratics with complex roots
Key points
- The imaginary unit i satisfies i² = −1. A complex number z = x + iy has real part Re(z) = x and imaginary part Im(z) = y.
- Add and subtract by combining real and imaginary parts; multiply by expanding brackets and replacing i² by −1: (2 + 3i)(1 − i) = 2 − 2i + 3i − 3i² = 5 + i.
- The conjugate of z = x + iy is z* = x − iy; zz* = x² + y² is real. Divide by multiplying top and bottom by the conjugate of the denominator.
- Quadratics with negative discriminant have complex conjugate roots: x² + 4x + 13 = 0 → x = (−4 ± √(−36))/2 = −2 ± 3i.
- Two complex numbers are equal only if their real parts are equal and their imaginary parts are equal; this lets you solve equations such as (a + bi)² = 3 + 4i.
In more detail
- Worked example: find √(3 + 4i). Let (a + bi)² = 3 + 4i: a² − b² = 3 and 2ab = 4. Then a = 2, b = 1 (or a = −2, b = −1), so ±(2 + i).
- Powers of i repeat every four: i, −1, −i, 1, so i¹⁰ = i² = −1.
- Exam tip: always give answers in the form x + iy with real x and y, and simplify i².
Practice
Complex numbers: arithmetic and solving quadratics with complex roots: 10 questions
Pick an answer to check it, or open the answer for the explanation.
1. What is i³?
- i
- −i
- −1
- 1
Show answer
B: −i. i³ = i² × i = −i.
2. Simplify (3 + 2i) + (1 − 5i).
- 4 + 7i
- 2 + 7i
- 3 − 10i
- 4 − 3i
Show answer
D: 4 − 3i. Add real and imaginary parts.
3. Expand (2 + 3i)(1 − i).
- −1 + i
- 2 − 3i
- 5 + i
- 5 − i
Show answer
C: 5 + i. 2 − 2i + 3i + 3 = 5 + i.
4. What is the conjugate of −4 + 7i?
- −4 − 7i
- 4 − 7i
- 4 + 7i
- 7 − 4i
Show answer
A: −4 − 7i. Change the sign of the imaginary part.
5. Solve x² + 4x + 13 = 0.
- 2 ± 3i
- −2 ± 3i
- −4 ± 6i
- −2 ± 9i
Show answer
B: −2 ± 3i. x = (−4 ± 6i)/2.
6. Calculate (1 + i)/(1 − i).
- 1
- −i
- 1 + i
- i
Show answer
D: i. Multiply by (1 + i): (1 + 2i − 1)/2 = i.
7. What is zz* for z = 3 + 4i?
- 7
- 9 − 16i
- 25
- −7
Show answer
C: 25. 3² + 4² = 25.
8. What is i⁴?
- 1
- −1
- i
- 0
Show answer
A: 1. (i²)² = 1.
9. If a + bi = 5 − 2i (a, b real), what is b?
- 2
- −2
- 5
- −5
Show answer
B: −2. Compare imaginary parts.
10. Solve z² = −9.
- ±3
- 3 ± i
- −3 only
- ±3i
Show answer
D: ±3i. √(−9) = 3i.
Keep going in Full Marks
Full Marks has 19 questions on Complex numbers: arithmetic and solving quadratics with complex roots (9 more than here), set out in practice levels. You also get notes and 3D models for all 4 subtopics of Complex numbers, plus games, timed exams and past papers.
On your exam board
- AQATopic 1: Complex numbers
- OCR ATopic 2: Complex numbers
- OCR BTopic 2: Complex numbers, Topic 4: Roots of polynomials
- EdexcelTopic 1: Complex numbers
Not in the Cambridge specification.