A-level Further Maths Year 13 · Topic 1
Complex numbers and series
At A level complex numbers become exponentials.
We will use the form r e to the i theta and de Moivre’s theorem to prove trigonometric identities.
Then we find the nth roots of unity and sum series using complex numbers.
- 4 subtopics
- 87 questions
- Free: 1.1 notes and 10 questions
In this topic
- 1.1Exponential form of complex numbers: e to the i thetaFree below · 17 questions
- 1.2De Moivre's theorem and multiple-angle identitiesOpen in Full Marks · 18 questions
- 1.3Roots of unity and nth roots of complex numbersOpen in Full Marks · 18 questions
- 1.4Summing series with complex numbers: the C + iS methodOpen in Full Marks · 34 questions
1.1 · Free notes
Exponential form of complex numbers: e to the i theta
Key points
- Euler's relation: e^(iθ) = cos θ + i sin θ, so any complex number can be written z = re^(iθ), where r = |z| and θ = arg z.
- Multiplying: r₁e^(iθ₁) × r₂e^(iθ₂) = r₁r₂e^(i(θ₁ + θ₂)); dividing: (r₁/r₂)e^(i(θ₁ − θ₂)).
- Special values: e^(iπ) = −1 (so e^(iπ) + 1 = 0), e^(iπ/2) = i, and e^(i2π) = 1.
- From Euler: cos θ = (e^(iθ) + e^(−iθ))/2 and sin θ = (e^(iθ) − e^(−iθ))/(2i).
In more detail
- Worked example: (1 + i)⁸ = (√2 e^(iπ/4))⁸ = 16e^(2πi) = 16.
- Beautiful identity: e^(iπ) + 1 = 0 links five fundamental constants: e, i, π, 1 and 0.
- Exam tip: give the argument in the range −π < θ ≤ π unless told otherwise.
Practice
Exponential form of complex numbers: e to the i theta: 10 questions
Pick an answer to check it, or open the answer for the explanation.
1. Write e^(iπ/2) in the form x + iy.
- 1
- −1
- −i
- i
Show answer
D: i. cos(π/2) + i sin(π/2).
2. What is e^(iπ)?
- 1
- −1
- i
- 0
Show answer
B: −1. cos π + i sin π.
3. Write 2(cos π/3 + i sin π/3) in exponential form.
- 2e^(iπ/3)
- e^(2iπ/3)
- 2e^(π/3)
- 3e^(iπ/2)
Show answer
A: 2e^(iπ/3). z = re^(iθ).
4. Calculate 3e^(iπ/4) × 2e^(iπ/2).
- 5e^(3iπ/4)
- 6e^(iπ/8)
- 6e^(3iπ/4)
- 6e^(iπ/4)
Show answer
C: 6e^(3iπ/4). Multiply moduli, add arguments.
5. What is |e^(iθ)| for real θ?
- θ
- e
- Depends on θ
- 1
Show answer
D: 1. cos²θ + sin²θ = 1.
6. Express cos θ using exponentials.
- (e^(iθ) − e^(−iθ))/2
- (e^(iθ) + e^(−iθ))/2
- (e^(iθ) − e^(−iθ))/(2i)
- e^(iθ)
Show answer
B: (e^(iθ) + e^(−iθ))/2. Add Euler for θ and −θ.
7. Write −2 in exponential form.
- 2e^(iπ)
- −2e^(i0)
- 2e^(iπ/2)
- 2e^(−iπ/2)
Show answer
A: 2e^(iπ). Modulus 2, argument π.
8. Calculate (4e^(iπ/2))/(2e^(iπ/6)).
- 2e^(2iπ/3)
- 8e^(iπ/3)
- 2e^(iπ/3)
- 2e^(iπ/12)
Show answer
C: 2e^(iπ/3). Divide moduli, subtract arguments.
9. What is the conjugate of re^(iθ)?
- −re^(iθ)
- re^(iθ)
- (1/r)e^(iθ)
- re^(−iθ)
Show answer
D: re^(−iθ). Reflect in the real axis.
10. Write 1 − i in exponential form.
- √2e^(iπ/4)
- √2e^(−iπ/4)
- 2e^(−iπ/4)
- √2e^(3iπ/4)
Show answer
B: √2e^(−iπ/4). Fourth quadrant.
Keep going in Full Marks
Full Marks has 17 questions on Exponential form of complex numbers: e to the i theta (7 more than here), set out in practice levels. You also get notes and 3D models for all 4 subtopics of Complex numbers and series, plus games, timed exams and past papers.