The complex number system — A/L Combined Mathematics

How do I practise The complex number system?

2 questions with worked solutions

Check before you answer: Equality Equating Technique: Equating real and imaginary parts separately from an equation like f(z) = 0 to form two independent real equations in x and y.

1. What is the principal value of the imaginary unit expression i⁴⁷, where i = √(-1)?
   1. A) -i
   2. B) -1
   3. C) +1
   4. D) +i
   Check my answer

   Correct answer: A) -i

   Since powers of i repeat in cycles of 4 (i¹ = i, i² = -1, i³ = -i, i⁴ = 1): 47 = 4 × 11 + 3. Therefore, i⁴⁷ = (i⁴)¹¹ · i³ = (1)¹¹ · (-i) = -i.

   Show worked solution

   Step 1: Divide 47 by 4: Quotient = 11, Remainder = 3. Step 2: i⁴⁷ = (i⁴)¹¹ · i³ = 1¹¹ · (-i) = -i. Answer: -i.
   Modelled on: the A/L Combined Mathematics examination pattern
2. Simplify i + i² + i³ + i⁴, where i = √(-1).
   1. A) 0
   2. B) i
   3. C) -1
   4. D) 1
   Check my answer

   Correct answer: A) 0

   Using i² = -1, the next two powers are i³ = -i and i⁴ = 1. The sum is therefore i + (-1) + (-i) + 1. The imaginary parts cancel (i - i = 0) and so do the real parts (-1 + 1 = 0), leaving 0.

   Show worked solution

   Step 1: i² = -1, i³ = i² · i = -i, i⁴ = (i²)² = 1. Step 2: Sum = i + (-1) + (-i) + 1. Step 3: Imaginary parts: i - i = 0; real parts: -1 + 1 = 0. Answer: 0.
   Modelled on: the A/L Combined Mathematics examination pattern

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Where does The complex number system fit in the syllabus?

Year 2 · Term 1