ACT · Math · Preparing for Higher Math: Number and Quantity

Complex numbers

Complex numbers is part of Preparing for Higher Math: Number and Quantity in the ACT Math section. The questions below are written in the style of the real test, each with the answer and a worked explanation.

Question types you will see here include powers of i, add or subtract complex numbers, multiply complex numbers and divide using the conjugate.

Common traps: not reducing the exponent by multiples of 4; mixing up i cubed and i; treating i squared as 1.

Sample questions

Question 1
If x = 3 − 2i, where i² = −1, what is the value of 2x + 5?
- A.
  1 − 4i
- B.
  6 − 4i
- C.
  8 − 4i
- D.
  11 − 4i
Answer: D. 11 − 4i
Multiply: 2(3 − 2i) = 6 − 4i. Then add 5 to the real part: 11 − 4i. Choice A (1 − 4i) subtracts 5 instead of adding it; Choice B (6 − 4i) forgets to add 5; Choice C (8 − 4i) adds 5 to x before doubling only the imaginary part.
Question 2
A marathon-training planner solves x² − 4x + 13 = 0. Which gives all solutions?
- A.
  −2 ± 3i
- B.
  2 ± 3i
- C.
  2 ± 6i
- D.
  4 ± 6i
Answer: B. 2 ± 3i
x = (4 ± √(16 − 52)) ÷ 2 = (4 ± 6i) ÷ 2 = 2 ± 3i. A flips the sign of −b. C divides only the 4 by 2, leaving 6i. D leaves out the division by 2a entirely.
Question 3
For i² = −1, what is (4 − 3i) − (7 + 2i) + (−1 + 6i)?
- A.
  −4 + i
- B.
  −4 + 5i
- C.
  −3
- D.
  10 + 5i
Answer: A. −4 + i
Real parts: 4 − 7 − 1 = −4. Imaginary parts: −3 − 2 + 6 = 1. So the result is −4 + i, keeping the real and imaginary parts separate. Choice B (−4 + 5i) adds the 2i instead of subtracting it. Choice C (−3) combines the real and imaginary parts, −4 + 1. Choice D (10 + 5i) adds the second parentheses instead of subtracting it.
Question 4
Which of the following is equivalent to (1 + 3i)(1 - 6i), where i² = -1?
- A.
  19 - 17i
- B.
  19 - 3i
- C.
  1 - 18i
- D.
  -17 - 3i
Answer: B. 19 - 3i
Multiply using FOIL: (1)(1) + (1)(-6)i + (3)(1)i + (3)(-6)i² = 1 + (-3)i + (-18)(-1) = 19 - 3i.
Question 5
A sound system at a gym in Pune is modeled with the quantity i¹⁷ + i³⁸ − i⁵⁹. Which expression equals it?
- A.
  −1 − 2i
- B.
  1 + 2i
- C.
  3 + 0i
- D.
  −1 + 2i
Answer: D. −1 + 2i
17 leaves remainder 1 so i¹⁷ = i; 38 gives i² = −1; 59 gives i³ = −i. So i − 1 − (−i) = −1 + 2i. Choice A (−1 − 2i) gets i⁵⁹ = i and subtracts it, instead of i³ = −i. Choice B (1 + 2i) treats i³⁸ as +1. Choice C (3 + 0i) treats every power as 1.

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