Log inSign up
← Back to Learn by Concept

ACT Math · Learn by Concept

ACT Math Help: Complex Numbers

Review real example questions for Complex Numbers in ACT Math.

Question 1 / 10

0 of 10 answered

For the complex number ii, where i2=−1i^2 = -1, what is the simplified form of 3+i1−i\dfrac{3 + i}{1 - i}?

All questions

Question 1

For the complex number ii, where i2=−1i^2 = -1, what is the simplified form of 3+i1−i\dfrac{3 + i}{1 - i}?

  1. 1+2i1 + 2i (correct answer)
  2. 2+i2 + i
  3. 1−2i1 - 2i
  4. 2−i2 - i

Explanation: The correct answer is A (1 + 2i). Multiply by the conjugate of the denominator: (3 + i)/(1 − i) × (1 + i)/(1 + i). Numerator: (3 + i)(1 + i) = 3 + 3i + i + i² = 3 + 4i − 1 = 2 + 4i. Denominator: (1 − i)(1 + i) = 1 − i² = 1 + 1 = 2. Result: (2 + 4i)/2 = 1 + 2i. B (2 + i) likely comes from incomplete multiplication or forgetting to divide. C (1 − 2i) comes from a sign error in the numerator expansion. D (2 − i) comes from multiplying by the wrong conjugate (1 − i) instead of (1 + i). The key technique: always multiply by the conjugate to eliminate i from the denominator, remembering that i² = −1.

Question 2

A complex impedance is given by −6+7i-6+7i. What is the complex conjugate of −6+7i-6+7i (flip the sign of the imaginary part only)?

  1. 6+7i6+7i
  2. −6−7i-6-7i (correct answer)
  3. 6−7i6-7i
  4. −6+7i-6+7i

Explanation: This problem asks for the complex conjugate of −6+7i-6 + 7i, which is found by changing the sign of the imaginary part only. The real part is −6-6, and the imaginary part 7i7i becomes −7i-7i. Thus, the conjugate is −6−7i-6 - 7i. Choice D might result from incorrectly flipping the sign of the real part instead of the imaginary part.

Question 3

Given i=−1i = \sqrt{-1}, what is the simplified form of (4+5i)−(1−2i)(4 + 5i) - (1 - 2i)?

  1. 3+3i3 + 3i
  2. 3+7i3 + 7i (correct answer)
  3. 5+3i5 + 3i
  4. 5+7i5 + 7i

Explanation: This is a complex numbers question testing subtraction with distribution. Choice B (3 + 7i) is correct — distribute the negative: (4 + 5i) − (1 − 2i) = 4 + 5i − 1 + 2i. Key step: −(−2i) = +2i. Combine real parts: 4 − 1 = 3. Combine imaginary parts: 5i + 2i = 7i. Result: 3 + 7i. Choice A (3 + 3i) correctly subtracts the real parts but fails to distribute the negative on the imaginary term: 5i − 2i = 3i instead of 5i + 2i = 7i. Choice C (5 + 3i) adds the real parts instead of subtracting: 4 + 1 = 5, and also gets the imaginary term wrong. Choice D (5 + 7i) adds real parts (correctly gets +7i from the imaginary) — two separate errors that partially cancel. Pro tip: When subtracting a complex number, rewrite it as addition of the negative first: (4 + 5i) + (−1 + 2i). This prevents sign errors by making every operation an addition. The most common mistake is treating −(−2i) as −2i instead of +2i.

Question 4

For the complex number ii, where i2=−1i^2 = -1, what is the value of (3+2i)−(5−4i)(3 + 2i) - (5 - 4i)?

  1. −2−2i-2 - 2i
  2. −2+6i-2 + 6i (correct answer)
  3. 8−2i8 - 2i
  4. −2−6i-2 - 6i

Explanation: This is a complex numbers question testing subtraction with distribution. Choice B (−2 + 6i) is correct — distribute the negative sign: (3 + 2i) − (5 − 4i) = 3 + 2i − 5 + 4i. The critical step: −(−4i) = +4i. Combine real parts: 3 − 5 = −2. Combine imaginary parts: 2i + 4i = 6i. Result: −2 + 6i. Choice A (−2 − 2i) gets the real part right but subtracts the imaginary parts without distributing the negative: treating it as 2i − 4i = −2i instead of 2i + 4i = 6i. Choice C (8 − 2i) adds the real parts instead of subtracting: 3 + 5 = 8, and also handles the imaginary term incorrectly. Choice D (−2 − 6i) gets the real part right but applies the sign error in the opposite direction — treating −(−4i) as −6i. Pro tip: When subtracting a complex number, rewrite the subtraction as adding the negative first: (3 + 2i) − (5 − 4i) becomes (3 + 2i) + (−5 + 4i). Then combine real and imaginary parts separately.

Question 5

What is (7+3i)−(2−5i)(7 + 3i) - (2 - 5i)?

  1. 5 + 8i (correct answer)
  2. 5 - 2i
  3. 9 + 8i
  4. 9 - 2i

Explanation: This problem requires subtracting two complex numbers. To subtract complex numbers, we subtract the real parts and subtract the imaginary parts. For (7 + 3i) - (2 - 5i), we get (7 - 2) + (3i - (-5i)) = 5 + (3i + 5i) = 5 + 8i. When subtracting a negative imaginary term, it becomes addition.

Question 6

You are asked to subtract two complex quantities and express the result in standard form. What is (8+4i)−(3−9i)(8 + 4i) - (3 - 9i)?

  1. 11−5i11 - 5i
  2. 5−5i5 - 5i
  3. 5+13i5 + 13i (correct answer)
  4. −5+13i-5 + 13i

Explanation: The operation is subtraction of two complex numbers: (8 + 4i) minus (3 - 9i). Distribute the negative: 8 + 4i - 3 + 9i. Combine real parts: 8 - 3 = 5. Combine imaginary parts: 4i + 9i = 13i. The result in standard form is 5 + 13i. Choice B might result from subtracting imaginary parts incorrectly.

Question 7

What is (4+2i)−(1−3i)(4 + 2i) - (1 - 3i)?

  1. 5 - i
  2. 5 + 5i
  3. 3 - i
  4. 3 + 5i (correct answer)

Explanation: This is subtraction of complex numbers, where we subtract corresponding parts. (4+2i)−(1−3i)=(4−1)+(2i−(−3i))=3+(2i+3i)=3+5i(4 + 2i) - (1 - 3i) = (4 - 1) + (2i - (-3i)) = 3 + (2i + 3i) = 3 + 5i. We subtract the real parts and subtract the imaginary parts separately.

Question 8

For a complex number −9+12i-9+12i, compute its magnitude to determine its distance from the origin. What is the absolute value of (−9+12i)(-9+12i)? Use ∣a+bi∣=a2+b2|a+bi|=\sqrt{a^2+b^2} and simplify.

  1. 33
  2. 63\sqrt{63}
  3. 2121
  4. 1515 (correct answer)

Explanation: The magnitude formula gives ∣a+bi∣=a2+b2|a + bi| = \sqrt{a^2 + b^2}. For (−9+12i)(-9 + 12i), we have a=−9a = -9 and b=12b = 12, so the magnitude is (−9)2+122=81+144=225=15\sqrt{(-9)^2 + 12^2} = \sqrt{81 + 144} = \sqrt{225} = 15. The absolute value represents the distance from the origin to the point (−9,12)(-9, 12) in the complex plane.

Question 9

What is (1+2i)(3+4i)(1 + 2i)(3 + 4i)?

  1. 11 - 2i
  2. 5 + 10i
  3. -5 - 10i
  4. -5 + 10i (correct answer)

Explanation: This is multiplication of complex numbers using FOIL. (1+2i)(3+4i)=1⋅3+1⋅4i+2i⋅3+2i⋅4i=3+4i+6i+8i2(1 + 2i)(3 + 4i) = 1 \cdot 3 + 1 \cdot 4i + 2i \cdot 3 + 2i \cdot 4i = 3 + 4i + 6i + 8i^2. Since i2=−1i^2 = -1, this becomes 3+10i+8(−1)=3+10i−8=−5+10i3 + 10i + 8(-1) = 3 + 10i - 8 = -5 + 10i.

Question 10

What is the complex conjugate of −3+7i-3 + 7i?

  1. 3 - 7i
  2. -3 - 7i (correct answer)
  3. -3 + 7i
  4. 3 + 7i

Explanation: The complex conjugate of a complex number changes the sign of the imaginary part. For a complex number a + bi, the conjugate is a - bi. The complex conjugate of -3 + 7i is -3 - 7i. We keep the real part the same and change the sign of the imaginary part from positive to negative.