Study on the flyACT

Study on the fly›ACT›Mathematics › Complex numbers

MathematicsFree

Complex numbers — ACT practice questions

On the ACT Mathematics test, Complex numbers covers products of a + bi numbers, powers of the imaginary unit i, and squares of complex binomials. A student must multiply complexes, reduce high powers of i from i squared equaling negative one, and expand a real-minus-imaginary square. Items often ask which statement cannot be true of a given product, which listed number equals a squared complex quantity, or the value of a difference of two powers of i. Traps include the wrong remainder when reducing an exponent and dropping a sign after replacing i squared. Choices are typically other a + bi numbers or competing claims about the factors.

  • Section: Mathematics 45 questions on the paper
  • 46 practice questions in the app

Sample questions

Question 1

Given that ii is the imaginary unit, which of the following numbers is equal to (4−3i)2(4-3i)^2?

  1. A25−24i25-24i
  2. B7+24i7+24i
  3. C7−24i7-24i
  4. D16−9i16-9i
Show answer

Answer: C — 7−24i7-24i

Expanding gives 16−24i+9i216-24i+9i^2. Since i2=−1i^2=-1, this becomes 16−24i−9=7−24i16-24i-9=7-24i.

Question 2

For complex numbers aa and bb, ab=−10ab=-\sqrt{10}. Which of the following statements cannot be true?

  1. ABoth aa and bb are real.
  2. BBoth aa and bb are pure imaginary.
  3. CExactly one of aa and bb is rational.
  4. DBoth aa and bb are rational.
Show answer

Answer: D — Both aa and bb are rational.

The product of two rational numbers must be rational, but −10-\sqrt{10} is irrational. Each other condition is possible with suitable factors.

Question 3

Given that i2=−1i^2=-1, what is the value of i17−i10i^{17}-i^{10}?

  1. A−1+i-1+i
  2. B1−i1-i
  3. C−1−i-1-i
  4. D1+i1+i
Show answer

Answer: D — 1+i1+i

Powers of ii repeat every 4 powers, so i17=ii^{17}=i and i10=−1i^{10}=-1. Therefore, i17−i10=i−(−1)=1+ii^{17}-i^{10}=i-(-1)=1+i.

Practice 46 Complex numbers questions in the app

More ACT topics

Mathematics

Everything on the ACT

Study on the fly is an independent study tool and is not affiliated with, sponsored by, or endorsed by any testing organization, school, or government body.

ACT® is a registered trademark of ACT Education Corp., which is not affiliated with, and does not endorse, this product.

All practice questions and passages are original works created for Study on the fly and are not reproduced from any official test.