Worksheets · Precalculus

Properties of complex numbers worksheet for Precalculus

Precalculus puts complex numbers on the complex plane: a + bi is the point (a, b), and its modulus is the distance to the origin, the first step toward polar form. Division questions clear an i from the denominator, including division by a pure imaginary like -6i.

Also taught in Algebra 2, at its own level and with its own examples.

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Pick more than one for a sheet that mixes them.

The preview can look cramped on a phone. The PDF and printed sheet come out normal.

Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Properties of Complex Numbers

Date Period

Simplify, or plot as asked.

  1. Simplify powers of ii

    i76
  2. i83
  3. i45
  4. i94
  5. i38

  1. Find the modulus or conjugate

    ∣6−5i∣
  2. Find the conjugate of −3+4i
  3. ∣8+2i∣
  4. ∣−3−8i∣
  5. Find the conjugate of −8−15i

  1. Divide by an imaginary number

    110−10i
  2. −45+45i−5i
  3. −42−21i7i
  4. −20+24i2i
  5. 84+84i−7i

  1. Plot on the complex plane

    Plot 1−6i in the complex plane.−10−10−5−5551010ReIm
  2. Plot −7+i in the complex plane.−10−10−5−5551010ReIm
  3. Plot 3i in the complex plane.−10−10−5−5551010ReIm
  4. Plot −4−6i in the complex plane.−10−10−5−5551010ReIm
  5. Plot 7+4i in the complex plane.−10−10−5−5551010ReIm

Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Properties of Complex NumbersAnswer keyVersion 2

Date Period

Simplify, or plot as asked.

  1. Simplify powers of ii

    i761
  2. i83−i
  3. i45i
  4. i94−1
  5. i38−1

  1. Find the modulus or conjugate

    ∣6−5i∣61
  2. Find the conjugate of −3+4i−3−4i
  3. ∣8+2i∣217
  4. ∣−3−8i∣73
  5. Find the conjugate of −8−15i−8+15i

  1. Divide by an imaginary number

    110−10i11i
  2. −45+45i−5i−9−9i
  3. −42−21i7i−3+6i
  4. −20+24i2i12+10i
  5. 84+84i−7i−12+12i

  1. Plot on the complex plane

    Plot 1−6i in the complex plane.−10−10−5−5551010ReImThe point (1,−6): 1 right, 6 down
  2. Plot −7+i in the complex plane.−10−10−5−5551010ReImThe point (−7,1): 7 left, 1 up
  3. Plot 3i in the complex plane.−10−10−5−5551010ReImThe point (0,3): 3 up
  4. Plot −4−6i in the complex plane.−10−10−5−5551010ReImThe point (−4,−6): 4 left, 6 down
  5. Plot 7+4i in the complex plane.−10−10−5−5551010ReImThe point (7,4): 7 right, 4 up

Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

How to do these

Find the modulus of 5−12i

  1. The modulus is √(a² + b²) with a = 5 and b = -12.
  2. √(25 + 144) = √169.

The answer is 13.

Where students go wrong

Putting the i into the square. b is -12, not -12i, and squaring -12i gives -144, which makes the modulus √(25 - 144) and is wrong.

Also called absolute value of a complex number, complex conjugate or rationalizing imaginary denominators.

What you can put on this worksheet

Powers of i
i19 → −i
Modulus and conjugate
∣8+15i∣ → 17
Clear an i from the denominator
33i → −i
Plot on the complex plane
Plot 1+6i in the complex plane. → The point (1,6): 1 right, 6 up
Read a point, divide by bi, letters
Simplify:
5+7i2i → 72−52i

Questions about these worksheets

Yes. Every properties of complex numbers sheet is free to print and free to download as a PDF. There is no account to make, no email to hand over and no limit on how many you take.

Yes. The answer key prints on a second page, and you can choose whether it shows each question beside its answer or just the answers in a list.

Yes. Print straight from the page, or download the PDF and print that. The sheet is laid out for paper rather than squeezed off a screen, so there is room to work under each question, and the answer key comes out on its own page.

Precalculus is usually taken around 11th or 12th grade, though schools vary and plenty of students meet it earlier or later. Pick the difficulty rather than the grade: the easy level suits a first lesson on it, the hard level suits review before a test.

Yes, as many as you like. The questions are built fresh each time rather than picked from a fixed set of files, so pressing Generate gives a new sheet. That is what you want for a second class, a retake, or two students sitting next to each other.

That is the closest comparison, and Kuta Software is good software. It is also paid software you install on a computer. These worksheets run in a browser tab for free, with no account and nothing to install. Like Kuta, the properties of complex numbers questions are generated when you ask for them rather than pulled from a fixed set of files, so two classes never get the same sheet, and the answer key comes with it.

Yes. There are three levels, and you can tick more than one for a sheet that mixes them, in which case the questions come out easiest first. The harder levels are not just larger numbers: they ask for something the easy ones do not.

You pick from 5: powers of i, modulus and conjugate, clear an i from the denominator, plot on the complex plane and read a point, divide by bi, letters. Tick as many as you want and set how many of each, or let it spread them evenly.

Working out the whole power. There is no need: only the remainder on dividing by four matters, so i to the 100 is 1 and i to the 99 is minus i, with no arithmetic in between.