Worksheets · Precalculus

Complex numbers worksheet for Precalculus

Precalculus treats complex numbers as points on the plane and as the roots polynomials have when the graph never crosses the x-axis. Students add, multiply and divide them, plot them, match real and imaginary parts, and solve quadratics whose solutions are not real.

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.

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

Sigma Prep · sigmaprep.io/worksheets

Name

Operations with Complex Numbers

Date Period

Answer each question. Write complex answers in the form a + bi.

  1. Add and subtract complex numbers

    (16−26i)+(31−18i)
  2. (6−3i)−(33+38i)
  3. (17+40i)+(−28+8i)

  1. Multiply (a+bi)(c+di)(a + bi)(c + di)

    (−27−15i)(20−23i)
  2. (2+24i)(6+7i)
  3. (−20+23i)(−25+14i)

  1. Divide complex numbers

    11+33i−11−11i
  2. −12+2i−5+7i
  3. 38−18i4−i

  1. Solve with the quadratic formula

    3y2+3y+30=0
  2. 2k2−5k+4=0
  3. 4b2+17=0

  1. Find xx and yy

    Find x and y: 2x+(y−5)i=−10−6i
  2. Find x and y: 3x+(2y−6)i=24−22i
  3. Find x and y: 2x+(y+2)i=−16+i

  1. Plot on the complex plane

    Plot −8+7i in the complex plane.−10−10−5−5551010ReIm
  2. Plot −6−4i in the complex plane.−10−10−5−5551010ReIm
  3. Plot 7+6i in the complex plane.−10−10−5−5551010ReIm

  1. Multiply by imaginary numbers, square, or multiply conjugates

    (−1−6i)2
  2. i⋅−2i⋅(1+3i)

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

Sigma Prep · sigmaprep.io/worksheets

Name

Operations with Complex NumbersAnswer keyVersion 2

Date Period

Answer each question. Write complex answers in the form a + bi.

  1. Add and subtract complex numbers

    (16−26i)+(31−18i)47−44i
  2. (6−3i)−(33+38i)−27−41i
  3. (17+40i)+(−28+8i)−11+48i

  1. Multiply (a+bi)(c+di)(a + bi)(c + di)

    (−27−15i)(20−23i)−885+321i
  2. (2+24i)(6+7i)−156+158i
  3. (−20+23i)(−25+14i)178−855i

  1. Divide complex numbers

    11+33i−11−11i−2−i
  2. −12+2i−5+7i1+i
  3. 38−18i4−i10−2i

  1. Solve with the quadratic formula

    3y2+3y+30=0{−12+i392,−12−i392}
  2. 2k2−5k+4=0{54+i74,54−i74}
  3. 4b2+17=0{i172,−i172}

  1. Find xx and yy

    Find x and y: 2x+(y−5)i=−10−6ix=−5, y=−1
  2. Find x and y: 3x+(2y−6)i=24−22ix=8, y=−8
  3. Find x and y: 2x+(y+2)i=−16+ix=−8, y=−1

  1. Plot on the complex plane

    Plot −8+7i in the complex plane.−10−10−5−5551010ReImThe point (−8,7): 8 left, 7 up
  2. Plot −6−4i in the complex plane.−10−10−5−5551010ReImThe point (−6,−4): 6 left, 4 down
  3. Plot 7+6i in the complex plane.−10−10−5−5551010ReImThe point (7,6): 7 right, 6 up

  1. Multiply by imaginary numbers, square, or multiply conjugates

    (−1−6i)2−35+12i
  2. i⋅−2i⋅(1+3i)2+6i

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

How to do these

7+i1−i

  1. Multiply the top and bottom by the conjugate 1 + i.
  2. Top: (7 + i)(1 + i) = 7 + 7i + i + i² = 6 + 8i. Bottom: (1 - i)(1 + i) = 1 - i² = 2.
  3. Divide both parts by 2.

The answer is 3+4i.

Where students go wrong

Forgetting that i² is -1. In (-1 - 6i)² the last term is 36i², which is -36, so the answer is -35 + 12i, not 37 + 12i.

Also called imaginary numbers, operations with i or arithmetic with complex numbers.

What you can put on this worksheet

Adding and subtracting
(4+i)−(−4−7i) → 8+8i
Multiplying, where i² becomes -1
(3+i)(−3−5i) → −4−18i
Dividing, using the conjugate
−5−15i3+i → −3−4i
Read a number off the complex plane
Point P stands for a complex number. Write it in a+bi form. → −1−3i
Square roots of negative numbers
−384 → 8i6
By square roots
v2+49=0 → {7i,−7i}
By the quadratic formula
v2+v+22=0 → {−12+i872,−12−i872}
By completing the square
Solve by completing the square: v2−8v+17=0 → {4+i,4−i}
Equal complex numbers: find x and y
Find x and y: x+(y+1)i=1+2i → x=1, y=1
Plot on the complex plane
Plot 1+6i in the complex plane. → The point (1,6): 1 right, 6 up
A monomial, a square, or three factors
(5−3i)2 → 16−30i
Strings of terms, mixed operations, roots with letters
Simplify: 6i⋅(−4i)+2(4+6i) → 32+12i
Solve for z
Solve for z: (−2+i)z+(6−2i)=2−10i → z=4i

Questions about these worksheets

Yes. Every 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 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 13: adding and subtracting, multiplying, where i² becomes -1, dividing, using the conjugate, read a number off the complex plane, square roots of negative numbers, by square roots, by the quadratic formula, by completing the square, equal complex numbers: find x and y, plot on the complex plane, a monomial, a square, or three factors, strings of terms, mixed operations, roots with letters and solve for z. Tick as many as you want and set how many of each, or let it spread them evenly.

Leaving i² in the answer. It is -1, always, and forgetting that costs a sign on the real part. The other one is dividing without the conjugate: an i in the denominator means the answer is not finished.