Complex Numbers Worksheet with Answers – Algebra 2 (Free PDF)

Problem 1 Sorting numbers into sets ★★★

Classify each number as natural, integer, rational, irrational, or non-real. List every set that applies.

1. 0.375
2. √45
3. −√64
4. 59
5. √(−25)

Problem 2 Simplifying negative roots ★★★

Write each as a multiple of i in simplest form.

1. √(−81)
2. √(−44)
3. −√(−72)

Problem 3 Reading a point on the complex plane ★★★

The figure shows point Z. Write Z in the form a + bi, name its real and imaginary parts, and give the conjugate a − bi.

Problem 4 Adding and subtracting ★★★

Perform each operation.

1. (6 + 2i) + (−9 + 7i)
2. (4 − 11i) − (−5 − 6i)

Problem 5 Identifying parts ★★★

State the real part and the imaginary part of each number.

1. 9 − 4i
2. −7i
3. 12
4. 12 + 34i

Problem 6 Plotting and naming points ★★★

Write the complex number for each labeled point in the figure.

Problem 7 Multiplying binomials ★★★

Multiply and write the product as a + bi: (5 + 3i)(4 − 6i).

Problem 8 Products with conjugates ★★★

Multiply.

1. (7i)(−3i)
2. (−2 + i)(−2 − i)
3. (5 + 6i)(5 − 6i)

Problem 9 Powers of an imaginary number ★★★

Simplify each power.

1. (3i)²
2. (2i)⁵
3. (−4i)³

Problem 10 Product of negative roots ★★★

Find each product. Rewrite the roots with i before multiplying.

1. √(−9) · √(−16)
2. √(−5) · √(−20)

Problem 11 Conjugate product ★★★

For each number, write its conjugate and find the product of the number and its conjugate.

1. 8 − 3i
2. 5i

Problem 12 Equal complex numbers ★★★

Find real numbers x and y so that (x + 3) + (2y − 1)i = 9 + 7i.

Problem 13 Adding on the complex plane ★★★

Let u = 4 + i and v = −2 + 3i. Find u + v and plot all three numbers on the grid.

Problem 14 Squaring a binomial ★★★

Show that (1 − i)² = −2i, then use this result to evaluate (1 − i)⁶.

Problem 15 Checking a solution ★★★

Decide whether 3 + 5i is a solution of x² − 6x + 34 = 0. Show the work.

Problem 16 Sum of four consecutive powers ★★★

Evaluate i⁵⁷ + i⁵⁸ + i⁵⁹ + i⁶⁰.

Problem 17 Series impedance ★★★

In a series circuit, a coil has impedance 12 + 5i ohms and a capacitor branch has 7 − 9i ohms. Find the total impedance.

Problem 18 Voltage in an AC circuit ★★★

A technician finds that the current in a circuit is 5 + 2i amps and the impedance is 3 − 6i ohms. Voltage equals current times impedance. Find the voltage in the form a + bi.

Problem 19 Quarter turns of a marker ★★★

A robotics club tracks a marker at the point 3 + 2i on a screen. Multiplying a number by i turns its point 90° counterclockwise about the origin. Find the marker position after one, two, and three such multiplications, and describe the pattern.

Problem 20 A real product ★★★

Find the value of k for which (3 + ki)(2 − 5i) is a real number, and give that real product.