Complex Number Arithmetic Practice Problems

All four complex-number operations get a full explanation in the Complex Number Arithmetic lesson. This practice set leans harder on two of them — powers of and conjugate products — since those are the ideas that keep resurfacing once the basics are down.

How the difficulty levels work

Difficulty	What it drills	Example
Easy	Add or subtract two complex numbers	Real or imaginary part of
Medium	A power of , or multiplying two complex numbers
Hard	Dividing, a bigger power of , or a conjugate product

Easy adds or subtracts two complex numbers, asking for either the real or the imaginary part of the result.

Medium alternates between simplifying a power of (always landing on or ) and multiplying two complex numbers with FOIL.

Hard rotates through three shapes: dividing two complex numbers using the conjugate, a much larger power of , and a genuine conjugate product, always fully real.

Worked examples

One example at each difficulty level, solved the way the practice set expects.

Easy. Simplify .

Subtract real parts and imaginary parts separately: .

Medium. Simplify and .

Powers of repeat every 4, and , so . FOIL: .

Hard. Simplify and .

A conjugate product is always real: . To divide, multiply top and bottom by : .

Using the practice problems

Set the difficulty and the question count, and a fresh set is built right away. Whole-number answers go in as-is; division problems accept a fraction or its decimal equivalent.

Check Answers scores every problem instantly and points out which ones need a second look
Show Answers reveals each solution at once, useful when grading a printed attempt
Print Worksheet produces a tidy printable version, good for homework or classroom use

The rules these practice problems drill

Rule	Statement
Add or subtract	Combine real parts and imaginary parts separately
Multiply	FOIL, then replace with
Divide	Multiply top and bottom by the denominator’s conjugate
Powers of	Cycle every four:

See the Complex Number Arithmetic lesson for the full derivation of each rule and worked examples for every operation.

Common mistakes to watch for

Forgetting to replace with after multiplying. A term like is not a finished answer; it has to become .

Losing a sign when turns a subtraction into an addition. becomes , not .

Miscounting the four-step cycle of . Divide the exponent by and use the remainder — a miscounted cycle is the most common error in the power-of- problems.

Writing a conjugate with the wrong sign. The conjugate of is — only the imaginary part’s sign flips.

Where to go next

For the same conjugate technique applied to a radical instead of a complex number, see the Rationalizing Denominators Practice Problems. For every rule’s full derivation, see the Complex Number Arithmetic lesson.

Frequently Asked Questions

Why does every problem ask for just the real or imaginary part?+

A full complex number like 7+i is two values in one, which can't be automatically checked as a single number. Asking for one part at a time still requires the complete calculation — the answer is just one checkable value instead of a pair.

How is this different from the Complex Number Practice Problems?+

Both cover the same four operations, but this practice set leans more heavily on powers of i, at every difficulty, and always uses a genuine conjugate product at Hard — more repetition on the two ideas the Complex Number Arithmetic lesson spends the most time on.

Why do powers of i only ever use even exponents in the answer key?+

An odd power of i is itself imaginary (i, −i, and so on), which isn't a single real number the checker can validate the usual way — every power-of-i problem here uses an even exponent, which always lands on a real result: 1 or −1.

Why is a conjugate product always a whole number?+

A complex number times its own conjugate, (a+bi)(a−bi), always simplifies to a²+b² — the imaginary parts cancel completely by construction, which is exactly why the conjugate trick clears i from a denominator.

How do I divide by a complex number?+

Multiply the numerator and denominator by the denominator's conjugate — the same complex number with the sign of its imaginary part flipped — which always turns the denominator into a real number.

Can I create as many sets as I want?+

There's no limit. Problems are generated fresh each time, completely free, with no account needed.

a+biComplex Number Arithmetic Practice Problems