Complex Numbers Practice Problems

Adding, multiplying, and dividing complex numbers are covered in full in the Complex Numbers lesson; this practice set is where that skill turns into speed. Since a complex answer like isn’t a single checkable number, every problem asks for one coordinate of it — the real part, the imaginary part, or (for a conjugate product, which is always fully real) the whole result.

How the difficulty levels work

Difficulty	What it drills	Example
Easy	Add or subtract, find one part
Medium	Multiply with FOIL, find one part
Hard	Conjugate products, powers of i, or division

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

Medium multiplies two complex numbers with FOIL, replacing with along the way, then asks for one part of the result.

Hard rotates between three shapes: a conjugate product (always a real number, ), a power of (always an even exponent here, so the answer is always or ), and dividing two complex numbers using the conjugate, asking for one part of the (often fractional) result.

Worked examples

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

Easy. Find the real part of .

Add real parts and imaginary parts: . The real part is .

Medium. Find the real part of .

FOIL: . Since , that’s , so the real part is .

Hard. Simplify , and .

. . For the quotient, multiply by the conjugate : .

Using the practice problems

Select a difficulty level and a count, and a new problem set loads instantly. Most answers are whole numbers; the Hard-tier division problems sometimes produce a fraction, entered the same way as any other fraction answer on this site.

Check Answers checks the full set in one click and highlights any that are still wrong
Show Answers uncovers every answer immediately, good for reviewing offline work
Print Worksheet lays out a print-ready page, suited for offline practice or handouts

The rules these practice problems drill

Rule	Statement
Definition
Addition
Multiplication
Conjugate Product

See the Complex Numbers lesson for the full derivation of each rule, plus the powers-of-i cycle and dividing by conjugate in detail.

Common mistakes to watch for

Treating as instead of . This single substitution drives almost every problem here — get it backward and the sign of the whole answer flips.

Reporting the imaginary part with an attached. The imaginary part of is , a plain number — not .

Multiplying straight across instead of using FOIL. needs all four cross products, exactly like multiplying two binomials.

Losing track of the power-of-i remainder. A remainder of always means ; double-check the division before answering.

Where to go next

Once these feel automatic, the Radical Practice Problems practices the exact same conjugate technique used here, aimed at radical denominators instead of imaginary ones. For the full reasoning behind every rule drilled here, see the Complex Numbers lesson.

Frequently Asked Questions

Why does every problem ask for the real part or imaginary part instead of the full answer?+

A complex number like 3-14i isn't a single number the way 7 or 3/4 is, so it can't be automatically checked the same way. Asking for just the real part or just the imaginary part still requires computing the full result correctly — it's checked one coordinate at a time instead of all at once.

What does "imaginary part" actually mean as a number?+

For a complex number a+bi, the imaginary part is just b — a real number, the coefficient of i, not b times i. The imaginary part of 3-14i is -14, not -14i.

Why do the Hard-tier conjugate problems ask for the whole answer instead of one part?+

Because (a+bi)(a-bi) always equals a²+b², a real number with no imaginary part at all — there's nothing to split into two coordinates, so the whole result is already a single checkable number.

How do I simplify i to a large power, like i to the 40th?+

Divide the exponent by 4 and use the remainder: 0 gives 1, 1 gives i, 2 gives -1, 3 gives -i. Every power-of-i problem here uses an even exponent, so the answer is always 1 or -1.

Do I need Radicals before this practice set?+

Rational Exponents and Radicals aren't required to practice the arithmetic here, but the Complex Numbers lesson explains i as an extension of the same radical ideas — worth reviewing if the conjugate technique feels unfamiliar.

Why do some answers come out with a fraction in them?+

Division by a complex number is done by multiplying top and bottom by the conjugate, and the resulting real denominator rarely divides evenly. An answer like is fully simplified — enter the real and imaginary parts as they stand.

a+biComplex Numbers Practice Problems