Complex Number Generator
Adding, multiplying, and dividing complex numbers are covered in full in the Complex Numbers lesson; this generator is where that skill turns into speed. Since a complex answer like \(5-2i\) 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 | \((3+4i)+(2-5i) \to \text{real} = 5\) |
| Medium | Multiply with FOIL, find one part | \((2+3i)(1+4i) \to \text{real} = -10\) |
| Hard | Conjugate products, powers of i, or division | \((4+i)(4-i) = 17\) |
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 \(i^{2}\) with \(-1\) along the way, then asks for one part of the result.
Hard rotates between three shapes: a conjugate product (always a real number, \(a^{2}+b^{2}\)), a power of \(i\) (always an even exponent here, so the answer is always \(1\) or \(-1\)), and dividing two complex numbers using the conjugate, asking for one part of the (often fractional) result.
Using the generator
Pick a difficulty and a question count, and a fresh set appears 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 scores the set and flags exactly which problems need another look
- Show Answers reveals every solution, useful for reviewing a paper attempt
- Print Worksheet outputs a clean page for offline practice or classroom handouts
The rules this generator drills
| Rule | Statement |
|---|---|
| Definition | \(i^{2} = -1\) |
| Addition | \((a+bi)+(c+di) = (a+c)+(b+d)i\) |
| Multiplication | \((a+bi)(c+di) = (ac-bd)+(ad+bc)i\) |
| Conjugate Product | \((a+bi)(a-bi) = a^{2}+b^{2}\) |
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 \(i^{2}\) as \(+1\) instead of \(-1\). 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 \(i\) attached. The imaginary part of \(3-14i\) is \(-14\), a plain number — not \(-14i\).
Multiplying straight across instead of using FOIL. \((a+bi)(c+di)\) needs all four cross products, exactly like multiplying two binomials.
Losing track of the power-of-i remainder. A remainder of \(2\) always means \(-1\); double-check the division before answering.
Where to go next
Once these feel automatic, the Radical Generator 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.