Complex Number Arithmetic Generator
The Complex Number Arithmetic lesson covers all four operations in full; this generator drills them with heavier repetition on powers of \(i\) and conjugate products — the two ideas that reappear most often once you move past the basics.
How the difficulty levels work
| Difficulty | What it drills | Example |
|---|---|---|
| Easy | Add or subtract two complex numbers | Real or imaginary part of \((4+3i)+(7+8i)\) |
| Medium | A power of \(i\), or multiplying two complex numbers | \(i^{18} = -1\) |
| Hard | Dividing, a bigger power of \(i\), or a conjugate product | \((5+2i)(5-2i) = 29\) |
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 \(i\) (always landing on \(1\) or \(-1\)) and multiplying two complex numbers with FOIL.
Hard rotates through three shapes: dividing two complex numbers using the conjugate, a much larger power of \(i\), and a genuine conjugate product, always fully real.
Using the generator
Pick a difficulty and a question count, and a fresh set appears instantly. Whole-number answers go in as-is; division problems accept a fraction or its decimal equivalent.
- 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 |
|---|---|
| Add or subtract | Combine real parts and imaginary parts separately |
| Multiply | FOIL, then replace \(i^{2}\) with \(-1\) |
| Divide | Multiply top and bottom by the denominator’s conjugate |
| Powers of \(i\) | Cycle every four: \(i, -1, -i, 1\) |
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 \(i^{2}\) with \(-1\) after multiplying. A term like \(6i^{2}\) is not a finished answer; it has to become \(-6\).
Losing a sign when \(i^{2}\) turns a subtraction into an addition. \(9-4i^{2}\) becomes \(9+4=13\), not \(9-4=5\).
Miscounting the four-step cycle of \(i\). Divide the exponent by \(4\) and use the remainder — a miscounted cycle is the most common error in the power-of-\(i\) problems.
Writing a conjugate with the wrong sign. The conjugate of \(2-3i\) is \(2+3i\) — 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 Generator. For every rule’s full derivation, see the Complex Number Arithmetic lesson.