Enter two complex numbers
Choose each number’s format independently. Expressions may include pi or sqrt. Invalid syntax produces a clear message.
Formula used
w = r₂(cos θ₂ + i sin θ₂)
zw = r₁r₂[cos(θ₁ + θ₂) + i sin(θ₁ + θ₂)]
Polar multiplication uses two simple operations. Multiply both magnitudes first. Then add both complex arguments together.
Rectangular verification uses standard complex multiplication. Combine real terms and imaginary terms. Both methods should produce matching coordinates.
How to use this calculator
- Choose rectangular, polar, or exponential input for z.
- Choose any supported input format for w.
- Enter values using decimals, pi, or square roots.
- Select angle units, precision, and principal range.
- Press calculate to view every supported result form.
- Copy, print, or export the completed calculation.
Use auto detection for mixed angle notation. Degree symbols force degree interpretation. Pi notation automatically indicates radians.
Example data
| Example | z | w | Expected product | Action |
|---|---|---|---|---|
| Rectangular values | 3 + 4i | 1 − 2i | 11 − 2i | |
| Degree angles | 2∠30° | 3∠45° | 6∠75° | |
| Radian angles | 4e^(iπ/3) | 2e^(−iπ/6) | 8e^(iπ/6) | |
| Zero product | 0 + 0i | 5 + 2i | 0, with undefined argument |
Frequently asked questions
What does zw mean?
It means multiply complex number z by w. The order does not change multiplication. Both products have identical polar values.
Why do polar magnitudes multiply?
Magnitude measures distance from the origin. Multiplication scales one distance by another. Therefore, their product gives the new magnitude.
Why are the arguments added?
Complex multiplication combines rotations around the origin. Each argument represents one rotation. Adding them gives the resulting direction.
Can z and w use different formats?
Yes, each number has independent format controls. One may be rectangular. The other may use polar notation.
How does automatic angle detection work?
A degree symbol selects degrees immediately. Pi notation selects radians automatically. Plain numbers use your chosen default.
What happens with a negative magnitude?
The calculator converts it to positive magnitude. It adds π to the argument. This produces standard polar representation.
Why can polar answers look different?
Coterminal angles represent the same complex number. They differ by complete rotations. The principal range selects one representative.
Is the argument of zero defined?
No, zero has no unique direction. Its argument is mathematically undefined. The calculator displays zero only for formatting.
Why does rectangular verification differ slightly?
Decimal trigonometric values require numerical rounding. Small differences can appear near zero. Higher precision usually reduces visible discrepancies.