Complex Number Calculator
Product, quotient and sum of two complex numbers, with the modulus and argument of the first.
Results
What this tool does
Division works by multiplying top and bottom by the conjugate of the denominator, which clears the imaginary part from the bottom — that is the step the third and fourth results carry out. The argument is given in degrees and measured anticlockwise from the positive real axis, using the two-argument arctangent so every quadrant comes out right.
Formula
(a+bi)(c+di) = (ac−bd) + (ad+bc)i
Variables
| Symbol | Meaning | Unit |
|---|---|---|
ar | Real part | — |
ai | Imaginary part | — |
cr | Real part | — |
ci | Imaginary part | — |
MR | Real part | — |
MI | Imaginary part | — |
DR | Real part | — |
DI | Imaginary part | — |
MO | Modulus | — |
AG | Argument | ° |
SR | Real part | — |
SI | Imaginary part | — |
Worked example
- Real part3
- Imaginary part4
- Real part1
- Imaginary part-2
- Real part11.000000
- Imaginary part-2.000000
- Real part-1.000000
- Imaginary part2.000000
- Modulus5.000000
- Argument53.13010 °
- Real part4.000000
- Imaginary part2.000000
Limitations
- The calculation runs at full precision and only the display is rounded. If you copy an intermediate value and retype it, small differences can appear.
Frequently asked questions
Why does multiplying by i turn things by ninety degrees?
Because multiplying complex numbers multiplies their moduli and adds their arguments. The number i has modulus 1 and argument 90 degrees, so multiplying by it leaves the length alone and adds a right angle. That is the whole reason complex numbers are used for alternating current and for waves: a phase shift becomes a multiplication, and calculus on oscillations becomes ordinary algebra.