What this calculator does
Complex arithmetic follows ordinary algebra with one extra rule: i squared is minus one. Addition and subtraction are componentwise, but multiplication mixes the parts, which is where the interesting behaviour starts.
Multiplying 3 + 2i by 1 − 4i gives 11 − 10i, and the four cross terms are what produce it. Division works by multiplying above and below by the conjugate, which clears the imaginary part from the denominator.
The formula
Addition and subtraction operate on real and imaginary parts separately. Multiplication expands all four products and uses i squared equals minus one. Division multiplies by the conjugate of the denominator.
| Term | Meaning |
|---|---|
| Modulus | The distance from the origin, being the square root of the sum of squared parts. |
| Argument | The angle from the positive real axis. |
| Conjugate | The same number with the imaginary part negated, which is what makes division work. |
The inputs explained
| Field | What to enter |
|---|---|
| a (real part 1) | Real part of the first number. |
| b (imaginary part 1) | Imaginary part of the first number. |
| Operation | Which operation to perform. |
| c (real part 2) | Real part of the second number. |
| d (imaginary part 2) | Imaginary part of the second number. |
When to use it
Working through algebra homework
Multiplication and division are where sign errors creep in.
AC circuit analysis
Impedance is complex, and circuit calculations are complex arithmetic.
Understanding the complex plane
Modulus and argument describe the same number in polar terms.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
What does each operation give?
The same pair under all four operations.
| Operation | Result | Modulus of result |
|---|---|---|
| Add | 4.000 − 2.000i | 4.472 |
| Subtract | 2.000 + 6.000i | 6.325 |
| Multiply | 11.000 − 10.000i | 14.866 |
| Divide | -0.2941 + 0.8235i | 0.8745 |
Questions
Why does multiplication mix the parts?
Because expanding the brackets produces four terms, and the product of the two imaginary parts becomes real once i squared is replaced by minus one. That one substitution is what makes complex multiplication a rotation as well as a scaling.
How does dividing by a complex number work?
By multiplying the top and bottom by the conjugate of the denominator. That makes the denominator real, since a number times its conjugate is the sum of the squared parts, and the division becomes ordinary.
What does the modulus represent?
The distance from the origin in the complex plane. Multiplying two complex numbers multiplies their moduli and adds their arguments, which is the geometric meaning of complex multiplication.
Are complex numbers actually used?
Extensively. AC circuit analysis, signal processing, quantum mechanics and control theory all rely on them. The name is historical and unfortunate: there is nothing imaginary about their applications.
For roots of complex numbers, see the complex nth roots calculator. For solving quadratics with complex roots, see the quadratic calculator.