Enter two complex numbers as z₁ = a + bi and z₂ = c + di, where i² = −1. Enter only the numeric coefficients: for 3 − 2i, use 3 for the real part and −2 for the imaginary part. Use 0 for any missing part.
The calculator returns separate real and imaginary components for z₁ + z₂, z₁ − z₂, z₁ × z₂ and z₁ ÷ z₂. Division is unavailable when c and d are both zero.
| Operation | Real part | Imaginary coefficient |
|---|---|---|
| z₁ + z₂ | a + c | b + d |
| z₁ − z₂ | a − c | b − d |
| z₁ × z₂ | ac − bd | ad + bc |
| z₁ ÷ z₂, when z₂ ≠ 0 | (ac + bd)/(c² + d²) | (bc − ad)/(c² + d²) |
Addition and subtraction combine matching components. Multiplication expands (a + bi)(c + di), then uses i² = −1 to obtain (ac − bd) + (ad + bc)i.
For division, multiply the numerator and denominator by c − di, the conjugate of the second number. This makes the denominator c² + d². The calculator computes each square by multiplication. If c = 0 and d = 0, the denominator is zero and the quotient is undefined.
Read each result as real part + imaginary coefficient × i. For example, real part 4 and imaginary coefficient −2 represent 4 − 2i. Decimal outputs use eight-digit precision; repeating fractions are approximate.
| Operation | Real part | Imaginary coefficient | Complex result |
|---|---|---|---|
| Sum | 4 | −2 | 4 − 2i |
| Difference | 2 | 6 | 2 + 6i |
| Product | 11 | −10 | 11 − 10i |
| Quotient | −5/17 ≈ −0.29411765 | 14/17 ≈ 0.82352941 | −5/17 + (14/17)i |
Can I enter negative numbers or decimals?
Yes. Enter each real part and imaginary coefficient as a number, including its sign. Do not enter i or a full expression in an input.
Can the second number have a zero real part?
Yes. Division by a purely imaginary number is defined when its imaginary coefficient is nonzero. Division is undefined only when both components of the second number are zero.
Does the order of the numbers matter?
The difference is the first number minus the second, and the quotient is the first number divided by the second. Reversing the inputs generally changes those results. Addition and multiplication give the same mathematical result in either order.
Does this calculator find complex roots or convert to polar form?
No. It performs arithmetic on two known complex numbers in rectangular form. It does not solve polynomial equations, process symbolic expressions, or calculate polar form, magnitude or argument.