Complex Number Calculator
Add, subtract, multiply, or divide two complex numbers and see a clear rectangular-form result.
Calculate with complex numbers
Enter z₁ = a + bi and z₂ = c + di, then choose an arithmetic operation.
About complex number arithmetic
A complex number combines a real part and an imaginary part in the form a + bi, where i² = -1. The real coefficient a locates the number horizontally on the complex plane, while the imaginary coefficient b locates it vertically. Complex numbers extend the real number system so equations such as x² + 1 = 0 have solutions, and they provide a compact language for oscillation, rotation, electrical impedance, waves, and many other mathematical models.
Addition and subtraction operate component by component. To add a + bi and c + di, add a to c and b to d. Subtraction uses the same structure with differences. This is analogous to adding or subtracting two-dimensional vectors. Keeping real and imaginary terms in separate columns is a reliable way to avoid sign errors, especially when one or more coefficients are negative.
Multiplication uses ordinary distribution together with i² = -1. Expanding (a + bi)(c + di) gives ac + adi + bci + bdi². Replacing i² with -1 and collecting terms produces the real part ac - bd and imaginary part ad + bc. Multiplication also has a geometric interpretation: moduli multiply and arguments add, so the operation combines scaling with rotation on the complex plane.
Division rationalizes the denominator with its conjugate. Multiplying numerator and denominator of (a + bi)/(c + di) by c - di makes the denominator c² + d². The result has real part (ac + bd)/(c² + d²) and imaginary part (bc - ad)/(c² + d²). Division is undefined only when both c and d are zero, because then the denominator's squared modulus is zero.
The calculator accepts whole numbers, negatives, and decimals for all four coefficients. Results are calculated with double-precision arithmetic and rounded to eight decimal places for display. Very small floating-point artifacts are shown as zero. The answer remains in rectangular form so its real and imaginary components are immediately visible; signs are formatted automatically, including a minus sign when the imaginary result is negative.
Complex arithmetic is used in algebra, control systems, communications, quantum mechanics, fluid dynamics, and signal processing. Even when software performs the calculation, understanding the component formulas helps you interpret the result and detect errors. Use the operation explanation under each answer as a compact reminder of the rule, and verify important division results by multiplying the quotient by the original divisor.
Complex number examples
| Calculation | Result | Method |
|---|---|---|
| (3 + 4i) + (2 - 5i) | 5 - 1i | Add real coefficients and imaginary coefficients separately. |
| (5 + 2i) - (1 + 7i) | 4 - 5i | Subtract each component of the second number. |
| (2 + 3i)(4 - i) | 11 + 10i | Distribute, replace i² with -1, and combine like terms. |
| (4 + 2i)/(1 - i) | 1 + 3i | Multiply by 1 + i and divide each component by 2. |
How to use the complex number calculator
- Choose addition, subtraction, multiplication, or division.
- Enter the real and imaginary coefficients of the first complex number.
- Enter the real and imaginary coefficients of the second complex number.
- Select Calculate Complex Result and review the rectangular-form answer and operation rule.
Complex number calculator FAQ
What does i mean in a complex number?
The symbol i is the imaginary unit defined by i² = -1. It allows square roots of negative real numbers to be represented consistently.
How do I add complex numbers?
Add the real parts together and add the imaginary coefficients together. Do not combine a real term directly with an imaginary term.
Why does complex multiplication use ac - bd?
The product of the two imaginary terms is bdi². Since i² equals -1, that contribution becomes the real term -bd.
How is complex division calculated?
Multiply the numerator and denominator by the denominator's conjugate. The denominator becomes its squared modulus, c² + d², which is real.
Can a complex result have a zero imaginary part?
Yes, complex arithmetic can produce an ordinary real number. The calculator displays it with a zero imaginary coefficient so both components remain explicit.