Complex Number Arithmetic Calculator
Complex numbers combine a real part and an imaginary part and are fundamental in engineering, physics and signal processing. This calculator takes two complex numbers, each entered as a real and an imaginary component, and returns their sum, difference, product and quotient, along with the modulus of each input. Addition and subtraction act on the parts separately, multiplication uses the rule that i squared equals minus one, and division multiplies by the conjugate of the denominator. All results are exact arithmetic in floating point.
Complex arithmetic formula
Sum = (a + c) + (b + d)i
Difference = (a - c) + (b - d)i
Product = (ac - bd) + (ad + bc)i
Quotient = ((ac + bd) + (bc - ad)i) / (c^2 + d^2)
Modulus = sqrt(real^2 + imaginary^2)
Division is undefined when z2 is zero (c and d both zero).
Complex number context
- The imaginary unit i satisfies i squared equal to minus one.
- Addition and subtraction combine real and imaginary parts independently.
- Multiplication and division rely on i squared equal to minus one and the conjugate.
- The modulus is the distance from the origin in the complex plane.
- Complex numbers are central to AC circuit analysis, control theory and Fourier methods.
Complex number arithmetic: frequently asked questions
How do you add and subtract complex numbers?
Add or subtract the real parts and the imaginary parts separately. For (a plus bi) and (c plus di), the sum is (a plus c) plus (b plus d)i and the difference is (a minus c) plus (b minus d)i. The calculator returns the real and imaginary parts of each result.
How do you multiply complex numbers?
Use the distributive rule with i squared equal to minus one. The product (a plus bi)(c plus di) equals (ac minus bd) plus (ad plus bc)i. The real part is ac minus bd and the imaginary part is ad plus bc.
How do you divide complex numbers?
Multiply the numerator and denominator by the conjugate of the denominator. The quotient (a plus bi) divided by (c plus di) equals ((ac plus bd) plus (bc minus ad)i) divided by (c squared plus d squared). Division is undefined when the second number is zero.
What is the modulus of a complex number?
The modulus, or absolute value, of a plus bi is the square root of (a squared plus b squared). It is the distance from the origin to the point in the complex plane and is always non-negative. The calculator reports the modulus of both input numbers.
What does i represent?
The imaginary unit i is defined by i squared equal to minus one. Every complex number can be written as a real part plus a real multiple of i. Complex numbers extend the real numbers and are essential in engineering, physics and signal processing.
Official sources
- NIST Digital Library of Mathematical Functions: Complex Numbers.
- U.S. National Institute of Standards and Technology: NIST home, mathematical references.
Reviewed by the CalculatorHub team, edited by James Graham, 17 June 2026. See our methodology.