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Complex numbers extend the real numbers with an imaginary unit i (where i² = -1), and arithmetic on them follows specific rules for combining real and imaginary parts.
This calculator adds, subtracts, multiplies, or divides two complex numbers and shows the modulus of the result.
How it works
Enter the real and imaginary parts of each complex number and choose an operation. The calculator applies the appropriate complex arithmetic formula and reports the result in a+bi form, along with its modulus (distance from the origin).
- Enter first number — real part (a₁).
- Enter first number — imaginary part (b₁).
- Enter second number — real part (a₂).
- Enter second number — imaginary part (b₂).
- Enter operation.
- Click Calculate to see your results.
Examples
(3+2i) × (1+4i)
Multiplying 3+2i by 1+4i gives -5+14i, using the distributive rule and i² = -1.
Who should use it
- Checking complex number arithmetic for algebra or engineering coursework.
- Working through electrical engineering calculations involving impedance (a common complex-number application).
Industry applications
- Electrical engineering (AC circuit analysis uses complex impedance)
- Mathematics and physics education
Advantages
- Supports all four basic arithmetic operations on complex numbers.
- Reports the modulus alongside the result automatically.
Limitations
- Limited to basic arithmetic — doesn't cover complex exponentials, roots, or polar form conversion.
Common mistakes to avoid
- Forgetting that i² = -1 when multiplying, which is the key rule that makes complex multiplication different from simple binomial expansion.
- Mixing up which number is "first" and "second" for subtraction and division, since order matters for those operations.
Best practices
- Double-check the modulus of your result as a sanity check — it should never be negative, and larger real/imaginary parts should produce a larger modulus.
Tips
- Complex numbers are essential in electrical engineering for AC circuit analysis, where impedance combines resistance (real part) and reactance (imaginary part) into a single complex number.