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How to use
Complex numbers combine real and imaginary parts as a+bi. Choose an operation on z₁ and z₂; the tool reports the result, modulus, argument, conjugate and polar form.
- Choose add, subtract, multiply or divide.
- Enter real and imaginary parts for both numbers.
- Calculate.
- Review rectangular and polar forms, modulus and argument.
How it is calculated
Example
For addition, combine real and imaginary parts separately: (3+4i)+(1−2i)=(3+1)+(4−2)i=4+2i. Its modulus is √(4²+2²)=√20≈4.472 and its argument is ≈26.565°. For multiplication, expand and use i²=−1: (3+4i)(1−2i)=3−6i+4i−8i²=11−2i.
Important notes
Division by 0+0i is undefined. The argument is shown in degrees using atan2, normally from −180° to 180°.
Worked examples and interpreting results
The complex number 3+4i has real part 3, imaginary part 4 and magnitude 5. Multiplication uses i²=−1, so i cannot be treated as an ordinary real variable.
| Case | Calculation | Result |
|---|---|---|
| Magnitude of 3+4i | √(3² + 4²) | 5 |
| Multiply by the conjugate | (3 + 4i)(3 − 4i) = 9 + 16 | 25 |
How to check the result
Conjugation changes only the imaginary part’s sign. A number times its conjugate equals its magnitude squared. Magnitude is real and nonnegative; it is not the real part or the sum of the two coefficients.
Frequently asked questions
What is modulus?
|a+bi|=√(a²+b²).
Does zero have a defined argument?
No. The number 0+0i has zero modulus and no direction, so its argument is mathematically undefined even if the display shows a numerical value.
What is polar form?
r∠θ, where r is modulus and θ is argument.
What is the complex conjugate used for?
The conjugate of a+bi is a−bi. Their product is a²+b², a non-negative real number, so conjugates help simplify division and calculate reciprocals and magnitudes.