Math
Complex Number Calculator
Work with complex numbers in rectangular and polar form. Do arithmetic, conversions, magnitude/argument, conjugates, integer powers, and n-th roots with clean results and history export.
Complex Math Workspace
Enter values as real (a) and imaginary (b) or use polar inputs. Choose an operation and calculate instantly.
Tip: z = a + bi. Use z₁ for single-number operations. For Polar → Rectangular, enter r and θ (z₂ is ignored).
Examples: (3+4i) has magnitude 5 and angle about 53.13°. Try Multiply: (3+4i)×(1−2i) or Roots: n=3 for cube roots of z₁.
| Feature | Formula | Example | Meaning |
|---|---|---|---|
| Add/Subtract | (a+bi) ± (c+di) = (a±c) + (b±d)i | (3+4i)+(1−2i)=4+2i | Combine real with real, imag with imag |
| Multiply | (a+bi)(c+di)=(ac−bd)+(ad+bc)i | (3+4i)(1−2i)=11−2i | FOIL with i²=−1 |
| Divide | (a+bi)/(c+di)=((ac+bd)+(bc−ad)i)/(c²+d²) | (3+4i)/(1−2i)=−1+2i | Use conjugate to remove i in denominator |
| Conjugate | conj(a+bi)=a−bi | conj(3+4i)=3−4i | Reflect across real axis |
| Magnitude | |a+bi|=√(a²+b²) | |3+4i|=5 | Distance from origin |
| Argument | arg(a+bi)=atan2(b,a) | arg(3+4i)≈53.13° | Angle from positive real axis |
| Polar Form | z=r(cosθ+i sinθ)=r∠θ | 5∠53.13° | Same value, different coordinates |
| Integer Powers | (r∠θ)^n = r^n ∠ (nθ) | (5∠θ)^2=25∠(2θ) | De Moivre’s theorem |
| n-th Roots | r^(1/n) ∠ ((θ+2πk)/n) | k=0…n−1 | Lists all distinct roots |
Quick Steps
- Choose an operation (arithmetic, conversion, magnitude/argument, power, or roots).
- Enter z₁ = a + bi. For operations involving z₂, enter z₂ = c + di.
- For Polar → Rectangular, enter r and θ (and pick degrees/radians).
- Click Calculate to see rectangular form, polar form, magnitude, and argument.
- For roots, scroll to the roots table to view all n solutions.
Best practice: keep your angle unit consistent. If your θ comes from calculus/physics, radians are usually the right choice.
Your calculation history will appear here after you run operations.