Complex Number Calculator
Work with numbers in the form a + bi. Enter both values, choose an operation, and click Calculate.
Number A
Number B
What is a complex number?
A complex number combines a real part and an imaginary part: z = a + bi, where i² = -1. Real numbers sit on a number line, but complex numbers live on a 2D plane: horizontal for the real axis and vertical for the imaginary axis.
How this calculator complex tool works
This calculator takes two complex numbers and applies the selected operation. It returns results in standard form and, when useful, in polar form as magnitude and angle.
Core formulas used
- Addition:
(a + bi) + (c + di) = (a + c) + (b + d)i - Subtraction:
(a + bi) - (c + di) = (a - c) + (b - d)i - Multiplication:
(a + bi)(c + di) = (ac - bd) + (ad + bc)i - Division:
(a + bi)/(c + di) = ((ac + bd) + (bc - ad)i) / (c² + d²) - Magnitude:
|a + bi| = √(a² + b²) - Argument:
arg(a + bi) = atan2(b, a) - Conjugate:
conj(a + bi) = a - bi
Why use a complex number calculator?
Complex arithmetic appears in electrical engineering, control systems, quantum mechanics, vibration analysis, signal processing, and computer graphics. A reliable calculator helps you:
- Check homework and exam preparation quickly.
- Validate simulation inputs before coding.
- Avoid sign mistakes in multiplication and division.
- Move between rectangular and polar interpretations.
Example workflow
Multiply two values
Suppose A = 3 + 4i and B = 1 - 2i. Select A × B. The tool computes: (3×1 - 4×-2) + (3×-2 + 4×1)i = 11 - 2i. You also get magnitude and angle of the result.
Find magnitude and angle
For A = 3 + 4i, choose |A| to get 5. Then choose arg(A) to view the phase angle in radians and degrees. This is useful when converting between rectangular and polar forms.
Common mistakes to avoid
- Forgetting that i² = -1 in multiplication.
- Mixing degrees and radians in phase calculations.
- Dividing by a complex number with zero magnitude.
- Dropping negative signs in imaginary components.
Final thoughts
A good calculator complex interface should be simple, accurate, and transparent about formulas. Use this page as both a practical tool and a quick refresher whenever you work with imaginary numbers, phasors, or two-dimensional number systems.