About this calculator
Complex Number Arithmetic Calculator supports addition, subtraction, multiplication and division between two complex numbers. After inputting z₁ = a + bi and z₂ = c + di, the tool will calculate the result according to the rules of complex number arithmetic and output the standard form.
Complex addition and subtraction operate on real and imaginary parts; complex multiplication uses i² = -1 expansion; and complex division is usually done by multiplying the complex conjugate of the denominator. Mastering these rules is the basis for learning complex equations, complex plane geometry, circuit phasors, and signal processing.
This calculator is suitable for quickly checking the hand calculation process, and is also suitable for converting complex expressions into the form of a + bi. Whether it is an integer, decimal or negative imaginary part, it can be entered and calculated directly.
What it calculates
The complex arithmetic calculator performs addition, subtraction, multiplication, and division for two complex numbers and returns the result in a + bi form.
Formula
- (a + bi) + (c + di) = (a + c) + (b + d)i
- (a + bi) - (c + di) = (a - c) + (b - d)i
- (a + bi)(c + di) = (ac - bd) + (ad + bc)i
- (a + bi) / (c + di) = ((ac + bd) + (bc - ad)i) / (c^2 + d^2)
Inputs
- Real and imaginary parts of the first complex number.
- Real and imaginary parts of the second complex number.
- The operation: add, subtract, multiply, or divide.
Example
| Operation | Result | Note |
|---|---|---|
| (3 + 4i) + (2 - i) | 5 + 3i | Add real parts and imaginary parts |
| (3 + 4i) - (2 - i) | 1 + 5i | Subtract matching parts |
| (1 + 2i)(3 + 4i) | -5 + 10i | Expand and use i^2 = -1 |
| (3 + 4i) / (1 - 2i) | -1 + 2i | Simplify with the denominator conjugate |
How to interpret the result
The real part is the horizontal coordinate on the complex plane, and the imaginary part is the vertical coordinate. Multiplication changes magnitude and angle; division is multiplication by a reciprocal.
Common mistakes
- Do not forget that i^2 = -1 when multiplying.
- Do not divide real parts and imaginary parts separately.
- Division by 0 + 0i is undefined.
How to use
Enter the real and imaginary parts of the first complex number first, then the real and imaginary parts of the second complex number. Select one of addition, subtraction, multiplication, or division, and then click Calculate.
For example, to calculate (2+3i)+(4-5i), enter the real part 2 and imaginary part 3 of z₁, the real part 4 and the imaginary part -5 of z₂, and select addition, the result is 6-2i.
When dividing, the second complex number cannot be 0 + 0i. Because dividing by zero is not defined for complex numbers, the calculator will prompt that the input is invalid or cannot be calculated.
Main features
Supports complex number addition, subtraction, multiplication and division.
Automatically handles imaginary units i² = -1 and complex conjugate simplification, supporting positive and negative numbers, decimals and zero imaginary part input.
Outputs standard a + bi form, suitable for mathematical learning, engineering phasors, signal processing, and complex expression simplification.
Use cases
In algebra courses, the four operations on complex numbers are the core content of the chapter on complex numbers. Students can use this tool to check whether the real and imaginary parts are combined correctly.
In circuit analysis, impedance is often written in complex form, and complex addition, multiplication, and division are used in series and parallel calculations.
In signal processing and control systems, frequency domain responses, poles and zeros, Fourier coefficients, etc. may contain complex operations, and fast calculation of standard forms can improve analysis efficiency.