Complex Number Calculator
Add, subtract, multiply or divide two complex numbers and see magnitude and polar information
Result
How we calculated it
About the Complex Number Calculator
Complex numbers extend ordinary real numbers by including an imaginary component built from i, where i² = −1. They are written in the form a + bi and are essential in algebra, electrical engineering, signal analysis and many equation-solving problems.
This calculator handles common arithmetic and representation tasks while keeping real and imaginary parts separate. Polar form can also describe the same number through magnitude and angle.
How to Use the Complex Number Calculator
- Enter the real and imaginary parts of each complex number.
- Choose addition, subtraction, multiplication, division or another supported operation.
- Calculate the result and review rectangular or polar form as needed.
- For division, confirm the divisor is not zero in both real and imaginary parts.
How the Calculation Works
Addition and subtraction combine like parts directly. Multiplication uses (a+bi)(c+di)=(ac−bd)+(ad+bc)i because i²=−1.
Division multiplies numerator and denominator by the complex conjugate of the denominator. Magnitude is |z|=√(a²+b²), while angle is commonly found from atan2(b,a).
Worked example
Complex Number Calculator Worked Example
(3+2i)(4−i)
Multiply: 12−3i+8i−2i² = 12+5i+2 = 14+5i because i²=−1.
Result guide
Understanding Your Complex Number Calculator Result
The result remains one complex number with real and imaginary components. A zero imaginary part means the result lies on the real number line even if complex arithmetic was used to obtain it.
Before you rely on the answer
Assumptions and Limitations
- Inputs are valid finite real and imaginary components.
- Angle conventions use the selected degree or radian unit.
- Division by 0+0i is invalid.
Practical tip
Using This Result in Practice
When checking complex multiplication by hand, expand every term before replacing i² with −1. This avoids the common mistake of treating i like an ordinary variable whose square stays positive.
Related calculations
Useful Calculations to Continue With
If a negative discriminant produced complex roots, use the Quadratic Equation Calculator to see where those values originated.
If your quantities are geometric components rather than complex-plane numbers, use the Vector Calculator instead.
Common questions
Complex Number Calculator FAQ
What does i mean in a complex number?
i is defined so that i²=−1.
What is the complex conjugate?
For a+bi, the conjugate is a−bi.
Can a complex number have zero imaginary part?
Yes. Then it is also an ordinary real number.