Complex Number Calculator
Add, subtract, multiply and divide complex numbers, then find real and imaginary parts, magnitude, conjugate and polar form.
Calculate with complex numbers in a + bi form
Enter complex numbers in a+bi form to perform arithmetic or inspect a single number.
The calculator separates real and imaginary parts and can return magnitude, conjugate and polar form.
Features
- Complex addition, subtraction, multiplication and division.
- Real and imaginary part output.
- Absolute value or modulus.
- Complex conjugate.
- Polar form with argument in radians and degrees.
Example expressions
(2+3i) + (1-4i)(2+3i) - (1-4i)(2+3i)(1-4i)(2+3i)/(1-4i)|3+4i|polar form of 1+i
Related formulas
(a+bi)(c+di) = (ac−bd) + (ad+bc)i.|a+bi| = √(a²+b²).Conjugate of a+bi is a−bi.Polar form uses r = √(a²+b²) and θ = atan2(b,a).
How to use this calculator
- Enter a complex number using i for the imaginary unit.
- Choose an arithmetic operation or a single-number property.
- Calculate and inspect the separated real and imaginary parts or polar details.
Common input mistakes
- Use i, not a variable name, for the imaginary unit.
- Complex division by 0+0i is undefined.
- Use parentheses when writing a complex number inside a larger expression.
FAQ
How should I enter a complex number?
Use a+bi form, such as 2+3i, -4i or 5.
Can I divide complex numbers?
Yes. The calculator multiplies by the conjugate of the denominator and rejects division by 0+0i.
What is the magnitude of 3+4i?
Its magnitude is √(3²+4²) = 5.
What is polar form?
Polar form represents a complex number by its modulus r and argument θ instead of separate real and imaginary parts.