Complex Number
Add, subtract, multiply, and divide complex numbers (a+bi) with this free online calculator. Get instant step-by-step results for arithmetic operations on imaginary numbers.
About This Calculator
This complex number calculator lets you perform basic arithmetic operations -- addition, subtraction, multiplication, and division -- on two complex numbers in the standard a+bi form. Simply enter the real and imaginary parts of each number, choose an operation, and get an instant result.
Complex numbers are fundamental in algebra, engineering, physics, and signal processing. The calculator handles all four arithmetic operations using standard complex arithmetic rules: addition combines real and imaginary parts separately; multiplication uses the FOIL-like method with i^2 = -1; and division uses the conjugate to eliminate the imaginary denominator.
The calculator supports zero, negative, fractional, and decimal values for all real and imaginary parts. When dividing, if both the real and imaginary parts of the denominator are zero (c = d = 0), the result shows undefined since division by zero is not possible.
Frequently Asked Questions
How do you add complex numbers?
To add two complex numbers (a+bi) and (c+di), add the real parts together and the imaginary parts together: (a+c) + (b+d)i. For example, (3+4i) + (1+2i) = 4 + 6i.
How do you multiply complex numbers?
Multiply complex numbers using the FOIL method just like binomials. For (a+bi) x (c+di), compute ac + adi + bci + bdi^2. Since i^2 = -1, this simplifies to (ac - bd) + (ad + bc)i.
How do you divide complex numbers?
To divide (a+bi) by (c+di), multiply the numerator and denominator by the conjugate (c-di). The result is ((ac + bd) / (c^2 + d^2)) + ((bc - ad) / (c^2 + d^2))i. Division by zero is undefined when c = d = 0.
What is the conjugate of a complex number?
The conjugate of a complex number a+bi is a-bi, where the sign of the imaginary part is flipped. The conjugate is used in division and to find the magnitude squared (a^2 + b^2) of a complex number.
What are real and imaginary parts of a complex number?
A complex number is written as a + bi, where a is the real part (a real number) and b is the imaginary part (the coefficient of i, where i = sqrt-1). Both a and b are real numbers; if b = 0, the number is purely real.
Is the Complex Number Calculator free to use?
Yes, all calculators on Calculy are completely free to use with no registration or download required. You can perform unlimited complex number calculations online.
Can I share my complex number calculation results?
Yes, after performing a calculation, you can copy the URL from your browser to share the exact inputs and results. The calculator encodes all values in the URL for easy sharing.
What is the difference between real and imaginary numbers?
Real numbers are numbers you use in everyday counting and measurement (1, -3, 2.5, pi). Imaginary numbers are multiples of i = sqrt-1. A complex number combines both, written as a + bi where a is real and bi is imaginary.