Complex Number Calculator

About Complex Number Calculator

Complex numbers have a real and imaginary part. Add/subtract component-wise. Multiply using (a+bi)(c+di) = (ac-bd)+(ad+bc)i.

How to use this calculator

1. Enter the complex number in the form \(a + bi\), for example \(3 + 4i\).
2. Choose the operation you want, such as addition, subtraction, multiplication, division, or a related complex-number calculation.
3. If needed, enter a second complex number, like \(1 - 2i\).
4. Click calculate and read the result in standard complex form.

The formula explained

The formula \(z = a + bi\) shows a complex number as a real part \(a\) plus an imaginary part \(bi\), where \(i^2 = -1\). This form lets you combine, simplify, and compare complex numbers correctly.

• z = the complex number
• a = the real part
• b = the coefficient of the imaginary part
• i = the imaginary unit, where \(i^2 = -1\)

Step by step method

1. Write each complex number in standard form \(a + bi\).
2. Combine the real parts and imaginary parts separately for addition and subtraction, or use distribution for multiplication and the conjugate for division.
3. Simplify powers of \(i\) using \(i^2 = -1\), \(i^3 = -i\), and \(i^4 = 1\).

Worked example

Problem. Add the complex numbers \((3 + 4i) + (2 - 5i)\).

1. Group the real parts and imaginary parts: \((3 + 2) + (4i - 5i)\).
2. Simplify: \(5 - i\).
3. Write the result in standard form.

Tips and common mistakes

• Keep the real part and imaginary part separate, and remember that \(i\) is not a variable here, it is the imaginary unit.
• When multiplying, use FOIL carefully and replace \(i^2\) with \(-1\) at the end.

Frequently asked questions