Complex Number Calculator

Why use a complex number calculator?

Working with complex numbers by hand can become tedious, especially when you repeat the same operations or need to convert between rectangular form (a + bi) and polar form (magnitude and angle). This calculator automates the core arithmetic, shows the result in standard a + bi form, and also reports its magnitude and angle so you can connect algebra with geometry on the complex plane.

Students can use this tool to check textbook exercises on complex arithmetic and to visualize how operations move points in the complex plane. Engineers and hobbyists working with AC circuits, phasor diagrams, or control systems can quickly evaluate sums, differences, products, and quotients of complex quantities without doing every step by hand.

Review of complex numbers

A complex number is any number that can be written in the form

where a and b are real numbers and i is the imaginary unit with

On the complex plane, the horizontal axis represents the real part a, and the vertical axis represents the imaginary part b. For example, the number 3 + 4i is plotted 3 units to the right and 4 units up from the origin. Thinking of complex numbers as points or arrows (vectors) in this plane makes the algebraic rules more intuitive.

Core arithmetic operations

Suppose you have two complex numbers

and .

Addition and subtraction

To add or subtract complex numbers, combine like parts:

• Addition: (a + bi) + (c + di) = (a + c) + (b + d)i
• Subtraction: (a + bi) − (c + di) = (a − c) + (b − d)i

Geometrically, addition corresponds to vector addition in the complex plane: place the arrow for w at the end of the arrow for z, and the sum is the arrow from the origin to the new endpoint.

Multiplication

To multiply, distribute the terms and use i² = −1:

(a + bi)(c + di) = (ac − bd) + (ad + bc)i

This formula comes from expanding (a + bi)(c + di) = ac + adi + bci + bdi², then replacing i² with −1 and collecting real and imaginary parts.

Division using the complex conjugate

Division is handled by multiplying numerator and denominator by the complex conjugate of the denominator. The complex conjugate of c + di is c − di. For

you multiply top and bottom by c − di:

The denominator simplifies to c² + d², a purely real number, so the result can again be written in standard a + bi form.

Polar form and Euler’s formula

Besides rectangular form a + bi, a complex number can also be written in polar form using its magnitude (also called modulus) and angle (also called argument). For

the magnitude r and angle θ are

The calculator uses this relationship internally to report the magnitude and angle for the result of your chosen operation.

Euler’s celebrated formula connects exponentials with trigonometric functions:

This shows that a complex number with magnitude r and angle θ can be written as

In polar form, multiplying two complex numbers multiplies their magnitudes and adds their angles. Dividing two complex numbers divides their magnitudes and subtracts their angles. The magnitude and angle that the calculator displays help you see these patterns in your own examples.

Worked example

Consider the complex numbers z = 3 + 4i and w = −1 + 2i.

Addition

Add real parts and imaginary parts separately:

• Real part: 3 + (−1) = 2
• Imaginary part: 4 + 2 = 6

So z + w = 2 + 6i.

Multiplication

Use the multiplication formula:

• ac − bd = (3)(−1) − (4)(2) = −3 − 8 = −11
• ad + bc = (3)(2) + (4)(−1) = 6 − 4 = 2

Therefore, (3 + 4i)(−1 + 2i) = −11 + 2i.

The magnitude of the product is

The angle is atan2(2, −11), which lies in the second quadrant because the real part is negative and the imaginary part is positive. If you enter these same values in the calculator and choose “Multiply”, you should see the same a + bi result along with its magnitude and angle.

Summary of formulas

The table below summarizes the four basic operations on

Interpreting the calculator output

For any selected operation, the calculator will typically provide:

• The result in standard a + bi form.
• The magnitude (distance from the origin in the complex plane).
• The angle, usually measured in radians via atan2(b, a).

The magnitude tells you how large the complex number is overall, while the angle tells you its direction relative to the positive real axis. Together, these describe the same point as the rectangular coordinates a and b. For applications like AC circuit analysis or phasor diagrams, it is often more natural to think in terms of magnitude and angle.

Notes, assumptions, and limitations

• Input format: Enter each complex number by specifying its real and imaginary parts as decimal numbers (for example, 3.5 and −1.2).
• Supported operations: This tool focuses on basic arithmetic (addition, subtraction, multiplication, and division). It does not perform powers, roots, exponentials, logarithms, or symbolic simplification.
• Division by zero: Division is only defined when the second complex number is non-zero (its real and imaginary parts are not both zero). If you attempt to divide by 0 + 0i, the calculator cannot produce a valid result.
• Precision and rounding: Results may be rounded to a reasonable number of decimal places for readability. Very large or very small inputs can lead to rounding errors or overflow/underflow, as with most floating-point calculations.
• Angle conventions: Angles are typically reported in radians using the standard atan2(b, a) function. Depending on implementation, angles may be shown in the range (−π, π] or [0, 2π).
• Educational use: The calculator is intended as a numerical and educational aid for students, teachers, and engineers. It should not be treated as a full computer algebra system or a substitute for rigorous analytic work when exact symbolic results are required.