Complex Arithmetic Calculator

Math Algebra • Complex Numbers

Written by STEM Calculators Team Published December 20, 2025 Updated February 24, 2026

1. Complex Arithmetic Calculator

Work with complex numbers in rectangular form \(a+bi\): add, subtract, multiply, divide, and conjugate — with steps and an Argand plot.

Operation:

Input z = a + bi

Re(z) = a Im(z) = b

Input w = c + di

Re(w) = c Im(w) = d

Equal axis units (square grid) Show point labels Draw vectors from origin

Ready

Drag to pan • wheel/pinch to zoom • Points are labeled with \((\mathrm{Re},\mathrm{Im})\). Division uses the conjugate method.

Batch operations on a list (optional)

Batch op: Constant c (Re): Constant c (Im): List (one per line or “;”):

Batch: idle

Batch results will appear here.

Argand plot

Top-right is \(+\mathrm{Re}\), up is \(+\mathrm{Im}\). Shows \(z\), \(w\), and the result.

Re: 0, Im: 0 sx: 60, sy: 60

Enter values and click Compute.

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Theory: Complex Arithmetic in Rectangular Form

A complex number is written as \(z=a+bi\), where \(a\) is the real part and \(b\) is the imaginary part. The imaginary unit satisfies \(i^2=-1\). This calculator works in rectangular form and visualizes points on the Argand plane (horizontal = Re, vertical = Im).

1) Basic operations

Addition / subtraction is done component-wise:

\[ (a+bi) \pm (c+di) = (a\pm c) + (b\pm d)i. \]

Multiplication uses FOIL and \(i^2=-1\):

\[ (a+bi)(c+di) = ac + adi + bci + bd i^2 = (ac-bd) + (ad+bc)i. \]

Conjugation flips the sign of the imaginary part:

\[ \overline{a+bi}=a-bi. \]

2) Division via the conjugate

To divide by \(w=c+di\), multiply numerator and denominator by \(\overline{w}=c-di\) so the denominator becomes real.

\[ \frac{a+bi}{c+di} = \frac{a+bi}{c+di}\cdot \frac{c-di}{c-di} = \frac{(a+bi)(c-di)}{c^2+d^2}. \]

Expanding the numerator:

\[ (a+bi)(c-di) = ac - adi + bci - bd i^2 = (ac+bd) + (bc-ad)i. \]

So the final rectangular form is:

\[ \frac{a+bi}{c+di} = \frac{ac+bd}{c^2+d^2} + \frac{bc-ad}{c^2+d^2}i, \qquad (c,d)\neq(0,0). \]

3) Modulus and quick identity checks

The modulus (distance from the origin) is:

\[ |a+bi| = \sqrt{a^2+b^2}. \]

Useful facts (true in exact arithmetic):

\(|\overline{z}| = |z|\) (conjugation keeps distance)
\(|zw| = |z|\,|w|\)
\(\left|\dfrac{z}{w}\right| = \dfrac{|z|}{|w|}\) for \(w\neq 0\)

The calculator reports numerical checks of these identities to help validate computations and spot input mistakes.

4) Worked example

Compute \((3+2i)(1-4i)\).

\[ (3+2i)(1-4i) = 3\cdot 1 + 3(-4i) + (2i)\cdot 1 + (2i)(-4i). \]

\[ = 3 - 12i + 2i - 8i^2 = 3 - 10i + 8 = 11 - 10i. \]

On the Argand plot, \(z\), \(w\), and the product are shown as points/vectors.

5) Batch operations idea

When you have many complex numbers (for example from exercises), you can process them as a list: conjugate each one, compute each modulus, or apply the same constant transformation \(z\mapsto z+c\), \(z\mapsto zc\), etc.

Frequently Asked Questions

How do you add and subtract complex numbers in a + bi form?

Add or subtract real parts and imaginary parts separately: (a+bi) ± (c+di) = (a±c) + (b±d)i. This keeps the result in rectangular form.

How does the calculator divide complex numbers?

Division uses the conjugate method: (a+bi)/(c+di) is computed by multiplying numerator and denominator by (c-di) so the denominator becomes c^2 + d^2. The result is then expressed as a real part plus an imaginary part.

What is the conjugate of a complex number and why is it useful?

The conjugate of a+bi is a-bi. It is useful for division because (c+di)(c-di) becomes the real number c^2 + d^2.

What does the Argand plot show for complex arithmetic?

It plots complex numbers as points where the horizontal axis is the real part and the vertical axis is the imaginary part. The plot shows z, w, and the computed result and can optionally draw vectors from the origin.

How can I apply the same operation to many complex numbers at once?

Use the batch panel to enter a list of complex numbers (one per line or separated by semicolons), choose a batch operation such as conjugate each or modulus each, and optionally set a constant c for z + c, z · c, or z / c. Then run the batch to generate the list of results.

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Frequently Asked Questions

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