Complex Number Arithmetic Calculator

Enter Calculation

First complex number (a + b i)

Second complex number (c + d i)

Results

Sum (Z1 + Z2)

6 + 8 i

Subtraction (Z1 - Z2)	-2 - 2 i
Multiplication (Z1 × Z2)	-7 + 22 i
Division (Z1 ÷ Z2)	0.560976 + 0.04878 i

What this calculator does

A complex number has the form \(a + b\,i\), where a is the real part, b is the imaginary part, and i is the imaginary unit defined by \(i^2 = -1\). This tool takes two complex numbers, \(Z_1 = a + b\,i\) and \(Z_2 = c + d\,i\), and instantly computes all four basic operations: their sum, difference, product, and quotient, each returned in standard \(a + b\,i\) form.

A complex number a+bi as a point on the complex plane.

How to use it

Type each number as a complex literal, for example 2+3i, 3-2i, -i, 5 (pure real), or 4i (pure imaginary). Spaces are ignored, and a bare i counts as 1i while -i counts as -1i. Press calculate to see addition, subtraction, multiplication, and division side by side.

The formulas explained

Addition and subtraction work component-wise: \((a+bi) \pm (c+di) = (a \pm c) + (b \pm d)\,i\). Multiplication uses the distributive law together with \(i^2 = -1\), giving \((ac - bd) + (bc + ad)\,i\). Division multiplies numerator and denominator by the conjugate of \(Z_2\), producing real part \((ac+bd)/(c^2+d^2)\) and imaginary part \((bc-ad)/(c^2+d^2)\).

$$\begin{gathered} z_1 = a + b\,i, \qquad z_2 = c + d\,i \\[1em] z_1 + z_2 = (a + c) + (b + d)\,i \\[0.4em] z_1 - z_2 = (a - c) + (b - d)\,i \\[0.4em] z_1 \cdot z_2 = (ac - bd) + (bc + ad)\,i \\[0.4em] \frac{z_1}{z_2} = \frac{ac + bd}{c^2 + d^2} + \frac{bc - ad}{c^2 + d^2}\,i \end{gathered}$$

Adding complex numbers follows the parallelogram rule.

Worked example

Let \(Z_1 = 2 + 3i\) and \(Z_2 = 4 + 5i\). Then the sum is \(6 + 8i\), the difference is \(-2 - 2i\), and the product is

$$(2 \cdot 4 - 3 \cdot 5) + (3 \cdot 4 + 2 \cdot 5)\,i = -7 + 22i.$$

For division the denominator is \(4^2 + 5^2 = 41\), so the quotient is

$$\frac{8+15}{41} + \frac{12-10}{41}\,i \approx 0.560976 + 0.048780\,i.$$

FAQ

What happens if I divide by zero? If \(Z_2 = 0\) (both c and d are zero) the quotient is undefined, so the division row shows "undefined" while the other three results stay valid.

Are the operations commutative? Addition and multiplication are commutative; subtraction and division are not, so the calculator preserves the order \(Z_1\) then \(Z_2\) exactly as entered.

Can I enter negative or decimal coefficients? Yes. Values like -1.5+0.5i are fully supported, and pure real or pure imaginary inputs are handled automatically.