Unlock Complex Number Division

Visualize and calculate complex number division with ease. Enter your complex numbers below!

Enter Complex Numbers

Specify the numerator and denominator in the form a + bi.

Numerator (z1 = a + bi)

+ i

Denominator (z2 = c + di)

+ i

Tip: Denominator cannot be zero.

Result

Quotient (z1 / z2):

Complex Plane Visualization

Understanding Complex Number Division

Complex number division involves dividing one complex number (numerator) by another (denominator). A complex number is of the form $$a + bi$$ , where $$a$$ is the real part, $$b$$ is the imaginary part, and $$i$$ is the imaginary unit ( $$i^2 = -1$$ ).

Formula

To divide two complex numbers $$z_1 = a + bi$$ and $$z_2 = c + di$$ , we use the formula:

\frac{z_1}{z_2} = \frac{a + bi}{c + di} = \frac{(a + bi)(c - di)}{(c + di)(c - di)} = \frac{(ac + bd) + (bc - ad)i}{c^2 + d^2} = \frac{ac + bd}{c^2 + d^2} + \frac{bc - ad}{c^2 + d^2}i

This involves multiplying the numerator and the denominator by the conjugate of the denominator ( $$c - di$$ ) to eliminate the imaginary part from the denominator.

Example

Divide $$z_1 = 4 + 2i$$ by $$z_2 = 1 - i$$ :

\frac{4 + 2i}{1 - i} = \frac{(4 + 2i)(1 + i)}{(1 - i)(1 + i)} = \frac{(4 - 2) + (4 + 2)i}{1^2 + (-1)^2} = \frac{2 + 6i}{2} = 1 + 3i

Thus, the quotient is $$1 + 3i$$ . Use this tool to easily calculate and visualize complex number division!