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Complex Number Division Calculator

Effortlessly divide complex numbers with our online calculator. Visualize results on the complex plane. Perfect for students and professionals.

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!

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