# Dividing complex numbers

Calculator and formula for dividing a complex number

## Dividing a complex number

Complex number division calculator

 Input Divident + i Divisor + i Decimal places 0 1 2 3 4 6 8 10 Result Quotient

## Formula for division of complex numbers

This article describes dividing complex numbers. In the next example we will divide the number $$3 + i$$ by the number $$1 - 2i$$. Wanted is so

$$\displaystyle(3+i)\,/\,(1-2i)=\frac{3+i}{1-2i}$$

According to the permanence principle, the calculation rules of the real numbers should be valid here. It bothers us that in the denominator of the fraction the $$i$$ occurs. Sharing by a real number would be quite easy.

This is where the complex conjugate comes into play. The fraction is extended by the conjugate complex number $$1 + 2i$$ of the denominator. This allows the $$i$$ to be truncated in the denominator and the denominator becomes a real number. Only in the numerator remains a complex number, which can be easily multiplied out.

The division looks like this

$$\displaystyle\frac{3+i}{1-2i}=\frac{(3+i)·(1+2i)}{(1-2i)·(1+2i)}=\frac{3+6i+i-2}{1+2i-2i+4}=\frac{1+7i}{5}=\frac{1}{5}+\frac{7}{5}i$$

The result is $$\displaystyle\frac{1}{5}+\frac{7}{5}i$$

This article described the division of complex numbers in normal form. Easier to calculate is the division of complex numbers in polar form.