# How do you simplify 8/(1+i)?

Jun 22, 2016

$\frac{8}{1 + i} = \frac{8}{1 + i} \cdot \frac{1 - i}{1 - i} = 4 - 4 i$

#### Explanation:

Simplifying a number with a complex denominator is sometimes called "rationalizing the denominator". The reason that it is considered simpler is that we can then see the real and imaginary portions more clearly. To do this, we simply multiply both the numerator and denominator by the complex conjugate of the denominator:

$\frac{8}{1 + i} = \frac{8}{1 + i} \cdot \frac{1 - i}{1 - i} = \frac{8 - 8 i}{1 + i - i - {i}^{2}}$

where ${i}^{2} = - 1$ so

$\frac{8}{1 + i} = \frac{8 - 8 i}{1 + 1} = \frac{8 - 8 i}{2} = 4 - 4 i$