# How do you divide (5/(1+i))?

Jul 28, 2018

$\frac{1}{2} \left(5 - 5 i\right)$

#### Explanation:

$\text{multiply the numerator/denominator by the complex}$
$\text{conjugate of the denominator}$

$\text{the conjugate of "1+i" is } 1 \textcolor{red}{-} i$

$= \frac{5 \left(1 - i\right)}{\left(1 + i\right) \left(1 - i\right)}$

$= \frac{5 - 5 i}{1 - {i}^{2}} \to {i}^{2} = - 1$

$= \frac{5 - 5 i}{2} = \frac{1}{2} \left(5 - 5 i\right)$

$\frac{5}{2} \left(1 - i\right)$

#### Explanation:

Given that

$\frac{5}{1 + i}$

$= \frac{5 \left(1 - i\right)}{\left(1 + i\right) \left(1 - i\right)}$

$= \frac{5 \left(1 - i\right)}{{1}^{2} - {i}^{2}}$

$= \frac{5 \left(1 - i\right)}{1 - \left(- 1\right)} \setminus \quad \left(\setminus \because \setminus {i}^{2} = - 1\right)$

$= \frac{5 \left(1 - i\right)}{2}$

$= \frac{5}{2} \left(1 - i\right)$

Jul 28, 2018

$\frac{5}{2} \left(1 - i\right)$

#### Explanation:

Dividing by complex numbers is the same as multiplying by the complex conjugate:

Complex Conjugate of a complex number $a + b i$ is $a - b i$.

Multiplying by the complex conjugate, we now have

$\frac{5 \left(1 - i\right)}{\left(1 + i\right) \left(1 - i\right)}$

$\frac{5 \left(1 - i\right)}{1 - {i}^{2}}$

Recall that ${i}^{2} = - 1$. This all simplifies to

$\frac{5}{2} \left(1 - i\right)$

Hope this helps!