# How do you calculate (1+i)/(1-i)?

Jan 5, 2018

$i$

#### Explanation:

We can compute this by multiplying both numerator and denominator by the conjugate, $1 + i$, of the denominator.

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

We know that ${i}^{2} = - 1$, so:

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