# How do you simplify (2i) / (1 - i)?

Mar 13, 2016

I found: $- 1 + i$

#### Explanation:

We can multiply and divide by the complex conjugate of the denominator, i.e., $1 + i$ to get:
$\frac{2 i}{1 - i} \cdot \frac{1 + i}{1 + i} = \frac{2 i - 2}{1 + 1} = \frac{2 i - 2}{2} =$
remembering that: ${i}^{2} = - 1$.
Finally:
$= 2 \frac{i}{2} - \frac{2}{2} = - 1 + i$

Mar 13, 2016

Multiply by the conjugate of the denominator to find that

$\frac{2 i}{1 - i} = i - 1$

#### Explanation:

Given a complex number $a + b i$ we call $a - b i$ the complex conjugate of that number. One useful property is that the product of a complex number and its conjugate is a real number. We will use that to remove the complex number from the denominator by multiplying the numerator and denominator by the denominator's conjugate.

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

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

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

$= i - 1$