Question

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

Answer 1

I found: `-1+i`

Explanation:

We can multiply and divide by the complex conjugate of the denominator, i.e., `1+i` to get:
`(2i)/(1-i)*(1+i)/(1+i)=(2i-2)/(1+1)=(2i-2)/2=`
remembering that: `i^2=-1`.
Finally:
`=2i/2-2/2=-1+i`

Answer 2

Multiply by the conjugate of the denominator to find that

`(2i)/(1-i) =i-1`

Explanation:

Given a complex number `a+bi` we call `a-bi` 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.

`(2i)/(1-i) = (2i(1+i))/((1-i)(1+i))`

`=(2(i-1))/(1+i-i+1)`

`=(2(i-1))/2`

`=i-1`