Simplify: (1 + i) / (1 - i)
Multiply both the numerator and the denominator by the conjugate of the denominator.
The conjugate of 1 - i is 1 + i.
[ (1 + i) / (1 - i) ] · [ (1 + i) / (1 + i) ]
---> [ (1 + i) ·(1 + i) ] / [ (1 - i) · (1 + i) ]
Numerator: (1 + i) ·(1 + i) = 1 + 2i + i2 = 1 + 2i - 1 = 2i
Denominator: (1 - i) · (1 + i) = 1 - i2 = 1 - -1 = 2
---> = 2i / 2 = i