SOLUTION: rationalize: (1+i)/(1-i)

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Question 805330: rationalize:
(1+i)/(1-i)
Answer by Edwin McCravy(20054) About Me  (Show Source):
You can put this solution on YOUR website!
%281%2Bi%29%2F%281-i%29

The denominator is 1-i.  Let's form the conjugate
of the denominator:

The conjugate of the denominator is formed by
Leaving the sign of the real part, 1, and
changing the sign of the imaginary part i, and
getting 1+i.

Now put 1+i over itself, like this %281%2Bi%29%2F%281%2Bi%29
which just equals to 1, and therefore we can multiply
it by the original fraction without changing the
value, like this:

%281%2Bi%29%2F%281-i%29%22%22%2A%22%22%281%2Bi%29%2F%281%2Bi%29

Put parentheses around each factor:

%28%281%2Bi%29%29%2F%28%281-i%29%29%22%22%2A%22%22%28%281%2Bi%29%29%2F%28%281%2Bi%29%29

Indicate the multiplication of the two fractions:

%28%281%2Bi%29%281%2Bi%29%29%2F%28%281-i%29%281%2Bi%29%29

Multiply out the top and bottom using "FOIL":

%281%2Bi%2Bi%2Bi%5E2%29%2F%281%2Bi-i-i%5E2%29

Combine terms.  The +i and the -i cancel in the
bottom

%281%2B2i%2Bi%5E2%29%2F%281-i%5E2%29

Now we replace each iČ by (-1)

%281%2B2i%2B%28-1%29%29%2F%281-%28-1%29%29

Simplify:

%281%2B2i-1%29%2F%281%2B1%29

2i%2F2

cross%282%29i%2Fcross%282%29

i

The answer is just simply "i".

Edwin