Question 805330
<pre>
{{{(1+i)/(1-i)}}}

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 {{{(1+i)/(1+i)}}}
which just equals to 1, and therefore we can multiply
it by the original fraction without changing the
value, like this:

{{{(1+i)/(1-i)}}}{{{""*""}}}{{{(1+i)/(1+i)}}}

Put parentheses around each factor:

{{{((1+i))/((1-i))}}}{{{""*""}}}{{{((1+i))/((1+i))}}}

Indicate the multiplication of the two fractions:

{{{((1+i)(1+i))/((1-i)(1+i))}}}

Multiply out the top and bottom using "FOIL":

{{{(1+i+i+i^2)/(1+i-i-i^2)}}}

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

{{{(1+2i+i^2)/(1-i^2)}}}

Now we replace each iČ by (-1)

{{{(1+2i+(-1))/(1-(-1))}}}

Simplify:

{{{(1+2i-1)/(1+1)}}}

{{{2i/2}}}

{{{cross(2)i/cross(2)}}}

i

The answer is just simply "i".

Edwin</pre>