Determine if i/(1+i) + i/ (1-i) is real or imaginary.

We have to determine the result of the sum of 2 ratios and
we'll have to decide if the result is a complex or real
number.

To calculate the sum of 2 ratios that do not have a
common denominator we'll have to calculate the LCD(least common denominator) of the 2
ratios.

We notice that LCD =
(1+i)(1-i)

We notice also that the product (1+i)(1-i) is
like:

(a-b)(a+b) = a^2 -
b^2

We'll write instead of product the difference of
squares, where a = 1 and b = i.

LCD =
(1+i)(1-i)

LCD = 1^2 -
i^2

We'll write instead of i^2 =
-1

LCD = 1 - (-1)

LCD =
2

Now, we'll multiply the first ratio by (1-i) and the
second ratio by (1+i):

 i(1-i)/2 + i(1+i)/
2

We'll remove the
brackets:

(i - i^2 + i +
i^2)/2

We'll eliminate like
terms:

2i/2

We'll
simplify:

 i(1-i)/2 + i(1+i)/ 2 =
i

The result is a complex
number, whose real part is 0 and imaginary part is
1.