What is the complex number z=i/(1-i)+i/(1+i)?

To determine the complex number, we'll have to determine
the result of the sum of 2 quotients.

To calculate the sum
of 2 quotients 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):

z = i(1+i)/2 + i(1-i)/
2

We'll remove the brackets:

z
= (i + i^2 + i - i^2)/2

We'll eliminate like
terms:

z = 2i/2

We'll
simplify:

z = 
i

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

The algebraic form of the complex number
z is z = i.

The polar form of the complex number
is:

z = |z|(cos a + i*sin
a)

We'll calculate the modulus and the argument of
z.

|z| = sqrt(Re(z)^2 +
Im(z)^2)

|z| = sqrt(0^2 +
1^2)

|z| =
1

tan a =
y/x

tan a = 1/0 =
+infinite

a =
pi/2

The polar form of the
complex number is:

z =
(cos pi/2 + i*sin pi/2)