Simplify Complex Fraction: 1/i + 1/(1+i) + 1/(1-i)

Published on August 4, 2024

Algebra

Simplify the complex number expression 1/i + 1/(1+i) + 1/(1-i). The solution is 1 - i.

Question

simplify it

Answer

To simplify the expression

, we’ll need to work with the imaginary unit

and manipulate these fractions. Here’s a step-by-step solution:

1. Rationalize the denominators:

For the first term, multiply the numerator and denominator by :

Since , we have:
For the second term, multiply the numerator and denominator by the conjugate of the denominator, :

Simplifying, we get:
For the third term, multiply the numerator and denominator by the conjugate of the denominator, :

Simplifying, we get:

2. Combine the terms:

Now our expression looks like this:

Combining the fractions, we get:

3. Simplify:

Notice that the

and

terms in the numerator cancel out:

This leaves us with:

Therefore, the simplified form of the expression is 1 - i.

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