(1+i)(3+i)/3-i-(1-i)(3-i)/3+i

3 min read Jun 16, 2024

Simplifying Complex Fractions: A Step-by-Step Guide

This article will walk you through the process of simplifying the complex fraction:

(1 + i)(3 + i) / (3 - i) - (1 - i)(3 - i) / (3 + i)

Before we dive into the simplification, let's refresh some key concepts:

Complex Numbers: Complex numbers are numbers of the form a + bi, where 'a' and 'b' are real numbers and 'i' is the imaginary unit (√-1).
Conjugates: The conjugate of a complex number a + bi is a - bi.
Rationalizing the Denominator: We use the conjugate of the denominator to eliminate the imaginary part from the denominator of a complex fraction.

Now, let's tackle the simplification:

Multiply each fraction by the conjugate of its denominator:
- For the first fraction:
  - (1 + i)(3 + i) / (3 - i) * (3 + i) / (3 + i)
- For the second fraction:
  - (1 - i)(3 - i) / (3 + i) * (3 - i) / (3 - i)
Expand the products in the numerator and denominator:
- First fraction:
  - [(1 + i)(3 + i)(3 + i)] / [(3 - i)(3 + i)]
- Second fraction:
  - [(1 - i)(3 - i)(3 - i)] / [(3 + i)(3 - i)]
Simplify using the distributive property and remembering that i² = -1:
- First fraction:
  - [(1 + 4i + 2i²)] / (9 - i²)
  - (1 + 4i - 2) / (9 + 1)
  - (-1 + 4i) / 10
- Second fraction:
  - [(1 - 4i + 2i²)] / (9 - i²)
  - (1 - 4i - 2) / (9 + 1)
  - (-1 - 4i) / 10
Subtract the simplified fractions:
- (-1 + 4i) / 10 - (-1 - 4i) / 10
- (-1 + 4i + 1 + 4i) / 10
- (8i) / 10
Simplify the result:
- (4i) / 5

Therefore, the simplified form of the given complex fraction is (4i) / 5.