(-3+3i)/(-5-4i)

Jasonbradley

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Jun 16, 2024

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Simplifying Complex Fractions: A Step-by-Step Guide

In mathematics, complex numbers are a fundamental concept with applications across various fields. A complex number consists of a real part and an imaginary part, represented as a + bi, where a and b are real numbers and i is the imaginary unit, defined as the square root of -1.

This article will guide you through the process of simplifying a complex fraction, specifically focusing on the expression (-3 + 3i) / (-5 - 4i).

Understanding the Process

To simplify a complex fraction, we need to eliminate the imaginary term from the denominator. This is achieved by multiplying both the numerator and denominator by the complex conjugate of the denominator.

Step-by-Step Solution

1. Find the Complex Conjugate: The complex conjugate of a complex number is obtained by changing the sign of the imaginary part. The complex conjugate of (-5 - 4i) is (-5 + 4i).

2. Multiply by the Complex Conjugate: Multiply both the numerator and denominator of the fraction by the complex conjugate:

   (-3 + 3i) / (-5 - 4i) * (-5 + 4i) / (-5 + 4i)

3. Expand: Apply the distributive property (FOIL method) to expand both the numerator and denominator:

   • Numerator: (-3 + 3i)(-5 + 4i) = 15 - 12i - 15i + 12i²
   • Denominator: (-5 - 4i)(-5 + 4i) = 25 - 20i + 20i - 16i²

4. Simplify: Remember that i² = -1. Substitute this value and simplify the expressions:

   • Numerator: 15 - 12i - 15i - 12 = 3 - 27i
   • Denominator: 25 - 20i + 20i + 16 = 41

5. Final Result: Express the simplified complex number in the form a + bi:

   (3 - 27i) / 41 = 3/41 - 27/41i

Conclusion

By multiplying the numerator and denominator by the complex conjugate of the denominator, we successfully eliminated the imaginary term from the denominator and simplified the complex fraction. This process allows us to express the complex fraction in a standard form, making it easier to work with and understand its properties.