(2+2i)/(5+4i)

Jasonbradley

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Jun 16, 2024

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Dividing Complex Numbers: A Step-by-Step Guide

Dividing complex numbers can seem daunting, but it's actually quite straightforward. We'll explore how to divide (2 + 2i) by (5 + 4i) using a technique called conjugate multiplication.

Understanding Conjugate Multiplication

The key to dividing complex numbers is to eliminate the imaginary part from the denominator. This is achieved by multiplying both the numerator and denominator by the conjugate of the denominator.

The conjugate of a complex number is formed by simply changing the sign of the imaginary part. For example, the conjugate of (5 + 4i) is (5 - 4i).

Step-by-Step Solution

1. Identify the conjugate: The conjugate of the denominator, (5 + 4i), is (5 - 4i).

2. Multiply the numerator and denominator by the conjugate:

   (2 + 2i) / (5 + 4i) * (5 - 4i) / (5 - 4i)
   

3. Expand the multiplication:

   (10 - 8i + 10i - 8i²) / (25 - 16i²)
   

4. Simplify using i² = -1:

   (10 + 2i + 8) / (25 + 16)
   

5. Combine real and imaginary terms:

   (18 + 2i) / 41
   

6. Express in standard form:

   18/41 + 2/41 i 
   

Therefore, (2 + 2i) / (5 + 4i) = 18/41 + 2/41 i.

Conclusion

Dividing complex numbers using conjugate multiplication is a simple process that allows you to express the result in the standard form of a complex number (a + bi). Remember, the key is to multiply both the numerator and denominator by the conjugate of the denominator.