## Understanding Complex Number Multiplication: (5-4i)(5+4i)

This article will explore the multiplication of complex numbers, specifically focusing on the example (5-4i)(5+4i). We'll delve into the process and highlight the interesting result.

### Complex Numbers: A Quick Recap

Complex numbers are numbers that can be expressed in the form **a + bi**, where 'a' and 'b' are real numbers, and 'i' is the imaginary unit, defined as the square root of -1 (i² = -1).

### Multiplying Complex Numbers

To multiply complex numbers, we use the distributive property, similar to multiplying binomials:

**FOIL Method:**

**F**irst: 5 * 5 = 25**O**uter: 5 * 4i = 20i**I**nner: -4i * 5 = -20i**L**ast: -4i * 4i = -16i²

**Simplify:**

- Remember that i² = -1. So, -16i² = -16 * (-1) = 16.
- Combine the real and imaginary terms: 25 + 20i - 20i + 16

**Final Result:**

- (5-4i)(5+4i) =
**41**

### Key Takeaway

Notice that the imaginary terms (20i and -20i) cancel each other out. This is a characteristic of multiplying complex conjugates, which are numbers of the form **a + bi** and **a - bi**.

The result of multiplying complex conjugates is always a **real number**. This is a crucial concept in various mathematical applications.

In our example, the product of (5-4i) and (5+4i) is the real number **41**.