## Exploring Complex Number Multiplication: (9 + 5i)(9 - 5i)

In the realm of complex numbers, multiplying two complex numbers can seem daunting at first. However, understanding the underlying concepts makes it a straightforward process. Let's delve into the multiplication of (9 + 5i) and (9 - 5i).

### Understanding Complex Numbers

Complex numbers consist of a real part and an imaginary part. The imaginary unit, **i**, is defined as the square root of -1.

**General Form:** A complex number is represented as **a + bi**, where 'a' is the real part and 'b' is the imaginary part.

### Multiplying Complex Numbers

To multiply two complex numbers, we treat them like binomials and apply the distributive property (or FOIL method).

Let's multiply (9 + 5i) and (9 - 5i):

**(9 + 5i)(9 - 5i)**

= 9(9) + 9(-5i) + 5i(9) + 5i(-5i)

= 81 - 45i + 45i - 25i²

Since i² = -1, we can substitute:

= 81 - 25(-1)

= 81 + 25

= **106**

### The Result

The product of (9 + 5i) and (9 - 5i) is **106**, a purely real number. This result is a consequence of multiplying a complex number by its conjugate.

### Conjugates and Their Significance

The conjugate of a complex number **a + bi** is **a - bi**. In our example, (9 - 5i) is the conjugate of (9 + 5i).

Multiplying a complex number by its conjugate always results in a real number. This property is crucial in various mathematical operations, particularly when dealing with complex fractions.

### Conclusion

The multiplication of (9 + 5i) and (9 - 5i) showcases the interesting behavior of complex numbers. While the initial multiplication involves both real and imaginary parts, the final result is a purely real number. This result highlights the importance of conjugates in simplifying complex number operations.