## Multiplying Complex Numbers: (7 + 3i)(7 - 3i)

This article explores the multiplication of two complex numbers: (7 + 3i) and (7 - 3i).

### Understanding Complex Numbers

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).

### The Multiplication Process

To multiply complex numbers, we use the distributive property (also known as FOIL method) like we would with any binomial multiplication:

**(7 + 3i)(7 - 3i) = (7 * 7) + (7 * -3i) + (3i * 7) + (3i * -3i)**

Simplifying the expression:

**49 - 21i + 21i - 9i²**

Since i² = -1, we can substitute:

**49 - 9(-1)**

Finally, simplifying the result:

**49 + 9 = 58**

### The Result

Therefore, the product of (7 + 3i) and (7 - 3i) is **58**.

### Important Observation

Notice that the result of multiplying (7 + 3i) and (7 - 3i) is a **real number**. This is because (7 + 3i) and (7 - 3i) are **complex conjugates**.

**Complex conjugates** are pairs of complex numbers that have the same real part and opposite imaginary parts. Multiplying complex conjugates always results in a real number. This property is often used in simplifying complex expressions and solving equations.