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Exploring the Product of Complex Conjugates: (3 + 5i)(3 - 5i)
In the realm of complex numbers, understanding the concept of conjugates is crucial. Complex conjugates are pairs of numbers that share the same real part but differ in the sign of their imaginary part.
Let's delve into the product of the complex conjugate pair (3 + 5i) and (3 - 5i):
1. Understanding Complex Conjugates
The complex conjugate of a number in the form of a + bi is a - bi. This simply means changing the sign of the imaginary component. In our case, (3 + 5i) and (3 - 5i) are complex conjugates.
2. The Product of Complex Conjugates
When multiplying complex conjugates, we can utilize the difference of squares pattern: (a + b)(a - b) = a² - b².
Applying this pattern to our problem:
(3 + 5i)(3 - 5i) = 3² - (5i)²
3. Simplifying the Expression
Remember that i² = -1. Substituting this value:
3² - (5i)² = 9 - 25(-1) = 9 + 25 = 34
4. The Result
Therefore, the product of the complex conjugates (3 + 5i) and (3 - 5i) is 34. Notice that the result is a real number, which is a characteristic of multiplying complex conjugates.
Key Takeaways:
- Complex conjugates are pairs that differ only in the sign of their imaginary parts.
- The product of complex conjugates always results in a real number.
- This property is often used to simplify expressions involving complex numbers.
This example illustrates the powerful interplay between complex numbers and their conjugates, providing a foundation for further exploration in the world of complex arithmetic.