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Exploring the Product of Complex Conjugates: (5 - 7i)(5 + 7i)
In the realm of complex numbers, understanding the concept of conjugates is crucial. The conjugate of a complex number is formed by simply changing the sign of its imaginary component. For instance, the conjugate of 5 - 7i is 5 + 7i.
Let's delve into the product of these complex conjugates: (5 - 7i)(5 + 7i).
The Significance of Complex Conjugates
The product of complex conjugates always results in a real number. This property holds a significant role in various mathematical operations, particularly in simplifying expressions involving complex numbers and finding the modulus of a complex number.
Expanding the Product
To determine the product of (5 - 7i)(5 + 7i), we expand it using the distributive property (FOIL method):
(5 - 7i)(5 + 7i) = 5(5 + 7i) - 7i(5 + 7i) = 25 + 35i - 35i - 49i²
Simplifying the Expression
Remembering that i² = -1, we can simplify the expression further:
25 + 35i - 35i - 49i² = 25 - 49(-1) = 25 + 49 = 74
The Result
Therefore, the product of (5 - 7i)(5 + 7i) equals 74, which is a real number, as expected.
Conclusion
This example demonstrates the fundamental property of complex conjugates: their product always yields a real number. This property is essential for various operations involving complex numbers and serves as a valuable tool in simplifying expressions and finding magnitudes.