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Multiplying Complex Numbers: (5 + 4i)(5 - 4i)
This article explores the multiplication of the complex numbers (5 + 4i) and (5 - 4i), illustrating the concept of complex conjugates and their significance.
Understanding Complex Conjugates
The complex conjugate of a complex number is formed by simply changing the sign of the imaginary part. For example, the complex conjugate of (5 + 4i) is (5 - 4i).
Complex conjugates have a unique property: when multiplied together, they result in a real number. This is because the imaginary terms cancel out during multiplication.
The Multiplication Process
Let's multiply (5 + 4i) and (5 - 4i):
(5 + 4i)(5 - 4i) = 5(5 - 4i) + 4i(5 - 4i)
Expanding the terms:
= 25 - 20i + 20i - 16i²
As i² = -1, we can substitute:
= 25 - 16(-1)
= 25 + 16
= 41
Conclusion
As expected, the product of the complex conjugates (5 + 4i) and (5 - 4i) is a real number, 41. This result highlights the key characteristic of complex conjugates and their application in simplifying complex number operations.