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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.