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Multiplying Complex Numbers: A Look at (9+2i)(9-2i)
This article will explore the multiplication of the complex numbers (9+2i) and (9-2i), illustrating the concept of complex conjugates and their significance in simplifying calculations.
Understanding Complex Numbers
Complex numbers are numbers that consist of a real part and an imaginary part. The imaginary part is a multiple of the imaginary unit i, where i is defined as the square root of -1.
For example, the complex number (9+2i) has a real part of 9 and an imaginary part of 2i.
Complex Conjugates
The complex conjugate of a complex number is formed by changing the sign of its imaginary part. In our case, the complex conjugate of (9+2i) is (9-2i).
The product of a complex number and its conjugate always results in a real number. This property is crucial in simplifying calculations involving complex numbers.
Multiplication of (9+2i) and (9-2i)
To multiply these two complex numbers, we can apply the distributive property or the FOIL method (First, Outer, Inner, Last).
Let's use the FOIL method:
- First: 9 * 9 = 81
- Outer: 9 * -2i = -18i
- Inner: 2i * 9 = 18i
- Last: 2i * -2i = -4i²
Adding these terms together:
81 - 18i + 18i - 4i²
Since i² is defined as -1, we can substitute:
81 - 4(-1) = 81 + 4 = 85
Conclusion
The product of (9+2i) and (9-2i) is 85, a real number. This demonstrates the property of complex conjugates, where multiplying a complex number by its conjugate eliminates the imaginary component, leaving only a real number.
This understanding is essential for various mathematical applications, including solving equations, simplifying expressions, and working with complex functions.