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Understanding Complex Number Multiplication: (3 + 8i)(3 - 8i)
This article will explore the multiplication of complex numbers, specifically focusing on the example of (3 + 8i)(3 - 8i).
Complex Numbers: A Quick Review
Complex numbers are numbers that can be expressed in the form a + bi, where:
- a and b are real numbers,
- i is the imaginary unit, defined as the square root of -1 (i² = -1).
Multiplication of Complex Numbers
To multiply complex numbers, we follow the same distributive property as with real numbers.
Let's break down the multiplication of (3 + 8i)(3 - 8i):
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Expand using FOIL (First, Outer, Inner, Last):
(3 + 8i)(3 - 8i) = 3(3) + 3(-8i) + 8i(3) + 8i(-8i)
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Simplify:
= 9 - 24i + 24i - 64i²
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Substitute i² with -1:
= 9 - 64(-1)
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Combine real terms:
= 9 + 64
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Final result:
= 73
The Significance of the Result
The product (3 + 8i)(3 - 8i) results in a real number (73). This is because the original complex numbers, (3 + 8i) and (3 - 8i), are complex conjugates.
Complex conjugates are pairs of complex numbers that have the same real part but opposite imaginary parts. When complex conjugates are multiplied, the imaginary terms cancel out, leaving only a real number.
This concept is crucial in various mathematical applications, particularly in areas like:
- Solving quadratic equations
- Simplifying complex fractions
- Working with electrical circuits
Conclusion
Understanding complex number multiplication, especially the interaction between complex conjugates, is essential for various mathematical and scientific applications. The example of (3 + 8i)(3 - 8i) demonstrates how multiplying complex conjugates results in a real number, highlighting the importance of this concept.