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Understanding Complex Number Multiplication: (4 - 2i)(4 + 2i)
This article will explore the multiplication of complex numbers, specifically focusing on the product of (4 - 2i) and (4 + 2i).
Complex Numbers: A Quick Refresher
Complex numbers are numbers of the form a + bi, where 'a' and 'b' are real numbers and 'i' is the imaginary unit, defined as the square root of -1 (i² = -1).
Multiplying Complex Numbers
Multiplying complex numbers is similar to multiplying binomials. We use the FOIL method (First, Outer, Inner, Last) to distribute terms:
- First: (4 * 4) = 16
- Outer: (4 * 2i) = 8i
- Inner: (-2i * 4) = -8i
- Last: (-2i * 2i) = -4i²
Simplifying the Expression
Now, we combine the terms and remember that i² = -1:
16 + 8i - 8i - 4(-1)
Simplifying further:
16 + 4 = 20
The Result
Therefore, the product of (4 - 2i) and (4 + 2i) is 20.
Important Observation
Notice that the result is a real number. This is because (4 - 2i) and (4 + 2i) 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 always cancel out, resulting in a real number.
This concept has significant applications in various fields like electrical engineering and physics.