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Multiplying Complex Numbers: (5 + 3i)(5 - 3i)
This article will demonstrate how to multiply two complex numbers: (5 + 3i) and (5 - 3i).
Understanding Complex Numbers
Complex numbers are numbers that can be expressed in 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
To multiply complex numbers, we use the distributive property, similar to multiplying binomials in algebra.
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FOIL Method: We expand the product using the First, Outer, Inner, and Last terms:
- First: 5 * 5 = 25
- Outer: 5 * (-3i) = -15i
- Inner: 3i * 5 = 15i
- Last: 3i * (-3i) = -9i²
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Simplify: Combine like terms and remember that i² = -1:
- 25 - 15i + 15i - 9(-1)
- 25 + 9 = 34
Result
Therefore, the product of (5 + 3i) and (5 - 3i) is 34.
Noteworthy Observation
The result, 34, is a real number. This is because (5 + 3i) and (5 - 3i) 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, resulting in a real number.