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Understanding Complex Number Multiplication: (5 - 2i)(5 + 2i)
This article will delve into the multiplication of complex numbers, specifically focusing on the expression (5 - 2i)(5 + 2i).
The Basics of Complex Numbers
Complex numbers are numbers that extend the real number system by including the imaginary unit 'i', where i² = -1. A complex number is typically represented as a + bi, where 'a' and 'b' are real numbers, and 'i' is the imaginary unit.
Multiplication of Complex Numbers
Multiplying complex numbers is similar to multiplying binomials, applying the distributive property (or FOIL method).
Here's how we multiply (5 - 2i)(5 + 2i):
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Expand the product: (5 - 2i)(5 + 2i) = 5(5 + 2i) - 2i(5 + 2i)
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Apply the distributive property: = 25 + 10i - 10i - 4i²
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Simplify using i² = -1: = 25 + 10i - 10i + 4
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Combine like terms: = 29
Result and Significance
The product of (5 - 2i)(5 + 2i) is 29, a real number. This result showcases an important property of complex numbers: the product of a complex number and its complex conjugate is always a real number.
Complex conjugate: The complex conjugate of a complex number a + bi is a - bi.
Why is this significant?
This property is particularly useful for simplifying expressions involving complex numbers and for dividing complex numbers. It helps to eliminate the imaginary unit from the denominator of a fraction, resulting in a simpler form.
Conclusion
The multiplication of complex numbers is an essential concept in mathematics. Understanding how to multiply complex numbers, particularly using the complex conjugate, is crucial for solving various problems and simplifying expressions involving complex numbers.