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Understanding Complex Numbers Multiplication: (4+i)(4-i)
This article will explore the multiplication of complex numbers, specifically focusing on the expression (4+i)(4-i).
What are Complex Numbers?
Complex numbers are numbers that extend the real number system by including the imaginary unit i, where i² = -1. They are expressed in the form a + bi, where a and b are real numbers.
Multiplying Complex Numbers
Multiplying complex numbers follows the distributive property, similar to multiplying binomials. Here's how it works:
(4+i)(4-i) = 4(4-i) + i(4-i)
Expanding the terms:
(4+i)(4-i) = 16 - 4i + 4i - i²
Remember, i² = -1:
(4+i)(4-i) = 16 - (-1)
Simplifying the expression:
(4+i)(4-i) = 17
Significance of the Result
The result 17 is a real number. This is a significant observation. When multiplying a complex number by its conjugate (formed by changing the sign of the imaginary part), the product always results in a real number.
Conjugates and Their Applications
The conjugate of a complex number a + bi is a - bi. This concept is crucial in various applications:
- Rationalizing denominators: Conjugates are used to eliminate complex numbers from the denominator of fractions, simplifying expressions.
- Finding magnitudes: The magnitude (or modulus) of a complex number a + bi is found by multiplying the complex number by its conjugate and then taking the square root: √((a + bi)(a - bi))
Conclusion
Understanding the multiplication of complex numbers, especially the concept of conjugates, provides a strong foundation for working with complex numbers in various mathematical contexts. The example (4+i)(4-i) = 17 highlights how the multiplication of a complex number with its conjugate leads to a real number, demonstrating a significant property in complex number arithmetic.