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Exploring Complex Numbers: (4 + 5i)(4 – 5i)
This expression involves multiplying two complex numbers: (4 + 5i) and (4 – 5i). Let's break down the process and explore the significance of the result.
Understanding 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.
Multiplication of Complex Numbers
To multiply complex numbers, we use the distributive property, similar to multiplying binomials:
(4 + 5i)(4 – 5i) = 4(4 – 5i) + 5i(4 – 5i)
Expanding the terms:
= 16 - 20i + 20i - 25i²
Since i² = -1, we can substitute:
= 16 - 20i + 20i + 25
The Result
Combining the real and imaginary terms:
= 41
Notice that the imaginary terms cancel out, leaving us with a purely real number. This is a common pattern when multiplying complex conjugates.
Complex Conjugates
Two complex numbers are conjugates if they have the same real part but opposite imaginary parts. In our example, (4 + 5i) and (4 – 5i) are complex conjugates.
Key takeaway: Multiplying a complex number by its conjugate always results in a real number. This property is frequently used in various mathematical and engineering applications to simplify expressions and manipulate equations involving complex numbers.