(4-i)-in-a%2Bbi-form.jpeg "(4+i)(4-i) In A+bi Form")
Multiplying Complex Numbers: (4 + i)(4 - i)
This article explores the multiplication of the complex numbers (4 + i) and (4 - i) and expresses the result in the standard form a + bi.
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.
Multiplying Complex Numbers
To multiply complex numbers, we use the distributive property, similar to multiplying binomials in algebra.
Here's how to multiply (4 + i)(4 - i):
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Expand using the distributive property: (4 + i)(4 - i) = 4(4 - i) + i(4 - i)
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Simplify by distributing: = 16 - 4i + 4i - i²
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Substitute i² with -1: = 16 - 4i + 4i - (-1)
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Combine real and imaginary terms: = 16 + 1
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The final result: = 17
Therefore, the product of (4 + i) and (4 - i) is 17, which can also be written as 17 + 0i in the standard form a + bi.
Key Observations
Notice that the result is a real number (17). This happens because (4 + i) and (4 - i) are complex conjugates.
Complex Conjugates: Two complex numbers are conjugates of each other if they have the same real part but opposite imaginary parts. For example, a + bi and a - bi are complex conjugates.
Important Property: The product of a complex number and its conjugate is always a real number. This property is useful in various applications, including simplifying complex fractions and solving equations.