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Multiplying Complex Numbers: (4 + 9i)(4 - 9i)
This article will explore the multiplication of complex numbers, specifically focusing on the product of (4 + 9i) and (4 - 9i).
Understanding Complex Numbers
Complex numbers are numbers of 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).
Multiplication of Complex Numbers
To multiply complex numbers, we use the distributive property (also known as FOIL method) similar to multiplying binomials.
Step 1: Apply the Distributive Property
(4 + 9i)(4 - 9i) = 4(4 - 9i) + 9i(4 - 9i)
Step 2: Expand the Multiplication
= 16 - 36i + 36i - 81i²
Step 3: Simplify by Replacing i² with -1
= 16 - 36i + 36i - 81(-1)
Step 4: Combine Real and Imaginary Terms
= 16 + 81
Step 5: Final Result
= 97
Conclusion
Therefore, the product of (4 + 9i) and (4 - 9i) is 97. This result highlights a key property of complex numbers: the product of a complex number and its conjugate (a complex number with the same real part but opposite imaginary part) always results in a real number.