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Multiplying Complex Numbers: (3 + 2i)(5 - 3i)
This article will guide you through the process of multiplying two complex numbers, (3 + 2i) and (5 - 3i).
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 (i² = -1).
Multiplication Process
To multiply two complex numbers, we use the distributive property, just like multiplying binomials.
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Expand the product: (3 + 2i)(5 - 3i) = 3(5 - 3i) + 2i(5 - 3i)
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Distribute: = 15 - 9i + 10i - 6i²
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Simplify using i² = -1: = 15 - 9i + 10i + 6
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Combine real and imaginary terms: = (15 + 6) + (-9 + 10)i
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Final Result: = 21 + i
Therefore, the product of (3 + 2i) and (5 - 3i) is 21 + i.
Visualizing the Multiplication
The multiplication of complex numbers can be visualized as a rotation and scaling operation in the complex plane. This is because the multiplication process involves both real and imaginary components.
Conclusion
Multiplying complex numbers involves using the distributive property and simplifying using the definition of the imaginary unit (i² = -1). The final result is another complex number with both real and imaginary parts.