(4-2i).jpeg "(2+3i)(4-2i)")
Multiplying Complex Numbers: (2 + 3i)(4 - 2i)
This article explores the multiplication of two complex numbers: (2 + 3i) and (4 - 2i).
Understanding Complex Numbers
Complex numbers are numbers that extend the real number system by including the imaginary unit, denoted by 'i', where i² = -1. A complex number is expressed in the form a + bi, where 'a' and 'b' are real numbers.
Multiplication Process
To multiply complex numbers, we use the distributive property, similar to multiplying binomials.
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Expand: (2 + 3i)(4 - 2i) = 2(4 - 2i) + 3i(4 - 2i)
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Distribute: = 8 - 4i + 12i - 6i²
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Simplify using i² = -1: = 8 - 4i + 12i + 6
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Combine real and imaginary terms: = (8 + 6) + (-4 + 12)i
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Final Result: = 14 + 8i
Therefore, the product of (2 + 3i) and (4 - 2i) is 14 + 8i.
Visual Representation
Complex numbers can be visualized as points in a complex plane, where the real part is represented on the horizontal axis and the imaginary part on the vertical axis. Multiplying complex numbers can be interpreted as a rotation and scaling of the complex plane.
Conclusion
This article demonstrated the process of multiplying complex numbers, illustrating the use of the distributive property and the fundamental property of i² = -1. The result, 14 + 8i, represents a complex number that combines a real component of 14 and an imaginary component of 8.