(4-2i).jpeg "(1+5i)(4-2i)")
Multiplying Complex Numbers: (1+5i)(4-2i)
This article will explore the process of multiplying two complex numbers: (1 + 5i) and (4 - 2i). We'll delve into the steps involved and the resulting complex number.
Understanding Complex Numbers
A complex number is a number 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 (i.e., i² = -1).
Multiplication Process
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Distribution: We multiply each term of the first complex number by each term of the second complex number, similar to multiplying binomials.
(1 + 5i)(4 - 2i) = (1 * 4) + (1 * -2i) + (5i * 4) + (5i * -2i)
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Simplifying: We perform the multiplication and simplify the terms.
= 4 - 2i + 20i - 10i²
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Substituting i²: Remember that i² = -1. Substitute this into the equation.
= 4 - 2i + 20i - 10(-1)
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Combining Real and Imaginary Terms: Combine the real terms and the imaginary terms separately.
= (4 + 10) + (-2 + 20)i
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Final Result: The product of the two complex numbers is:
= 14 + 18i
Therefore, (1 + 5i)(4 - 2i) = 14 + 18i.
Conclusion
Multiplying complex numbers involves applying the distributive property and simplifying the expression by substituting i² with -1. The result of this multiplication is another complex number, in this case, 14 + 18i.