(2-5i).jpeg "(4+3i)(2-5i)")
Multiplying Complex Numbers: (4 + 3i)(2 - 5i)
This article will explore the process of multiplying two complex numbers, specifically (4 + 3i)(2 - 5i).
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 (i² = -1).
Multiplying Complex Numbers
To multiply complex numbers, we use the distributive property, similar to how we multiply binomials in algebra.
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Distribute: (4 + 3i)(2 - 5i) = 4(2 - 5i) + 3i(2 - 5i)
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Simplify: = 8 - 20i + 6i - 15i²
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Substitute i² with -1: = 8 - 20i + 6i - 15(-1)
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Combine real and imaginary terms: = (8 + 15) + (-20 + 6)i
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Final result: = 23 - 14i
Therefore, the product of (4 + 3i) and (2 - 5i) is 23 - 14i.
Visualizing Complex Numbers
Complex numbers can be visualized on a complex plane, where the horizontal axis represents the real part and the vertical axis represents the imaginary part. Multiplying complex numbers can be viewed as a geometric transformation on this plane.
Conclusion
Multiplying complex numbers involves applying the distributive property and simplifying the expression by substituting i² with -1. The resulting product is also a complex number, expressed in the form a + bi. Understanding complex number multiplication is crucial in various fields like electrical engineering, physics, and mathematics.