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Multiplying Complex Numbers: (5 + 6i)(-4 + 7i)
This article will guide you through multiplying the complex numbers (5 + 6i) and (-4 + 7i).
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 complex numbers, we use the distributive property, just like with real numbers.
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Expand the product: (5 + 6i)(-4 + 7i) = 5(-4 + 7i) + 6i(-4 + 7i)
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Distribute: = -20 + 35i - 24i + 42i²
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Simplify using i² = -1: = -20 + 35i - 24i - 42
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Combine real and imaginary terms: = (-20 - 42) + (35 - 24)i
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Final Result: = -62 + 11i
Therefore, the product of (5 + 6i) and (-4 + 7i) is -62 + 11i.
Visualizing the Result
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 seen as a rotation and scaling operation on this plane.
By visualizing the multiplication of (5 + 6i) and (-4 + 7i), you can see how the resulting complex number (-62 + 11i) is positioned on the complex plane relative to the original numbers.
Conclusion
Multiplying complex numbers involves applying the distributive property and simplifying using the fact that i² = -1. The process is similar to multiplying binomials, and the result is another complex number. By understanding the properties of complex numbers and their operations, we can effectively work with these numbers in various mathematical contexts.