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Multiplying Complex Numbers: (9 + 5i)(4 - 2i)
This article will guide you through the process of multiplying complex numbers, specifically focusing on the example of (9 + 5i)(4 - 2i).
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 (also known as FOIL method) similar to multiplying binomials in algebra.
Step 1: Expand the product
(9 + 5i)(4 - 2i) = 9(4) + 9(-2i) + 5i(4) + 5i(-2i)
Step 2: Simplify each term
= 36 - 18i + 20i - 10i²
Step 3: Substitute i² with -1
= 36 - 18i + 20i - 10(-1)
Step 4: Combine real and imaginary terms
= 36 + 10 + (-18i + 20i)
Step 5: Final result
= 46 + 2i
Therefore, the product of (9 + 5i)(4 - 2i) is 46 + 2i.
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 seen as a rotation and scaling operation on this plane.
Conclusion
Multiplying complex numbers involves applying the distributive property and substituting i² with -1. The result is another complex number expressed in the form a + bi. Understanding complex number multiplication is fundamental in various fields like electrical engineering, quantum mechanics, and signal processing.