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Exploring the Complex Multiplication: (4 + 5i)(x + yi)
This article delves into the multiplication of complex numbers, specifically exploring the product of (4 + 5i) and (x + yi). We will unravel the process, understand the resulting complex number, and highlight key takeaways.
Understanding Complex Numbers
Complex numbers are a fundamental concept in mathematics, extending the real number system by including the imaginary unit 'i', where i² = -1. They are represented in the form a + bi, where 'a' and 'b' are real numbers, and 'i' is the imaginary unit.
Multiplying Complex Numbers
Multiplying complex numbers follows the distributive property, similar to multiplying binomials. We expand the expression:
(4 + 5i)(x + yi) = 4(x + yi) + 5i(x + yi)
Applying the distributive property further:
= 4x + 4yi + 5ix + 5iy²
Remember that i² = -1, so we substitute:
= 4x + 4yi + 5ix + 5(-1)
Combining real and imaginary terms:
= (4x - 5) + (4y + 5x)i
Therefore, the product of (4 + 5i) and (x + yi) is (4x - 5) + (4y + 5x)i.
Key Takeaways
- Real and Imaginary Components: The result of multiplying complex numbers is also a complex number with both a real and imaginary component.
- Distributive Property: The multiplication process relies on the distributive property, similar to multiplying binomials.
- Understanding i²: The key to simplifying the expression lies in recognizing and substituting i² = -1.
By understanding these concepts, you can confidently multiply complex numbers and interpret their results in the complex plane.