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Understanding Complex Number Multiplication: (5-4i)(5+4i)
This article will explore the multiplication of complex numbers, specifically focusing on the example (5-4i)(5+4i). We'll delve into the process and highlight the interesting result.
Complex Numbers: A Quick Recap
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).
Multiplying Complex Numbers
To multiply complex numbers, we use the distributive property, similar to multiplying binomials:
- FOIL Method:
- First: 5 * 5 = 25
- Outer: 5 * 4i = 20i
- Inner: -4i * 5 = -20i
- Last: -4i * 4i = -16i²
- Simplify:
- Remember that i² = -1. So, -16i² = -16 * (-1) = 16.
- Combine the real and imaginary terms: 25 + 20i - 20i + 16
- Final Result:
- (5-4i)(5+4i) = 41
Key Takeaway
Notice that the imaginary terms (20i and -20i) cancel each other out. This is a characteristic of multiplying complex conjugates, which are numbers of the form a + bi and a - bi.
The result of multiplying complex conjugates is always a real number. This is a crucial concept in various mathematical applications.
In our example, the product of (5-4i) and (5+4i) is the real number 41.