(8-2i).jpeg "(8+2i)(8-2i)")
Multiplying Complex Numbers: (8+2i)(8-2i)
This article will walk through the process of multiplying the complex numbers (8+2i) and (8-2i), and explain the significance of the result.
Understanding Complex Numbers
Complex numbers are numbers of the form a + bi, where a and b are real numbers, and i is the imaginary unit, defined as the square root of -1.
Multiplying Complex Numbers
To multiply complex numbers, we use the distributive property, just as we would with real numbers:
(8 + 2i)(8 - 2i) = 8(8 - 2i) + 2i(8 - 2i)
Expanding this expression gives us:
= 64 - 16i + 16i - 4i²
Since i² = -1, we can substitute to simplify:
= 64 - 4(-1)
= 64 + 4
= 68
The Significance of the Result
Notice that the product of (8+2i) and (8-2i) is a real number (68). This is not a coincidence. The complex numbers (8+2i) and (8-2i) are complex conjugates.
Complex conjugates are pairs of complex numbers that have the same real part but opposite imaginary parts. When you multiply complex conjugates, the imaginary terms always cancel out, leaving only a real number.
In Conclusion
Multiplying (8+2i) and (8-2i) results in 68. This demonstrates the concept of complex conjugates and how their multiplication always produces a real number.