(3-%E2%80%93-8i).jpeg "(3 + 8i)(3 – 8i)")
Multiplying Complex Numbers: (3 + 8i)(3 – 8i)
This article will explore the multiplication of two complex numbers, specifically (3 + 8i)(3 – 8i). Understanding this process is crucial for working with complex numbers and their applications in various fields such as electrical engineering, physics, and mathematics.
What are 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. Complex numbers extend the real number system, allowing for the representation of solutions to equations that cannot be represented with real numbers alone.
Multiplying Complex Numbers
To multiply complex numbers, we use the distributive property (also known as FOIL method).
(3 + 8i)(3 – 8i) = (3)(3) + (3)(-8i) + (8i)(3) + (8i)(-8i)
This expands to:
9 - 24i + 24i - 64i²
Since i² = -1, we can substitute to simplify the equation:
9 - 24i + 24i + 64
Combining the real and imaginary terms, we get:
73
Conclusion
Therefore, the product of (3 + 8i)(3 – 8i) is 73. This result showcases a key feature of complex numbers. When multiplying a complex number by its conjugate (a complex number with the opposite sign of the imaginary term), the result is always a real number. This is a valuable property for simplifying complex expressions and solving equations.