(3-4i).jpeg "(3+4i)(3-4i)")
Exploring the Multiplication of Complex Numbers: (3 + 4i)(3 - 4i)
In the realm of mathematics, complex numbers are an intriguing extension of real numbers. They are represented 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. Let's delve into the multiplication of complex numbers, focusing on the expression (3 + 4i)(3 - 4i).
Understanding Complex Conjugates
The expression (3 + 4i)(3 - 4i) involves the multiplication of two complex numbers that are complex conjugates. Complex conjugates are pairs of complex numbers that have the same real part but opposite imaginary parts. In our case, the conjugate of (3 + 4i) is (3 - 4i).
The Power of Complex Conjugates
A crucial property of complex conjugates is that their product always results in a real number. This is because the imaginary terms cancel out. Let's see how it works:
(3 + 4i)(3 - 4i) = 3(3 - 4i) + 4i(3 - 4i)
Expanding the product:
= 9 - 12i + 12i - 16i²
Since i² = -1, we can simplify the expression:
= 9 - 16(-1)
= 9 + 16
= 25
Significance of the Result
The product of (3 + 4i) and (3 - 4i) is 25, a real number. This demonstrates the crucial role complex conjugates play in simplifying complex expressions and obtaining real-valued results. This property is widely used in various mathematical and engineering applications.
Key Takeaways
- Complex conjugates are pairs of complex numbers with the same real part but opposite imaginary parts.
- The product of complex conjugates always results in a real number.
- The concept of complex conjugates is essential for simplifying complex expressions and obtaining real-valued solutions.