(2-6i).jpeg "(2+6i)(2-6i)")
Exploring the Multiplication of Complex Numbers: (2 + 6i)(2 - 6i)
This article delves into the multiplication of complex numbers, specifically focusing on the product of (2 + 6i) and (2 - 6i). We'll explore the process of multiplication and uncover the intriguing results.
Understanding Complex Numbers
Before we dive into the multiplication, let's briefly recap complex numbers. They are numbers of the form a + bi, where:
- a is the real part, a regular number.
- b is the imaginary part, multiplied by the imaginary unit i, where i² = -1.
Multiplying Complex Numbers
To multiply complex numbers, we can use the distributive property, similar to multiplying binomials in algebra.
(2 + 6i)(2 - 6i) = 2(2 - 6i) + 6i(2 - 6i)
Expanding the terms:
= 4 - 12i + 12i - 36i²
Remember, i² = -1. Substituting this value:
= 4 - 12i + 12i - 36(-1)
Combining the real and imaginary terms:
= 4 + 36 = 40
The Significance of the Result
The result, 40, is a real number. This outcome reveals a significant property of complex numbers. When multiplying a complex number with its conjugate (the same number with the opposite sign for the imaginary part), the result is always a real number.
In general:
(a + bi)(a - bi) = a² - (bi)² = a² + b²
The conjugate of (2 + 6i) is (2 - 6i), which explains why the multiplication resulted in a real number.
Conclusion
Through this exploration, we've seen how multiplying complex numbers can lead to surprising results. We discovered that multiplying a complex number by its conjugate always yields a real number. This concept has important applications in various fields, including mathematics, physics, and engineering.