(1-2i).jpeg "(1+2i)(1-2i)")
Understanding Complex Number Multiplication: (1+2i)(1-2i)
This article will explore the multiplication of complex numbers, specifically focusing on the product of (1+2i) and (1-2i). We will delve into the process of multiplication, examine the result, and discuss the significance of this particular product.
What are Complex Numbers?
Complex numbers are numbers that extend the real number system by incorporating the imaginary unit i, where i² = -1. A complex number is generally expressed in the form a + bi, where a and b are real numbers.
Multiplying Complex Numbers
To multiply complex numbers, we treat them as binomials and distribute each term of the first complex number to every term of the second. In essence, we apply the FOIL method (First, Outer, Inner, Last) that we use for multiplying real binomials.
Calculating (1+2i)(1-2i)
Let's break down the multiplication:
(1 + 2i)(1 - 2i) = (1 * 1) + (1 * -2i) + (2i * 1) + (2i * -2i)
Simplifying:
= 1 - 2i + 2i - 4i²
Since i² = -1, we can substitute:
= 1 - 4(-1)
= 1 + 4
= 5
Result and Significance
The product of (1+2i) and (1-2i) is 5, a real number. This result highlights an important concept in complex numbers: The product of a complex number and its conjugate is always a real number.
The conjugate of a complex number a + bi is a - bi. In our example, (1-2i) is the conjugate of (1+2i).
Conclusion
The multiplication of complex numbers like (1+2i)(1-2i) demonstrates the intriguing properties of complex numbers. By understanding the process of multiplication and the concept of conjugates, we gain deeper insight into the world of complex numbers and their applications in various fields, including mathematics, physics, and engineering.