(4-2i).jpeg "(4+2i)(4-2i)")
Multiplying Complex Numbers: (4 + 2i)(4 - 2i)
This article explores the multiplication of the complex numbers (4 + 2i) and (4 - 2i). We'll cover the process, the result, and the significance of this type of multiplication.
Understanding 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. The imaginary unit is defined as the square root of -1 (i² = -1).
The Multiplication Process
To multiply complex numbers, we use the distributive property (also known as FOIL - First, Outer, Inner, Last) just like we do with binomials:
(4 + 2i)(4 - 2i) = 4(4) + 4(-2i) + 2i(4) + 2i(-2i)
Simplifying the terms:
= 16 - 8i + 8i - 4i²
The Result and Significance
Since i² = -1, we can substitute:
= 16 - 8i + 8i - 4(-1)
= 16 + 4
= 20
The product of (4 + 2i) and (4 - 2i) is 20.
This result is significant because it demonstrates a key property of complex numbers: the product of a complex number and its conjugate always results in a real number. The conjugate of a complex number is formed by changing the sign of the imaginary part. In this case, the conjugate of (4 + 2i) is (4 - 2i).
This property is essential in various mathematical applications, particularly in simplifying complex expressions and solving equations involving complex numbers.