(4%2B2i).jpeg "(2-3i)(4+2i)")
Multiplying Complex Numbers: (2-3i)(4+2i)
This article explores the multiplication of two complex numbers: (2-3i) and (4+2i). We'll use the distributive property, often referred to as FOIL (First, Outer, Inner, Last), to solve this problem.
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 defined as the square root of -1 (i² = -1).
Multiplying (2-3i)(4+2i)
Applying the FOIL method, we multiply each term in the first complex number with each term in the second complex number:
- First: (2)(4) = 8
- Outer: (2)(2i) = 4i
- Inner: (-3i)(4) = -12i
- Last: (-3i)(2i) = -6i²
Now, we combine the terms and simplify using the fact that i² = -1:
8 + 4i - 12i - 6(-1)
Simplifying further:
8 + 4i - 12i + 6
Combining like terms:
14 - 8i
Therefore, the product of (2-3i) and (4+2i) is 14 - 8i.
Conclusion
Multiplying complex numbers involves using the distributive property and understanding the fundamental property of the imaginary unit, i². This process allows us to simplify the product and express it in the standard form of a complex number (a + bi).