(3%2B2i).jpeg "(3-2i)(3+2i)")
Multiplying Complex Numbers: (3 - 2i)(3 + 2i)
This article will demonstrate how to multiply two complex numbers, specifically (3 - 2i)(3 + 2i).
Understanding Complex Numbers
Complex numbers are numbers that consist of a real part and an imaginary part. The imaginary part is denoted by the imaginary unit 'i', where i² = -1. Complex numbers are written in the form a + bi, where 'a' is the real part and 'b' is the imaginary part.
The Multiplication Process
To multiply complex numbers, we use the distributive property, just like we do with binomials.
Step 1: Expand the Expression
(3 - 2i)(3 + 2i) = 3(3 + 2i) - 2i(3 + 2i)
Step 2: Distribute
= 9 + 6i - 6i - 4i²
Step 3: Simplify
= 9 - 4i²
Step 4: Substitute i² with -1
= 9 - 4(-1)
Step 5: Final Result
= 9 + 4 = 13
Conclusion
Therefore, the product of (3 - 2i) and (3 + 2i) is 13, a real number. This result highlights an important property of complex numbers: The product of a complex number and its conjugate is always a real number. In this case, (3 - 2i) and (3 + 2i) are conjugates of each other.