(3-2i).jpeg "(3+2i)(3-2i)")
Multiplying Complex Numbers: (3 + 2i)(3 - 2i)
This article explores the multiplication of complex numbers, specifically focusing on the product of (3 + 2i) and (3 - 2i).
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.
Multiplication Process
To multiply complex numbers, we can use the distributive property (also known as FOIL method):
(3 + 2i)(3 - 2i) = 3(3) + 3(-2i) + 2i(3) + 2i(-2i)
Simplifying the expression:
= 9 - 6i + 6i - 4i²
Since i² = -1, we can substitute it in the equation:
= 9 - 4(-1)
= 9 + 4
= 13
Result and Significance
The product of (3 + 2i) and (3 - 2i) is 13, a real number. This demonstrates an important property of 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, (3 - 2i) is the conjugate of (3 + 2i).
This property is widely used in simplifying complex expressions and solving equations involving complex numbers.