(5%2B3i).jpeg "(5-3i)(5+3i)")
Understanding Complex Numbers and Multiplication
This article will explore the multiplication of complex numbers, focusing on the example (5-3i)(5+3i).
Complex Numbers
Complex numbers are numbers that can be expressed in the form a + bi, where:
- a and b are real numbers.
- i is the imaginary unit, defined as the square root of -1 (i² = -1).
Understanding the Example
The expression (5-3i)(5+3i) involves multiplying two complex numbers. This multiplication can be seen as a binomial expansion, similar to how we multiply (x-y)(x+y) in algebra.
Expanding the Expression
Let's use the distributive property (or FOIL method) to expand the expression:
(5-3i)(5+3i) = 5(5) + 5(3i) - 3i(5) - 3i(3i)
Simplifying, we get:
= 25 + 15i - 15i - 9i²
Since i² = -1, we can substitute:
= 25 + 15i - 15i + 9
The Final Result
Combining the real and imaginary terms, we obtain the final result:
(5-3i)(5+3i) = 34
Important Observation
Notice that the result of the multiplication is a real number (34). This is a common occurrence when multiplying complex conjugates.
Complex Conjugate
The complex conjugate of a complex number a + bi is a - bi. In our example, (5-3i) and (5+3i) are complex conjugates.
The product of a complex number and its conjugate always results in a real number. This property is often used in simplifying expressions involving complex numbers.