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Multiplying Complex Numbers: (2 - 3i)(2 + 3i)
This article explores the multiplication of complex numbers, specifically the product of (2 - 3i) and (2 + 3i).
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 of Complex Numbers
To multiply two complex numbers, we use the distributive property, similar to multiplying binomials in algebra.
(2 - 3i)(2 + 3i)
Step 1: Expand using the distributive property
(2 - 3i)(2 + 3i) = 2(2 + 3i) - 3i(2 + 3i)
Step 2: Simplify by multiplying
= 4 + 6i - 6i - 9i²
Step 3: Substitute i² with -1
= 4 + 6i - 6i - 9(-1)
Step 4: Combine like terms
= 4 + 9 = 13
Result
Therefore, the product of (2 - 3i) and (2 + 3i) is 13.
Interesting Observation
The result is a real number. This is not a coincidence. The complex numbers (2 - 3i) and (2 + 3i) are called complex conjugates. The product of any two complex conjugates always results in a real number.
Conclusion
This example demonstrates how to multiply complex numbers and highlights the important concept of complex conjugates. Understanding these concepts is essential for working with complex numbers in various mathematical fields.