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Exploring Complex Number Multiplication: (6 + 3i)(6 − 3i)
This expression involves the multiplication of two complex numbers. Let's break down the steps and explore the interesting result:
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).
Multiplication Process
To multiply complex numbers, we can use the distributive property (or the FOIL method):
(6 + 3i)(6 − 3i) = 6(6) + 6(−3i) + 3i(6) + 3i(−3i)
Simplifying the Expression
Let's simplify the terms:
- 36 - 18i + 18i - 9i²
Since i² = -1, we can substitute:
- 36 - 18i + 18i + 9
The Result
Combining like terms, we arrive at the final result:
(6 + 3i)(6 − 3i) = 45
Key Observations
- The result is a real number. This is because the imaginary terms cancel each other out.
- The expression (6 + 3i) and (6 − 3i) are conjugates of each other. Conjugates always result in a real number when multiplied.
Significance
This calculation demonstrates a crucial concept in complex numbers: conjugates provide a way to eliminate imaginary terms and obtain real number results. This property is widely used in various applications, including solving equations, simplifying expressions, and working with complex functions.