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Multiplying Complex Numbers: (7 + 3i)(7 - 3i)
This article explores the multiplication of two complex numbers: (7 + 3i) and (7 - 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 (i² = -1).
The Multiplication Process
To multiply complex numbers, we use the distributive property (also known as FOIL method) like we would with any binomial multiplication:
(7 + 3i)(7 - 3i) = (7 * 7) + (7 * -3i) + (3i * 7) + (3i * -3i)
Simplifying the expression:
- 49 - 21i + 21i - 9i²
Since i² = -1, we can substitute:
- 49 - 9(-1)
Finally, simplifying the result:
- 49 + 9 = 58
The Result
Therefore, the product of (7 + 3i) and (7 - 3i) is 58.
Important Observation
Notice that the result of multiplying (7 + 3i) and (7 - 3i) is a real number. This is because (7 + 3i) and (7 - 3i) are complex conjugates.
Complex conjugates are pairs of complex numbers that have the same real part and opposite imaginary parts. Multiplying complex conjugates always results in a real number. This property is often used in simplifying complex expressions and solving equations.