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Simplifying Complex Number Multiplication: (5 + 7i)(5 - 7i)
This article explores the multiplication of two complex numbers, (5 + 7i) and (5 - 7i), and demonstrates how the result leads to a real number.
Understanding Complex Numbers
Complex numbers are numbers of 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).
Multiplying Complex Numbers
To multiply complex numbers, we use the distributive property, similar to multiplying binomials.
(5 + 7i)(5 - 7i) = 5(5 - 7i) + 7i(5 - 7i)
Expanding the product:
= 25 - 35i + 35i - 49i²
Notice that the terms with 'i' cancel each other out. Replacing i² with -1, we get:
= 25 - 49(-1)
The Result
Simplifying the expression:
= 25 + 49
= 74
Therefore, the product of (5 + 7i) and (5 - 7i) is 74, a real number.
Key Takeaway
The multiplication of complex numbers of the form (a + bi) and (a - bi) always results in a real number. This is because the imaginary terms cancel out, leaving only the real components. This pattern is known as the difference of squares, which can be generalized as:
(a + bi)(a - bi) = a² - (bi)² = a² + b²