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Exploring the Multiplication of Complex Numbers: (7 + 5i)(7 - 5i)
This article delves into the multiplication of complex numbers, focusing on the specific example of (7 + 5i)(7 - 5i). We will explore the process and the interesting outcome that arises.
Understanding Complex Numbers
A complex number is a number 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.
Multiplying Complex Numbers
The multiplication of complex numbers follows the distributive property, much like multiplying binomials in algebra.
Let's break down the multiplication of (7 + 5i)(7 - 5i):
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FOIL Method: We can use the FOIL method (First, Outer, Inner, Last) to expand the product:
- First: 7 * 7 = 49
- Outer: 7 * -5i = -35i
- Inner: 5i * 7 = 35i
- Last: 5i * -5i = -25i²
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Simplifying: Combine like terms and remember that i² = -1.
- 49 - 35i + 35i - 25(-1)
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Final Result: The imaginary terms cancel out, leaving us with a real number:
- 49 + 25 = 74
The Significance of the Outcome
The multiplication of (7 + 5i) and its conjugate (7 - 5i) results in a purely real number. This is a general property of complex conjugates. The product of a complex number and its conjugate always yields a real number. This property is particularly useful in simplifying expressions and solving equations involving complex numbers.
In Summary
The multiplication of (7 + 5i)(7 - 5i) demonstrates a key concept in complex number arithmetic: the product of a complex number and its conjugate is a real number. This property simplifies calculations and has applications in various mathematical and scientific fields.