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):

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

Simplifying: Combine like terms and remember that i² = 1.
 49  35i + 35i  25(1)

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.