Verification Of Commutative Property Of Addition With Rational Numbers

In the realm of mathematics, the beauty of numbers lies not only in their individual values but also in the relationships and properties that govern their behavior. Among these fundamental properties, the commutative property holds a significant place, particularly when dealing with operations like addition. This property, simply put, states that the order in which we add numbers does not affect the sum. In this article, we delve deep into the commutative property of addition, specifically in the context of rational numbers. We aim to verify the given equation: (a) ${ \frac{9}{-14} + \frac{17}{-21} = \frac{17}{-21} + \frac{9}{-14} }$, providing a comprehensive understanding and step-by-step verification process.

Before we dive into the verification, it's crucial to understand what rational numbers are. A rational number is any number that can be expressed as the quotient or fraction ${ \frac{p}{q} }$ of two integers, where ${ p }$ is the numerator and ${ q }$ is the denominator, and ${ q }$ is not equal to zero. Examples of rational numbers include ${ \frac{1}{2} }$, ${ \frac{-3}{4} }$, 5 (which can be written as ${ \frac{5}{1} }$), and even terminating or repeating decimals (since they can be converted into fractions). In our equation, ${ \frac{9}{-14} }$ and ${ \frac{17}{-21} }$ are both rational numbers.

The commutative property of addition is a fundamental concept in arithmetic. It asserts that for any two numbers, say ${ a }$ and ${ b }$, the sum ${ a + b }$ is equal to the sum ${ b + a }$. In mathematical notation, this is expressed as: ${ a + b = b + a }$ This property is not limited to integers; it extends to all real numbers, including rational numbers. The commutative property simplifies many calculations and is a cornerstone of algebraic manipulations. It allows us to rearrange terms in an expression without altering the outcome, which is particularly useful when dealing with complex equations or expressions.

To verify the given equation, we need to evaluate both sides of the equation separately and then compare the results. If both sides yield the same value, the commutative property is verified for the given rational numbers.

Step 1: Evaluate the Left-Hand Side (LHS)

The left-hand side (LHS) of the equation is: ${ \frac{9}{-14} + \frac{17}{-21} }$ To add these fractions, we need to find a common denominator. The least common multiple (LCM) of 14 and 21 is 42. We then convert each fraction to an equivalent fraction with the denominator 42. ${ \frac{9}{-14} = \frac{9 \times -3}{-14 \times -3} = \frac{-27}{42} }$ ${ \frac{17}{-21} = \frac{17 \times -2}{-21 \times -2} = \frac{-34}{42} }$ Now, we can add the fractions: ${ \frac{-27}{42} + \frac{-34}{42} = \frac{-27 + (-34)}{42} = \frac{-61}{42} }$ So, the LHS evaluates to ${ \frac{-61}{42} }$.

Step 2: Evaluate the Right-Hand Side (RHS)

The right-hand side (RHS) of the equation is: ${ \frac{17}{-21} + \frac{9}{-14} }$ Notice that this is the same as the LHS but with the order of the fractions reversed. We already found the equivalent fractions with a common denominator in the previous step: ${ \frac{17}{-21} = \frac{-34}{42} }$ ${ \frac{9}{-14} = \frac{-27}{42} }$ Now, we add the fractions: ${ \frac{-34}{42} + \frac{-27}{42} = \frac{-34 + (-27)}{42} = \frac{-61}{42} }$ Thus, the RHS also evaluates to ${ \frac{-61}{42} }$.

Step 3: Compare LHS and RHS

We found that:

• LHS = ${ \frac{-61}{42} }$
• RHS = ${ \frac{-61}{42} }$ Since LHS = RHS, we have verified that: ${ \frac{9}{-14} + \frac{17}{-21} = \frac{17}{-21} + \frac{9}{-14} }$

The verification of the commutative property for the given rational numbers demonstrates a fundamental principle in mathematics. This property is not just a theoretical concept; it has practical implications in various mathematical operations and problem-solving scenarios. For instance, when simplifying complex expressions, we can rearrange terms using the commutative property to make the calculation easier. In algebra, this property is essential for manipulating equations and solving for unknowns. Moreover, the commutative property serves as a building block for understanding more advanced mathematical concepts.

When working with rational numbers and the commutative property, there are a few common mistakes that students often make. One frequent error is forgetting to find a common denominator before adding fractions. This can lead to incorrect results. Always ensure that the fractions have the same denominator before performing addition or subtraction. Another mistake is mishandling negative signs. It's crucial to pay close attention to the signs of the numerators and denominators when converting fractions and performing arithmetic operations. A simple sign error can significantly alter the outcome. Additionally, some students may confuse the commutative property with other properties, such as the associative or distributive properties. It's important to understand the distinct characteristics of each property to apply them correctly.

In conclusion, we have successfully verified that ${ \frac{9}{-14} + \frac{17}{-21} = \frac{17}{-21} + \frac{9}{-14} }$, demonstrating the commutative property of addition for these specific rational numbers. This exercise not only reinforces our understanding of the commutative property but also highlights the importance of careful arithmetic and attention to detail when working with fractions. The commutative property is a fundamental concept in mathematics, and its verification underscores its validity and applicability in various mathematical contexts. By understanding and applying such properties, we can enhance our mathematical skills and problem-solving abilities.