Find The Additive Inverse Of The Following Rational Numbers: (a) -3/7 (b) 0 (c) -5 4/7 (d) -31/51 (e) -27/-83 (f) -2/5 (g) -(-7/12)
In the realm of mathematics, understanding additive inverses is a cornerstone concept, particularly when dealing with rational numbers. The additive inverse, also known as the opposite or negation, is a number that, when added to the original number, results in a sum of zero. This concept is fundamental in various mathematical operations and problem-solving scenarios. In this comprehensive guide, we will delve into the intricacies of finding additive inverses, exploring different types of numbers, and providing detailed explanations with examples. This article will explore the additive inverses of various rational numbers, ensuring a clear and thorough understanding of this crucial mathematical principle. Whether you're a student grappling with the basics or a math enthusiast seeking a refresher, this guide aims to illuminate the path to mastering additive inverses.
What is an Additive Inverse?
At its core, the additive inverse of a number is the value that, when added to the original number, yields zero. This property is formally known as the additive inverse property, which states that for any number 'a', there exists a number '-a' such that a + (-a) = 0. The number '-a' is the additive inverse of 'a', and vice versa. This concept is not confined to integers; it extends to rational numbers, real numbers, and even complex numbers. Understanding additive inverses is crucial for simplifying expressions, solving equations, and performing various mathematical operations. For instance, in algebra, additive inverses are used to isolate variables in equations. Similarly, in arithmetic, they play a key role in subtraction, which can be thought of as adding the additive inverse. The concept is straightforward yet powerful, underpinning many advanced mathematical principles. To truly grasp the significance of additive inverses, it's essential to see them in action through various examples and applications. This guide will provide numerous examples, ensuring that the concept becomes second nature.
Additive Inverses of Rational Numbers
Rational numbers, which can be expressed in the form p/q, where p and q are integers and q ≠ 0, have additive inverses that are equally rational. The additive inverse of a rational number p/q is simply -p/q. This means that adding p/q and -p/q together will always result in zero. This property holds true regardless of whether the rational number is positive, negative, or a fraction. The sign change is the key operation here. If the original number is positive, its additive inverse is negative, and vice versa. For example, the additive inverse of 3/7 is -3/7, and the additive inverse of -5/8 is 5/8. When dealing with mixed numbers, it's often helpful to convert them to improper fractions before finding the additive inverse. For example, to find the additive inverse of 2 1/3, first convert it to 7/3, and then the additive inverse is -7/3. Understanding how to find additive inverses of rational numbers is not only crucial for basic arithmetic but also for more advanced topics like solving linear equations and inequalities. Rational numbers are the building blocks of many mathematical concepts, making the mastery of their additive inverses a fundamental skill. This section will further explore various examples to solidify your understanding.
Now, let's apply the concept of additive inverses to a series of practical examples. We will systematically find the additive inverses of the given numbers, providing a step-by-step explanation for each case. This hands-on approach will help solidify your understanding and build confidence in solving similar problems. Each example is carefully chosen to cover a range of scenarios, including fractions, negative numbers, and mixed numbers. By working through these examples, you will gain a practical understanding of how to apply the concept of additive inverses in different contexts. Remember, the key is to change the sign of the number while keeping its magnitude the same. This simple rule is the foundation for finding additive inverses. The following examples will demonstrate this rule in action, providing you with a clear and concise method for solving such problems.
(a) -3/7
The given number is -3/7, which is a rational number. To find its additive inverse, we simply change the sign. The additive inverse of -3/7 is -(-3/7), which simplifies to 3/7. To verify, we can add the number and its additive inverse: -3/7 + 3/7 = 0. This confirms that 3/7 is indeed the additive inverse of -3/7. This example illustrates the basic principle of finding additive inverses for negative fractions. The process is straightforward: negate the negative sign to obtain the positive counterpart. Understanding this simple step is crucial for mastering additive inverses. The ability to quickly find the additive inverse of a fraction is a valuable skill in various mathematical contexts.
(b) 0
Zero is a unique number in mathematics, possessing the special property of being its own additive inverse. This is because 0 + 0 = 0. There is no other number that, when added to zero, results in zero, other than zero itself. This makes zero a special case in the realm of additive inverses. It's neither positive nor negative, and its additive inverse is simply itself. Understanding this unique property of zero is important for various mathematical operations and concepts. For instance, in the context of number lines, zero is the central point, and additive inverses are equidistant from zero on either side. This visual representation further reinforces the concept of zero as its own additive inverse. Therefore, the additive inverse of 0 is 0.
(c) -5 4/7
To find the additive inverse of -5 4/7, we first need to convert the mixed number to an improper fraction. -5 4/7 can be converted as follows: (-5 * 7) + 4 = -35 + 4 = -39. So, the improper fraction is -39/7. Now, to find the additive inverse, we change the sign. The additive inverse of -39/7 is -(-39/7), which simplifies to 39/7. We can also express this as a mixed number: 39 ÷ 7 = 5 with a remainder of 4, so 39/7 is equal to 5 4/7. Thus, the additive inverse of -5 4/7 is 5 4/7. This example demonstrates the importance of converting mixed numbers to improper fractions before finding the additive inverse. This step ensures accuracy and simplifies the process. The ability to convert between mixed numbers and improper fractions is a fundamental skill in mathematics, particularly when dealing with additive inverses and other operations involving fractions.
(d) -31/51
For the rational number -31/51, finding the additive inverse involves a straightforward sign change. The additive inverse of -31/51 is -(-31/51), which simplifies to 31/51. This example is a clear illustration of how changing the sign of a negative fraction yields its additive inverse. The process is simple yet crucial for understanding the concept of additive inverses. By negating the negative sign, we obtain the positive counterpart, which, when added to the original number, results in zero. This fundamental principle applies to all rational numbers, making it a cornerstone of arithmetic and algebra. The ability to quickly identify the additive inverse of a fraction is a valuable skill in various mathematical contexts.
(e) -27/-83
The number -27/-83 presents a slightly different scenario. First, we observe that both the numerator and the denominator are negative. A fraction with a negative numerator and a negative denominator is equivalent to a positive fraction. Therefore, -27/-83 simplifies to 27/83. Now, to find the additive inverse, we change the sign of the simplified fraction. The additive inverse of 27/83 is -27/83. This example highlights the importance of simplifying fractions before finding their additive inverses. By recognizing that a negative divided by a negative results in a positive, we can accurately determine the additive inverse. This step-by-step approach ensures clarity and minimizes errors. Understanding the properties of fractions and their signs is crucial for mastering additive inverses and other mathematical operations.
(f) -2/5
To find the additive inverse of -2/5, we apply the same principle of changing the sign. The additive inverse of -2/5 is -(-2/5), which simplifies to 2/5. This example reinforces the basic concept of finding additive inverses for negative fractions. By negating the negative sign, we obtain the positive counterpart, which, when added to the original number, equals zero. This simple process is fundamental to understanding additive inverses and their applications. The ability to quickly identify the additive inverse of a fraction is a valuable skill in various mathematical contexts, including algebra and calculus.
(g) -(-7/12)
This example involves a double negative, which requires careful attention. The expression -(-7/12) indicates the negation of a negative number. When we negate a negative number, we obtain its positive counterpart. Therefore, -(-7/12) simplifies to 7/12. Now, to find the additive inverse, we change the sign of the simplified fraction. The additive inverse of 7/12 is -7/12. This example emphasizes the importance of simplifying expressions before finding additive inverses. By addressing the double negative first, we can accurately determine the additive inverse. Understanding the rules of signs and their application is crucial for mastering additive inverses and other mathematical operations. This step-by-step approach ensures clarity and minimizes errors.
In conclusion, finding additive inverses is a fundamental concept in mathematics that is essential for various operations and problem-solving scenarios. Through the examples provided, we have demonstrated how to find additive inverses for different types of numbers, including fractions, mixed numbers, and negative numbers. The key principle is to change the sign of the number while maintaining its magnitude. This simple yet powerful concept underpins many advanced mathematical principles and is crucial for building a strong foundation in mathematics. By understanding and mastering additive inverses, you will be well-equipped to tackle more complex mathematical problems and concepts. This guide has provided a comprehensive overview of additive inverses, ensuring that you have a clear and thorough understanding of this crucial mathematical principle. Continue practicing and applying this concept to various problems to further solidify your knowledge and skills.