Identifying Rational Numbers An In-Depth Explanation

• a) $\sqrt{5}$
• b) 3.456733244
• c) $3 . \overline{4}$
• d) $\infty$

In the realm of mathematics, understanding the different types of numbers is crucial. Among these, rational numbers hold a significant place. To answer the question, "Which of the following is an example of a rational number?" we need to delve into the definition and characteristics of rational numbers and then evaluate each option accordingly. This article will provide a comprehensive explanation of rational numbers, their properties, and examples, allowing you to confidently identify them. We will examine each option provided in the question, explaining why some are rational and others are not. By the end of this discussion, you will have a clear understanding of what constitutes a rational number and be able to distinguish it from other types of numbers, such as irrational numbers. This knowledge is essential not only for answering specific questions but also for building a solid foundation in mathematical concepts.

Defining Rational Numbers

To effectively identify a rational number, it is essential to first understand its definition. A rational number is any number that can be expressed as a fraction ${\frac{p}{q}}$, where p and q are integers, and q is not equal to zero. This definition is fundamental to differentiating rational numbers from other types of numbers, such as irrational numbers. The key components of this definition are integers and the fractional form. Integers are whole numbers (positive, negative, or zero), which means that both the numerator (p) and the denominator (q) must be whole numbers. The condition that q cannot be zero is crucial because division by zero is undefined in mathematics. This definition encompasses a wide range of numbers, including integers themselves, terminating decimals, and repeating decimals. For example, the number 5 is a rational number because it can be written as ${\frac{5}{1}}$. Similarly, the decimal 0.75 is rational as it can be expressed as ${\frac{3}{4}}$. Understanding this foundational definition is the first step in identifying rational numbers in various forms. The ability to convert a number into the fractional form ${\frac{p}{q}}$ is the ultimate test of its rationality. This concept is not only important in basic arithmetic but also plays a crucial role in more advanced mathematical fields such as algebra and calculus.

Characteristics of Rational Numbers

Beyond the basic definition, rational numbers exhibit specific characteristics that help in their identification. One of the most important characteristics is their decimal representation. Rational numbers can be expressed as either terminating decimals or repeating decimals. A terminating decimal is a decimal that ends after a finite number of digits, such as 0.5, 0.75, or 3.125. These decimals can easily be converted into fractions, confirming their rationality. For example, 0.5 is equivalent to ${\frac{1}{2}}$, 0.75 is ${\frac{3}{4}}$, and 3.125 is ${\frac{25}{8}}$. Repeating decimals, on the other hand, have a pattern of digits that repeats indefinitely, such as 0.333... (which is ${\frac{1}{3}}$) or 1.142857142857... (which is ${\frac{8}{7}}$). These repeating patterns indicate that the number can also be expressed as a fraction. In contrast, irrational numbers have non-repeating, non-terminating decimal representations, which means they cannot be written as a simple fraction. Another characteristic of rational numbers is their density on the number line. Between any two rational numbers, there exists another rational number. This property highlights the infinite nature of rational numbers and their widespread presence within the real number system. Understanding these characteristics – terminating or repeating decimal representations and density on the number line – is essential for correctly identifying rational numbers and distinguishing them from irrational numbers. These concepts are foundational in mathematics and are frequently used in various mathematical operations and problem-solving scenarios.

Evaluating the Options

Now, let's evaluate each option provided in the question to determine which one represents a rational number.

a) $\sqrt{5}$

The first option is $\sqrt{5}$, which represents the square root of 5. To determine if this is a rational number, we need to consider whether it can be expressed as a fraction of two integers. The square root of 5 is approximately 2.236067977... This decimal representation is non-repeating and non-terminating. Numbers with non-repeating and non-terminating decimal expansions are classified as irrational numbers. Therefore, $\sqrt{5}$ cannot be expressed as a fraction of two integers and is not a rational number. This is a classic example of an irrational number, similar to $\sqrt{2}$ and $\pi$. Understanding that square roots of non-perfect squares are irrational is crucial in mathematics. This concept is often encountered in algebra and geometry, where irrational numbers play a significant role.

b) 3.456733244

The second option is 3.456733244. This is a decimal number, and to determine if it's rational, we need to see if it terminates or repeats. In this case, the decimal terminates after a finite number of digits. Terminating decimals can be written as fractions, making them rational numbers. To convert 3.456733244 into a fraction, we can write it as ${\frac{3456733244}{1000000000}}$. Since both the numerator and the denominator are integers, this number fits the definition of a rational number. Therefore, 3.456733244 is a rational number. This example illustrates how terminating decimals are inherently rational. The ability to recognize and convert terminating decimals into fractions is a fundamental skill in mathematics, especially when dealing with real numbers and their properties.

c) $3 . \overline{4}$

The third option is $3 . \overline{4}$, which represents a repeating decimal. The bar over the 4 indicates that the digit 4 repeats infinitely (3.4444...). Repeating decimals are a key characteristic of rational numbers. They can be expressed as a fraction of two integers. To convert $3 . \overline{4}$ into a fraction, let x = 3.4444... Multiplying both sides by 10, we get 10x = 34.4444... Subtracting the original equation from this, we have 10x - x = 34.4444... - 3.4444..., which simplifies to 9x = 31. Dividing both sides by 9, we get x = ${\frac{31}{9}}$. Since ${\frac{31}{9}}$ is a fraction of two integers, $3 . \overline{4}$ is a rational number. This conversion process highlights the method for converting repeating decimals into fractions, a common technique in mathematics. Understanding that repeating decimals are rational is essential for distinguishing them from irrational numbers.

d) $\infty$

The fourth option is $\infty$, which represents infinity. Infinity is not a real number; it is a concept representing a quantity without bound. Since infinity cannot be expressed as a fraction of two integers, it is not a rational number. Rational numbers are a subset of real numbers, and infinity does not fall within this set. Therefore, $\infty$ is not a rational number. This distinction is crucial in mathematical analysis and set theory, where the concept of infinity is used extensively but must be differentiated from real numbers.

Conclusion: Identifying the Rational Number

After evaluating all the options, we can definitively identify which one is a rational number. Option a) $\sqrt{5}$ is an irrational number because it has a non-repeating, non-terminating decimal representation. Option d) $\infty$ is not a real number and thus not rational. Options b) 3.456733244 and c) $3 . \overline{4}$ are both rational numbers. The former is a terminating decimal, and the latter is a repeating decimal, both of which can be expressed as a fraction of two integers. Therefore, the correct examples of rational numbers from the given options are 3.456733244 and $3.\overline{4}$. This exercise underscores the importance of understanding the definition and characteristics of rational numbers and the ability to distinguish them from irrational numbers and other non-real number concepts. Mastering these concepts is crucial for building a strong foundation in mathematics and for tackling more advanced topics in algebra, calculus, and beyond. The ability to identify rational numbers is not just a theoretical exercise but a practical skill that applies to various mathematical and real-world scenarios.