Which Of The Following Expressions Represents A Rational Number? $3 imes \pi$; $\frac{2}{3}+9.26$; $\sqrt{45}+ \sqrt{36}$; $14.\overline{3}+5.78765239$

In mathematics, understanding the nature of numbers is crucial. Numbers can be broadly classified into rational and irrational numbers. 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 includes integers, terminating decimals, and repeating decimals. An irrational number, conversely, cannot be expressed in this form. These numbers have non-repeating, non-terminating decimal expansions. This article delves into the process of identifying rational numbers from a set of given expressions, focusing on the key characteristics that differentiate rational numbers from their irrational counterparts. Identifying rational numbers requires a clear understanding of their definition and properties. Rational numbers can be expressed as a fraction of two integers, meaning they can be written in the form ${ \frac{p}{q} }$, where p and q are both integers and q is not zero. This definition encompasses several types of numbers, including integers, terminating decimals, and repeating decimals. For instance, the number 5 is rational because it can be written as ${ \frac{5}{1} }$. Similarly, 0.75 is rational because it can be expressed as ${ \frac{3}{4} }$, and 0.333... (a repeating decimal) is rational because it can be written as ${ \frac{1}{3} }$. The ability to convert a number into a fraction of integers is the definitive test for rationality. This involves recognizing patterns in decimal representations, such as repeating sequences, and understanding how to manipulate expressions to fit the ${ \frac{p}{q} }$ format. In contrast, irrational numbers cannot be expressed as a fraction of integers. Their decimal representations are non-repeating and non-terminating, meaning they go on forever without any discernible pattern. Common examples of irrational numbers include ${ \sqrt{2} }$ and ${ \pi }$. These numbers have decimal expansions that never repeat or terminate, making it impossible to write them as a simple fraction. When dealing with expressions involving sums, differences, products, or quotients of numbers, it is essential to evaluate each component carefully to determine the overall rationality of the expression. For example, the sum of two rational numbers is always rational, but the sum of a rational and an irrational number is always irrational. Similarly, the product of two rational numbers is rational, but the product of a non-zero rational number and an irrational number is irrational. Understanding these rules helps in quickly assessing the rationality of complex expressions without having to perform extensive calculations. In the following sections, we will apply these principles to specific examples to illustrate the process of identifying rational numbers.

Evaluating the Expressions

Expression 1: $3 imes ext{${\pi}$}$

The first expression we need to analyze is the product of 3 and ${ \pi }$ (pi). To determine whether this expression results in a rational number, we need to understand the nature of both numbers involved. The number 3 is an integer, and as discussed earlier, all integers are rational numbers because they can be expressed as a fraction with a denominator of 1 (e.g., ${ 3 = \frac{3}{1} }$). However, ${ \pi }$ (pi) is a well-known irrational number. It is a transcendental number, meaning it is not the root of any non-zero polynomial equation with integer coefficients. The decimal representation of ${ \pi }$ is non-repeating and non-terminating, approximately 3.14159265358979323846..., and it continues infinitely without any repeating pattern. This characteristic makes it impossible to express ${ \pi }$ as a fraction of two integers. When a rational number (other than zero) is multiplied by an irrational number, the result is always an irrational number. This is because multiplying an irrational number by a rational number simply scales the irrational number, but it does not change its fundamental property of having a non-repeating, non-terminating decimal expansion. Therefore, the product of 3 and ${ \pi }$ will also have a non-repeating, non-terminating decimal expansion, making it an irrational number. In simpler terms, if you try to write ${ 3 \pi }$ as a fraction, you would need to incorporate the infinite, non-repeating decimal of ${ \pi }$ into the fraction, which is not possible according to the definition of rational numbers. Thus, the expression ${ 3 \times \pi }$ is not a rational number. The result of this multiplication will always retain the irrationality of ${ \pi }$, no matter how much it is scaled by a rational number. This principle is fundamental in understanding how irrational numbers behave in mathematical operations. By recognizing that ${ \pi }$ is irrational and applying the rule that the product of a non-zero rational number and an irrational number is irrational, we can confidently classify ${ 3 \pi }$ as an irrational number. This understanding is crucial in various mathematical contexts, including geometry, trigonometry, and calculus, where ${ \pi }$ frequently appears in formulas and calculations. Recognizing the irrational nature of ${ 3 \pi }$ helps in avoiding misconceptions and ensuring accurate mathematical reasoning. This foundational concept is essential for students and professionals alike in navigating more complex mathematical problems. Therefore, the final conclusion is that ${ 3 \times \pi }$ is an irrational number due to the presence of ${ \pi }$ in the expression.

Expression 2: $rac{2}{3} + 9.26$

The second expression we need to evaluate is the sum of ${ \frac{2}{3} }$ and 9.26. To determine whether this expression results in a rational number, we need to examine both terms individually and then consider their sum. The first term, ${ \frac{2}{3} }$, is a fraction where both the numerator (2) and the denominator (3) are integers, and the denominator is not zero. According to the definition of rational numbers, any number that can be expressed as a fraction of two integers is a rational number. Therefore, ${ \frac{2}{3} }$ is a rational number. It is also worth noting that ${ \frac{2}{3} }$ has a repeating decimal representation (0.666...), which is another characteristic of rational numbers. The second term, 9.26, is a decimal number. To determine if it is rational, we need to check if it can be expressed as a fraction of two integers. The decimal 9.26 can be written as ${ \frac{926}{100} }$. Both 926 and 100 are integers, so 9.26 is also a rational number. It is a terminating decimal, which is another indicator of rationality. Now that we have established that both ${ \frac{2}{3} }$ and 9.26 are rational numbers, we need to consider their sum. A fundamental property of rational numbers is that the sum of two rational numbers is always rational. This can be shown algebraically. Let ${ \frac{a}{b} }$ and ${ \frac{c}{d} }$ be two rational numbers, where a, b, c, and d are integers and b and d are not zero. Their sum is: ${ \frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd} }$ Since a, b, c, and d are integers, ${ ad + bc }$ and ${ bd }$ are also integers (because the sum and product of integers are integers). Furthermore, since b and d are not zero, ${ bd }$ is not zero. Therefore, the sum ${ \frac{ad + bc}{bd} }$ is a fraction of two integers, which means it is a rational number. Applying this principle to our expression, we have the sum of two rational numbers, ${ \frac{2}{3} }$ and 9.26. To find the sum, we first convert 9.26 to a fraction, which is ${ \frac{926}{100} }$ or ${ \frac{463}{50} }$ when simplified. Now we add the two fractions: ${ \frac{2}{3} + \frac{463}{50} = \frac{2 \times 50 + 463 \times 3}{3 \times 50} = \frac{100 + 1389}{150} = \frac{1489}{150} }$ The result, ${ \frac{1489}{150} }$, is a fraction where both the numerator (1489) and the denominator (150) are integers. Thus, the sum is a rational number. Therefore, the expression ${ \frac{2}{3} + 9.26 }$ is a rational number because it is the sum of two rational numbers, and the result can be expressed as a fraction of two integers. This confirms that the expression satisfies the definition of a rational number.

Expression 3: $\sqrt{45} + \sqrt{36}$

The third expression under consideration is the sum of two square roots: ${ \sqrt{45} + \sqrt{36} }$. To determine whether this expression results in a rational number, we need to evaluate each square root individually and then consider their sum. The first term is ${ \sqrt{45} }$. We need to determine if the square root of 45 is a rational number. To do this, we can simplify the square root by factoring 45 into its prime factors. The prime factorization of 45 is ${ 3^2 \times 5 }$, so we can write ${ \sqrt{45} }$ as ${ \sqrt{3^2 \times 5} }$. Using the property of square roots that ${ \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} }$, we have: ${ \sqrt{45} = \sqrt{3^2} \times \sqrt{5} = 3\sqrt{5} }$ Here, we see that ${ \sqrt{45} }$ simplifies to ${ 3\sqrt{5} }$. The number ${ \sqrt{5} }$ is irrational because 5 is a prime number and is not a perfect square. The square root of any non-perfect square integer is irrational. Therefore, ${ \sqrt{5} }$ has a non-repeating, non-terminating decimal expansion. Since 3 is a rational number and ${ \sqrt{5} }$ is an irrational number, their product, ${ 3\sqrt{5} }$, is also irrational. This is because multiplying an irrational number by a non-zero rational number results in an irrational number. The second term in the expression is ${ \sqrt{36} }$. We need to determine if the square root of 36 is a rational number. The number 36 is a perfect square because it is the square of an integer (6). Therefore, ${ \sqrt{36} = 6 }$. The number 6 is an integer, and all integers are rational numbers because they can be expressed as a fraction with a denominator of 1 (e.g., ${ 6 = \frac{6}{1} }$). So, ${ \sqrt{36} }$ is a rational number. Now we consider the sum of ${ \sqrt{45} }$ and ${ \sqrt{36} }$. We have: ${ \sqrt{45} + \sqrt{36} = 3\sqrt{5} + 6 }$ We know that ${ 3\sqrt{5} }$ is irrational and 6 is rational. The sum of a rational number and an irrational number is always irrational. This is because adding a rational number to an irrational number does not eliminate the non-repeating, non-terminating decimal expansion of the irrational number. Therefore, the sum ${ 3\sqrt{5} + 6 }$ is an irrational number. In conclusion, the expression ${ \sqrt{45} + \sqrt{36} }$ is an irrational number because it is the sum of an irrational number (${ 3\sqrt{5} }$) and a rational number (6). This confirms that the expression does not satisfy the definition of a rational number.

Expression 4: $14.\overline{3} + 5.78765239$

The fourth expression we need to analyze is the sum of ${ 14.\overline{3} }$ and 5.78765239. To determine whether this expression results in a rational number, we must examine each term individually and then consider their sum. The first term, ${ 14.\overline{3} }$, represents a repeating decimal. The overline above the 3 indicates that the digit 3 repeats infinitely. Repeating decimals are rational numbers because they can be expressed as a fraction of two integers. To convert ${ 14.\overline{3} }$ to a fraction, we can use the following method: Let ${ x = 14.\overline{3} = 14.3333... }$ Multiply both sides by 10 to move one repeating digit to the left of the decimal point: ${ 10x = 143.3333... }$ Now, subtract the original equation from this new equation: ${ 10x - x = 143.3333... - 14.3333... }$ ${ 9x = 129 }$ Divide both sides by 9: ${ x = \frac{129}{9} }$ Simplify the fraction: ${ x = \frac{43}{3} }$ Thus, ${ 14.\overline{3} }$ can be expressed as the fraction ${ \frac{43}{3} }$, which means it is a rational number. The second term, 5.78765239, is a decimal number with a finite number of digits. Such decimals are known as terminating decimals. Terminating decimals are also rational numbers because they can be expressed as a fraction of two integers. To convert 5.78765239 to a fraction, we write it as: ${ 5.78765239 = \frac{578765239}{100000000} }$ Both the numerator (578765239) and the denominator (100000000) are integers, so 5.78765239 is a rational number. Now that we have established that both ${ 14.\overline{3} }$ and 5.78765239 are rational numbers, we need to consider their sum. As mentioned earlier, the sum of two rational numbers is always rational. Therefore, the expression ${ 14.\overline{3} + 5.78765239 }$ will also result in a rational number. To find the sum, we add the two numbers: ${ 14.\overline{3} + 5.78765239 = \frac{43}{3} + \frac{578765239}{100000000} }$ To add these fractions, we need to find a common denominator, which is 300000000: ${ \frac{43}{3} + \frac{578765239}{100000000} = \frac{43 \times 100000000}{3 \times 100000000} + \frac{578765239 \times 3}{100000000 \times 3} }$ ${ = \frac{4300000000}{300000000} + \frac{1736295717}{300000000} }$ ${ = \frac{4300000000 + 1736295717}{300000000} }$ ${ = \frac{6036295717}{300000000} }$ The result, ${ \frac{6036295717}{300000000} }$, is a fraction where both the numerator and the denominator are integers. Thus, the sum is a rational number. Therefore, the expression ${ 14.\overline{3} + 5.78765239 }$ is a rational number because it is the sum of two rational numbers, and the result can be expressed as a fraction of two integers. This confirms that the expression satisfies the definition of a rational number.

Conclusion

In summary, we have analyzed four expressions to determine which are rational numbers. A rational number can be expressed as a fraction ${ \frac{p}{q} }$, where p and q are integers and q is not zero. This includes integers, terminating decimals, and repeating decimals. Based on our analysis:

• ${ 3 \times \pi }$ is irrational because it is the product of a rational number (3) and an irrational number (${ \pi }$).
• ${ \frac{2}{3} + 9.26 }$ is rational because it is the sum of two rational numbers, and the result can be expressed as a fraction of two integers.
• ${ \sqrt{45} + \sqrt{36} }$ is irrational because it is the sum of an irrational number (${ 3\sqrt{5} }$) and a rational number (6).
• ${ 14.\overline{3} + 5.78765239 }$ is rational because it is the sum of two rational numbers, both of which can be expressed as fractions of integers.

Therefore, the rational expressions are ${ \frac{2}{3} + 9.26 }$ and ${ 14.\overline{3} + 5.78765239 }$. This exercise highlights the importance of understanding the properties of rational and irrational numbers, and how these properties are preserved under basic arithmetic operations. Recognizing the nature of numbers is crucial in various mathematical contexts, ensuring accurate calculations and logical reasoning. Grasping these fundamental concepts empowers students and professionals to tackle more complex mathematical problems with confidence and precision. Understanding the distinction between rational and irrational numbers is not only essential for mathematical proficiency but also for broader analytical thinking. The ability to identify and classify numbers correctly underpins numerous applications in science, engineering, and economics, making it a valuable skill in diverse fields. By mastering these concepts, individuals can enhance their problem-solving capabilities and contribute effectively to quantitative analysis in their respective domains.

Final Answer

The rational expressions are:

• $$\frac{2}{3} + 9.26$$

• $$14.\overline{3} + 5.78765239$$