Rational And Irrational Numbers Classification Exercise 1.4

Classifying Numbers: Rational or Irrational

In the realm of mathematics, numbers can be broadly classified into two main categories: rational and irrational. Understanding the distinction between these types of numbers is crucial for grasping more advanced mathematical concepts. Rational numbers are those that can be expressed as a fraction ${ \frac{p}{q} }$, where p and q are integers and q is not zero. These numbers have decimal representations that either terminate (e.g., 0.5) or repeat in a pattern (e.g., 0.333...). On the other hand, irrational numbers cannot be expressed as a simple fraction. Their decimal representations are non-terminating and non-repeating, meaning they go on infinitely without any discernible pattern. Common examples of irrational numbers include ${ \sqrt{2} }$ and ${ \pi }$.

The process of classifying numbers as either rational or irrational involves examining their fundamental properties and determining whether they fit the criteria for rational numbers. This often requires algebraic manipulation, simplification, and a deep understanding of number theory. For instance, a number presented in a complex form may need to be simplified to reveal its true nature. Operations involving irrational numbers, such as addition, subtraction, multiplication, and division, can sometimes result in rational numbers, but this is not always the case. The sum or difference of a rational and an irrational number is always irrational, while the product or quotient of a non-zero rational number and an irrational number is also irrational. However, the sum or product of two irrational numbers can be either rational or irrational, depending on the specific numbers involved.

In the following sections, we will delve into specific examples and classify them as either rational or irrational, providing detailed explanations for each classification. This exercise will not only solidify your understanding of rational and irrational numbers but also enhance your ability to apply mathematical principles in real-world scenarios. As we progress, remember that the key to accurate classification lies in a thorough analysis of the number's properties and its relationship to the definitions of rational and irrational numbers. By mastering these concepts, you will be well-equipped to tackle more complex mathematical problems and appreciate the elegance and precision of the number system.

Let's proceed to classify the given numbers:

(i) $2-\sqrt{5}$

To classify the number ${ 2 - \sqrt{5} }$, we need to understand the nature of its components. The number 2 is a rational number because it can be expressed as the fraction ${ \frac{2}{1} }$. The number ${ \sqrt{5} }$ is an irrational number because it cannot be expressed as a fraction of two integers, and its decimal representation is non-terminating and non-repeating (approximately 2.236...). When we subtract an irrational number from a rational number, the result is always an irrational number. This is because if the result were rational, we could express ${ 2 - \sqrt{5} }$ as ${ \frac{p}{q} }$, where p and q are integers, and then rearrange the equation to isolate ${ \sqrt{5} }$. This would imply that ${ \sqrt{5} = 2 - \frac{p}{q} }$, which simplifies to ${ \sqrt{5} = \frac{2q - p}{q} }$. Since p and q are integers, ${ 2q - p }$ is also an integer, making ${ \frac{2q - p}{q} }$ a rational number. This contradicts the fact that ${ \sqrt{5} }$ is irrational. Therefore, the difference between a rational number and an irrational number must be irrational.

In this case, subtracting ${ \sqrt{5} }$ from 2 yields an irrational number. The decimal representation of ${ 2 - \sqrt{5} }$ is approximately -0.236..., which is non-terminating and non-repeating, further confirming its irrationality. Understanding this principle is crucial for classifying more complex expressions involving both rational and irrational components. The key takeaway is that the presence of an irrational component in a sum or difference generally makes the entire expression irrational, unless the irrational parts cancel out in some way. This concept is fundamental in number theory and is essential for simplifying and classifying various mathematical expressions. The ability to quickly identify and classify numbers as rational or irrational is a valuable skill in mathematics, particularly in algebra and calculus.

Therefore, 2 - √5 is an irrational number.

(ii) $(3+\sqrt{23})-\sqrt{23}$

To determine whether the number ${ (3 + \sqrt{23}) - \sqrt{23} }$ is rational or irrational, we need to simplify the expression first. The given expression involves the addition and subtraction of numbers, including a square root. By carefully applying the rules of arithmetic, we can reduce the expression to its simplest form and then classify it accordingly. The key to simplifying this expression is recognizing that ${ \sqrt{23} }$ appears twice, once with a positive sign and once with a negative sign. This suggests that these terms might cancel each other out, which would significantly simplify the expression.

When we expand the expression, we have ${ 3 + \sqrt{23} - \sqrt{23} }$. Notice that the ${ +\sqrt{23} }$ and ${ -\sqrt{23} }$ terms are additive inverses, meaning they sum up to zero. Therefore, the expression simplifies to just 3. Now, we need to classify the simplified number, which is 3. The number 3 can be expressed as a fraction ${ \frac{3}{1} }$, where both the numerator and the denominator are integers. According to the definition of rational numbers, any number that can be expressed in this form is a rational number. Therefore, 3 is a rational number. This example illustrates an important principle: expressions that initially appear to be irrational due to the presence of square roots or other irrational components can sometimes simplify to rational numbers. This often occurs when irrational terms cancel each other out or combine in such a way that the irrationality is eliminated. It is crucial to always simplify expressions before attempting to classify them as rational or irrational to avoid making incorrect judgments based on their initial appearance.

In summary, by simplifying the given expression, we found that ${ (3 + \sqrt{23}) - \sqrt{23} = 3 }$, which is a rational number. This highlights the importance of simplification in mathematical problem-solving and demonstrates that even expressions involving irrational numbers can sometimes result in rational numbers.

Therefore, (3 + √23) - √23 is a rational number.

(iii) $\frac{2 \sqrt{7}}{7 \sqrt{7}}$

To classify the number ${ \frac{2 \sqrt{7}}{7 \sqrt{7}} }$, we again need to simplify the expression. This involves recognizing common factors in the numerator and the denominator that can be canceled out. The presence of ${ \sqrt{7} }$ in both the numerator and the denominator is a key observation. If we can cancel out these terms, the expression will become much simpler and easier to classify. In this case, we have a fraction where both the numerator and the denominator contain the term ${ \sqrt{7} }$. This means that ${ \sqrt{7} }$ is a common factor that can be divided out from both the numerator and the denominator. When we perform this cancellation, we are left with ${ \frac{2}{7} }$.

Now, we need to determine whether ${ \frac{2}{7} }$ is rational or irrational. By definition, a rational number can be expressed as a fraction ${ \frac{p}{q} }$, where p and q are integers and q is not zero. In this case, 2 and 7 are both integers, and 7 is not zero, so ${ \frac{2}{7} }$ fits the definition of a rational number. The decimal representation of ${ \frac{2}{7} }$ is approximately 0.285714..., which is a repeating decimal, further confirming that it is a rational number. This example reinforces the importance of simplifying expressions before classifying them. What initially appears to be an expression involving irrational numbers can, after simplification, turn out to be a rational number. This is a common theme in mathematical problem-solving, where seemingly complex expressions can often be reduced to simpler, more manageable forms through careful manipulation and application of algebraic principles.

Therefore, 2√7 / 7√7 is a rational number.

(iv) $\frac{1}{\sqrt{2}}$

To classify the number ${ \frac{1}{\sqrt{2}} }$, we need to consider the nature of the denominator. The denominator, ${ \sqrt{2} }$, is an irrational number because it cannot be expressed as a fraction of two integers. A fraction with an irrational denominator is typically irrational itself. However, to be completely sure, we often rationalize the denominator to put the expression in a more standard form. Rationalizing the denominator means eliminating any square roots (or other radicals) from the denominator of a fraction. This is done by multiplying both the numerator and the denominator by a suitable factor that will eliminate the radical in the denominator. In this case, the suitable factor is ${ \sqrt{2} }$ itself.

When we multiply both the numerator and the denominator of ${ \frac{1}{\sqrt{2}} }$ by ${ \sqrt{2} }$, we get ${ \frac{1 \times \sqrt{2}}{\sqrt{2} \times \sqrt{2}} }$, which simplifies to ${ \frac{\sqrt{2}}{2} }$. Now, we have a fraction with a rational denominator (2), but the numerator is ${ \sqrt{2} }$, which is irrational. The product of a rational number (${ \frac{1}{2} }$) and an irrational number (${ \sqrt{2} }$) is always irrational, as long as the rational number is non-zero. To understand why, suppose ${ \frac{\sqrt{2}}{2} }$ were rational. Then we could write it as ${ \frac{p}{q} }$ for integers p and q. This would mean ${ \frac{\sqrt{2}}{2} = \frac{p}{q} }$, which implies ${ \sqrt{2} = \frac{2p}{q} }$. Since 2p and q are integers, this would make ${ \sqrt{2} }$ rational, which is a contradiction. Therefore, ${ \frac{\sqrt{2}}{2} }$ must be irrational. This example illustrates a common technique in simplifying expressions and classifying numbers: rationalizing the denominator. By removing the radical from the denominator, we can more easily analyze the nature of the number and determine whether it is rational or irrational.

Therefore, 1/√2 is an irrational number.

(v) $2 \pi$

To classify the number ${ 2\pi }$, we need to understand the nature of ${ \pi }$ (pi). The number ${ \pi }$ is one of the most famous irrational numbers in mathematics. It represents the ratio of a circle's circumference to its diameter and has a non-terminating, non-repeating decimal representation (approximately 3.14159...). The exact value of ${ \pi }$ cannot be expressed as a fraction of two integers, which is the defining characteristic of an irrational number. Now, let’s consider the number 2. The number 2 is a rational number because it can be expressed as the fraction ${ \frac{2}{1} }$. When we multiply a rational number by an irrational number, the result is always an irrational number (as long as the rational number is not zero). This is a fundamental property of rational and irrational numbers. To understand why this is the case, suppose that ${ 2\pi }$ were a rational number. Then we could express it as a fraction ${ \frac{p}{q} }$, where p and q are integers. This would mean ${ 2\pi = \frac{p}{q} }$, which implies ${ \pi = \frac{p}{2q} }$. Since p and q are integers, 2q is also an integer, making ${ \frac{p}{2q} }$ a rational number. This contradicts the fact that ${ \pi }$ is irrational. Therefore, the product of 2 and ${ \pi }$ must be irrational.

In this case, multiplying the rational number 2 by the irrational number ${ \pi }$ yields an irrational number. The number ${ 2\pi }$ is approximately 6.28318..., which is a non-terminating and non-repeating decimal, further confirming its irrationality. This example highlights the principle that multiplying a rational number by an irrational number results in an irrational number. This principle is crucial for classifying numbers and simplifying expressions involving both rational and irrational components. The ability to quickly identify and classify numbers as rational or irrational is a valuable skill in mathematics, particularly in algebra, calculus, and number theory.

Therefore, 2π is an irrational number.