- \(\frac{11}{3}\) is rational as it is a ratio of integers.
- \(\sqrt{48} = 4\sqrt{3}\) is irrational because \(\sqrt{3}\) is irrational.
- \(6.25 = \frac{25}{4}\) and \(0.01045 = \frac{209}{20000}\) are rational as they are terminating decimals.
- \(\sqrt{\frac{16}{81}} = \frac{4}{9}\) is rational.
- \(\sqrt{\frac{3}{16}} = \frac{\sqrt{3}}{4}\) is irrational.
- \(0.\overline{42} = \frac{14}{33}\) is rational as it is a repeating decimal.
Explanation
- Definitions and Approach
A rational number is any number that can be expressed as a fraction \(\frac{p}{q}\), where \(p\) and \(q\) are integers and \(q
eq 0\). This includes integers, terminating decimals, and repeating decimals. An irrational number cannot be represented as such a ratio, meaning its decimal representation is non-terminating and non-repeating. We will evaluate each given number to determine its classification.
- Classification of Numbers
- \(\frac{11}{3}\): This is already expressed as a ratio of two integers, \(11\) and \(3\). Thus, it is a rational number.
- \(\sqrt{48}\): We simplify this by factoring the radicand: \(\sqrt{48} = \sqrt{16 \times 3} = \sqrt{16} \times \sqrt{3} = 4\sqrt{3}\). Since \(\sqrt{3}\) is an irrational number, multiplying it by \(4\) results in an irrational number.
- \(6.25\): This is a terminating decimal. It can be written as \(\frac{625}{100}\), which simplifies to \(\frac{25}{4}\). Since it is expressed as a ratio of integers, it is a rational number.
- \(0.01045\): This is a terminating decimal. It can be written as \(\frac{1045}{100000} = \frac{209}{20000}\). Since it is a ratio of integers, it is a rational number.
- \(\sqrt{\frac{16}{81}}\): Using the property \(\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}\), we get \(\frac{\sqrt{16}}{\sqrt{81}} = \frac{4}{9}\). As this is a ratio of integers, it is a rational number.
- \(\sqrt{\frac{3}{16}}\): Similarly, \(\frac{\sqrt{3}}{\sqrt{16}} = \frac{\sqrt{3}}{4}\). Because \(\sqrt{3}\) is irrational, \(\frac{\sqrt{3}}{4}\) is an irrational number.
- \(0.\overline{42}\): This is a repeating decimal. Any repeating decimal can be converted into a fraction. If \(x = 0.\overline{42}\), then \(100x = 42.\overline{42}\). Subtracting \(x\) from \(100x\) gives \(99x = 42\), so \(x = \frac{42}{99} = \frac{14}{33}\). Since it can be expressed as a ratio of integers, it is a rational number.
- Final Classification
Based on the analysis, the numbers are classified as follows:
- Rational: \(\frac{11}{3}\), \(6.25\), \(0.01045\), \(\sqrt{\frac{16}{81}}\), \(0.\overline{42}\).
- Irrational: \(\sqrt{48}\), \(\sqrt{\frac{3}{16}}\).
Examples
Understanding the difference between rational and irrational numbers is essential in fields like engineering and finance. For instance, in structural engineering, many calculations involve square roots of non-perfect squares, which are irrational. If an engineer uses an approximate rational value like \(\frac{22}{7}\) for \(\pi\) or a truncated decimal for \(\sqrt{2}\), they must be aware of the potential margin of error introduced in their models. Similarly, in computing, knowing if a number is rational or irrational helps programmers choose appropriate data types and algorithms for precision-critical tasks.