Rational Numbers
Rational numbers are numbers that can be expressed as the ratio of two integers. Rational numbers follow the rules of arithmetic and all rational numbers can be reduced to the form \(\frac{a}{b}\), where \(b\neq0\) and \(\gcd(a,b)=1\).
Rational numbers are often denoted by \(\mathbb{Q}\). These numbers are a subset of the real numbers, which comprise the complete number line and are often denoted by \(\mathbb{R}\). Real numbers that cannot be expressed as the ratio of two integers are called irrational numbers.
The decimal expansion of a rational number always terminates after a finite number of digits or repeats a sequence of finite digits over and over. E.g \(2.5\) has a terminating decimal expansion. Thus it is a rational number.
\(\Rightarrow\) Every integer is a rational number.
\(\Rightarrow\) Every fraction where the denominator \(\neq 0\) is a rational number.
Determine the rational representation of \( 0.\overline{238095}\). The line over \( 238095\) denotes that it is a repeating decimal of the form \( 0.238095238095238095\ldots\).
Let \( S = 0.\overline{238095}\). Then \(1000000S = 238095.\overline{238095}\), and taking the sum, we obtain
\[\begin{aligned} - S & = -000000.\overline{238095} \\ 1000000S & = 238095.\overline{238095}\\ \hline \\ 999999S & = 238095. \end{aligned} \]
Hence, \(S = \frac {238095}{999999} = \frac {5}{21}\). \(_\square\)
\[ \begin{array} {r r r r r } 0, & 1, & \frac{2}{3}, & \frac{ 1.2}{3.4}, & \frac{5}{6} + \frac{7}{8}, \\\\ \sqrt{10}, & \sqrt{16}, & \pi, & \frac{\pi}{\pi + \pi}, & \sqrt{ 1+ \frac{5}{4} } \end{array} \]
How many of the above 10 numbers are rational?
What is the smallest rational number larger than 1 such that its square root is also a rational number?
Is \(0.285714285714\overline {285714}\) rational?
\(\)
Note: The notation \(“\, \overline{285714}" \) indicates that these digits in the decimal are being repeated. For example, \( 0.1\overline{2} = 0.12222 \ldots. \)