1.3.1

If \(x=0.37373737\cdots\), express \(x\) as a ratio of integers.

$$ \begin{align} x=0.37373737\cdots\Rightarrow& 100x=37.373737\cdots\\ \Rightarrow& 99x=37\\ \Rightarrow& x=\frac{37}{99} \end{align} $$

1.3.2

If \(y=0.519191919\cdots\), express \(y\) as a ratio of integers.

$$ \begin{align} y=0.5191919\cdots\Rightarrow& 10y=5.191919\cdots\\ \Rightarrow&1000y=519.191919\cdots\\ \Rightarrow&990y=514\\ \Rightarrow&y=\frac{514}{990} \end{align} $$

1.3.3

By generalizing the idea of the previous exercises, explain why each ultimately periodic decimal represents a rational number.

Any periodic decimal number can be written as

$$ r=0.b_1b_2\cdots b_k\overline{a_1a_2\cdots a_n} $$

$$ \begin{align} \Rightarrow&10^k r=b_1b_2\cdots b_k.\overline{a_1a_2\cdots a_n}\\ \Rightarrow&10^{k+n}r=b_1b_2\cdots b_ka_1a_2\cdots a_n.\overline{a_1a_2\cdots a_n}\\ \Rightarrow&(10^{k+n}-10^k)r=b_1b_2\cdots b_ka_1a_2\cdots a_n-b_1b_2\cdots b_k\\ \Rightarrow& r=\frac{b_1b_2\cdots b_ka_1a_2\cdots a_n-b_1b_2\cdots b_k}{10^{k+n}-10^k} \end{align} $$

Since both \(b_1b_2\cdots b_ka_1a_2\cdots a_n-b_1b_2\cdots b_k\) and \(10^{k+n}-10^k\) are integers (integers are closed under addition, subtraction, and multiplication), any periodic decimal number represents a ratio of two integers (i.e., rational number).

1.3.4

Find the decimals for \(1/6\) and \(1/7\).

$$ \begin{align} \frac{1}{6}=&1\times10^{-1}+\frac{4}{6}\times10^{-1}\\ =&1\times10^{-1}+6\times10^{-2}+\frac{4}{6}\times10^{-2}\\ =&0.1\overline{6} \end{align} $$

$$ \begin{align} \frac{1}{7}=&1\times10^{-1}+\frac{3}{7}\times10^{-1}\\ =&1\times10^{-1}+4\times10^{-2}+\frac{2}{7}\times10^{-2}\\ =&1\times10^{-1}+4\times10^{-2}+2\times10^{-3}+\frac{6}{7}\times10^{-3}\\ =&1\times10^{-1}+4\times10^{-2}+2\times10^{-3}+8\times10^{-4}+\frac{4}{7}\times10^{-4}\\ =&1\times10^{-1}+4\times10^{-2}+2\times10^{-3}+8\times10^{-4}+5\times10^{-5}+\frac{5}{7}\times10^{-5}\\ =&1\times10^{-1}+4\times10^{-2}+2\times10^{-3}+8\times10^{-4}+5\times10^{-5}+7\times10^{-6}+\frac{1}{7}\times10^{-6}\\ =&0.\overline{142857} \end{align} $$

1.3.5

By means of the division processes, or otherwise, explain why each rational number has an ultimately periodic decimal.

1.3.6

If each rational point in the plane is surrouned by a disk of fixed size \(\varepsilon\), show that there is no line from \(\langle0,0\rangle\) that misses all other disks.

A line from \(\langle 0,0\rangle\) that passes a point \(\langle m,n\rangle\) is

$$ nx-my=0 $$

The distance between the line and a point \(\langle a,b\rangle\) is

$$ d=\frac{|na-mb|}{\sqrt{n^2+m^2}} $$

\(\forall\varepsilon>0\), \(\exists a,b\in\mathbb{N}\) such that \(\frac{|na-mb|}{\sqrt{n^2+m^2}}<\varepsilon ~~~~~ \Leftarrow\) (Is this clear????) Therefore, there is always a point that is arbitrarily close to the line.

1.3.7

Conclude that, if space were filled uniformly with stars of uniform size, the whole sky would be filled with light (the Olbers paradox).