We have introduced a decimal number by a finite number of digits with a decimal point in-between, which represent a fraction according to the decimal number system. But with a finite number of digits, you cannot write every fraction like this; then you need the concept of a repeating decimal, also called a nonterminating decimal number.
A repeating decimal is a fraction that cannot be written in a finite decimal notation, but can only be written as a nonterminating decimal number in which a repetitive part occurs.
In the normal notation, a repeating decimal is rounded. Another way of writing is to strike through the first digit and the last digit of the repetitive part .
Examples
\[\begin{aligned}\frac{4}{3}&\approx 1.667\\ &=1.66666\ldots\\&=1{.}\!\!\not\!6\\\\\frac{69}{70}&\approx 0.9857\\ &=0{.}9\overbrace{857142}\overbrace{857142}\ldots\\[0.25cm] &=0.9\!\!\not\!85714\!\!\not\!2\end{aligned}\]
What exactly do we mean when we write \(\frac{7}{6}=1.1\!\!\not\!6\)?
The simplest answer is an infinitely large scheme of which a small piece is written below \[\begin{aligned} 1\le &\frac{7}{6} \le 2\\ 1.1\le &\frac{7}{6} \le 1.2\\ 1.16\le &\frac{7}{6} \le 1.67\\ 1.166\le &\frac{7}{6} \le 1.667 \\ 1.1666\le &\frac{7}{6} \le 1.6667\\\ldots\,&..\ldots\end{aligned}\]
In a more mathematical sense, a repeating decimal is understood as a sequence: \[1.1\!\!\not\!6=1.1+0.06+0.006+0.0006+\cdots =1.1+6\sum_{k=2}^{\infty}10^{-k}\]
The repetitive part of a repeating decimal is also indicated by a line above or below it, or by enclosing it in square brackets. \[\begin{aligned} 0.9\!\!\not\!85714\!\!\not\!2 &= 0.9\underline{857142} \\[0.25cm] &= 0.9\overline{857142}\\[0.25cm] &= 0.9[857142]\end{aligned}\]
Fractions whose denominators have no divisors other than \(1\), \(2\) and \(5\) can be written as repeating decimal in two ways: with a tail of only zeros and with a tail of only nines. For example \[\begin{aligned}\frac{33}{20}=1.65&=1.65000000\ldots =1.65\!\!\not\!0\\&=1.64999999\ldots =1.64\!\!\not\!9\end{aligned}\]
Every rational number can be written as at least one repeating decimal, and conversely, every repeating decimal belongs to a rational number.
As an example, we write \(0{.}\!\!\not\!56\!\!\not\!7\) as a rational number.
Suppose \(0{.}\!\!\not\!56\!\!\not\!7=x\), then \(1000x=567{.}\!\!\not\!56\!\!\not\!7\).
Then \(1000x-x=567\) because the repetitive parts cancel each other out.
In other words, \(999x=567\), or \(x=567\div 999\). So: \[0{.}\!\!\not\!56\!\!\not\!7=\frac{567}{999}\]
If there is also a non-repetitive part, then conversion has to be done. For example: \[\begin{aligned}1.234\!\!\not\!56\!\!\not\!7&=1.234+0.000\!\!\not\!56\!\!\not\!7\\[0.25cm] &=1.234+\frac{0{.}\!\!\not\!56\!\!\not\!7}{1000}\\[0.25cm] &= \frac{1234}{1000}+\frac{567}{999000} \\[0.25cm] &=\frac{1232766}{999000}+\frac{567}{999000}\\[0.25cm] &= \frac{1233333}{999000} \\[0.25cm] &= \frac{45679}{37000}\end{aligned}\]
Repeating decimals
