Repeating decimals

Calculating with numbers: Decimal numbers

Repeating decimals

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.

Repeating decimal

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:

$1.1\!\!\not\!6=1.1+0.06+0.006+0.0006+\cdots =1.1+6\sum_{k=2}^{\infty}10^{-k}$

Alternative notations 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 with two notations of a repeating decima 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:

$\begin{aligned}1.234\!\!\not\!56\!\!\not\!7&=1.234+0.000\!\!\not\!56\!\!\not\!7\\[0.25cm] &=1.234+\frac{0{.}\!\!\not\!56\!\!\not\!7}{1000}\\[0.25cm] &= \frac{1234}{1000}+\frac{567}{999000} \\[0.25cm] &=\frac{1232766}{999000}+\frac{567}{999000}\\[0.25cm] &= \frac{1233333}{999000} \\[0.25cm] &= \frac{45679}{37000}\end{aligned}$