# Thread: Show that rational numbers correspond to decimals

1. ## Show that rational numbers correspond to decimals

Show that rational numbers correspond to decimals which are either repeating or terminating.

Hint: If q = m|n, then when dividing m by n to put q into decimal form there are at most n different remainders. Conversley, if d is a repeating decimal, then find s,t such that 10^sd - 10^td is an integer.

I took an example and understand, but don't know how to write it in a general form.

My example:

Part I
2/7
0.2857142...
72.000...
-14
60
-56
40
-35
50
-49
10
- 7
30
-28
20
-14
6....
So the my bold numbers can only be 1-6

Part II
R = 12.34545...
100R = 1234.54545...
100R - R = 1222.2
99R = (12222 / 10)
R = (12222 / 990)

Any help on writing this in a general form would be greatly appreciated.

2. I would recommend that you interpret "terminating decimal" as a "repeating decimal" where the repeating part happens to be 0. That way, you can simply show that any rational number is a "repeating" decimal.

let m and n be two integers. Then $\frac{m}{n}$ means "m divided by n". If m> n, then there exist q and r ("quotient" and "remainder") such that m= qn+ r with r< n. That is, $\frac{m}{n}= q+ \frac{r}{n}$ so it is sufficient to prove this for $\frac{r}{n}$ where r is less than n. When we "long divide" n into r, we will have, at each decimal place, a "remainder" that is less than n and larger than or equal to 0, then bring down a "0" and continue. The critical point is that there can only be n different remainders, 0 to n-1 as I said. After at most n steps, a remainder must repeat, we bring down a "0" again so we are dividing n into exactly the same number as before and everything repeats.

3. What Halls has shown is that for any rational there is some repeating decimal which is equal to it. We can also show that any repeating decimal is rational:

Let $x$ be a repeating decimal, we can split it into two parts:

$x=a+10^{-l} p$

where $a$ is the terminating non periodic part of length $l$ digits and $p\in[0,1]$ is the purely periodic.

Let the period be $d$ digits long and the terminating decimal with $d$ digits which are equal to $p$ terminated after the $d$-th be $u$ digit then:

$\displaystyle p=\sum_{n=0}^{\infty} u \times 10^{-n\times d}=u\sum_{n=0}^{\infty} 10^{-n\times d}$

Now the sum on the right is a convergent geometric series and so we can write down its sum, and it is rational, and as $u$ is rational $p$ is rational. Hence as $a$ is rational $x$ is rational.

CB

4. On a different forum, many years ago, a person asked how to show that any rational number could be written as a fraction with integer numerator and denominator. It took me a moment to realize that he must have been taught that the definition of "rational number" is a number that can be written as a terminating or repeating decimal. Of course, it can be done either way.