Proof: Numbers with repeating blocks of digits are rational

by e(ho0n3
Tags: blocks, digits, numbers, proof, rational, repeating
 P: 1,370 Hi everyone, I need to prove that any number with a repeating block of digits is a rational number. Someone told me I should first find a method of constructing a rational number in the form a/b from a number with repeating blocks of digits (and to do it with very 'easy' numbers first). I'm still stumped though. For example, given 0.33333..., how do I show that it equals 1/3?
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P: 2,188
 Quote by e(ho0n3 Hi everyone, I need to prove that any number with a repeating block of digits is a rational number. Someone told me I should first find a method of constructing a rational number in the form a/b from a number with repeating blocks of digits (and to do it with very 'easy' numbers first). I'm still stumped though. For example, given 0.33333..., how do I show that it equals 1/3?
You can express 0.333.. as a geometric series:

$$\sum^{\infty}_{n=1} \frac{3}{10^n} = 3\sum^{\infty}_{n=1} \left(\frac{1}{10}\right)^n$$

use:

$$\sum^{\infty}_{n=1}r^n = \frac{r}{1-r}$$

$$3\left(\frac{\frac{1}{10}}{1 - \frac{1}{10}}\right) = \frac{1}{3}$$
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P: 11,137

Proof: Numbers with repeating blocks of digits are rational

n=0.3333...
10n=3.3333...
=>10n-n=9n=3.0
=>n=3/9=1/3
 PF Patron Sci Advisor Emeritus P: 11,137 Let n == 0.abc...kabc...kabc... repeating blocks of (abc...k), each block having r digits Then n*10^r == abc...k(point)abc...kabc...kabc... i.e. move the decimal point r places to the right. Now subtract, n*(10^r - 1)==abc...k digits after the decimal point vanish So n== abc...k/(10^r - 1)= p/q, a rational number QED Plz excuse the freedom I've exercised with notation.
 PF Patron Sci Advisor Emeritus P: 11,137 Alternatively, jcsd's geometric sum method can be generalized for repeating blocks of any size. PS: Also look at recent post on n/7, n=1,2,...6
 P: 20 .ABC...Z (with repeating length L)= $$A*\sum_{k=1}^\infty \frac{1}{10^k^L}+B*\sum_{k=1}^\infty \frac{1}{10^k^L*10}+C*\sum_{k=1}^\infty \frac{1}{10^k^L*10^2}+...+$$$$Z*\sum_{k=1}^\infty \frac{1}{10^k^L*10^L/10}$$ Because A, B,..., Z, are rational and $$\sum_{k=1}^\infty \frac{1}{10^k^X}$$ is rational for any X, the above sum is also rational.
 P: 101 have you tried using different bases other than 10?

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