jasen Posted November 20, 2015 Report Share Posted November 20, 2015 I think I can write every rational number in repeating decimals1 = 0.99999999........4 = 3.99999999.......17.5 = 17.4999999....and so on.wheather my statement is right or wrong, prove your agreement or objection. Quote Link to comment Share on other sites More sharing options...

0 jasen Posted January 22, 2016 Author Report Share Posted January 22, 2016 (edited) There is 1 number can't written like that. The number is 0. So the statement is wrong. Edited January 22, 2016 by jasen Quote Link to comment Share on other sites More sharing options...

0 Rainman Posted November 21, 2015 Report Share Posted November 21, 2015 Yes. For any denominator b, let R = {0, 1, 2, ..., b-1} be the remainder set of b. Let D = {0, 1, 2, ...., 9} be the set of digits in the decimal system. Define a function f : R → D, such that f(r) is the number of times b goes into 10r. Define a function g : R → R, such that g(r) is the remainder for the division 10r/b. For any numerator a, to perform the division a/b, start by writing down the number of times b goes into a, let r be the remainder. If r = 0, you are done. Otherwise write down the decimal point. Then repeat this algorithm:1) Write down f(r).2) If g(r) = 0, stop.3) Let r = g(r).If g(r) = 0, the process terminates. Otherwise, as soon as g(r) takes a value it has previously had, the process repeats. Since there are only b different values g(r) could have, the process must terminate or repeat in at most b-1 steps. Meanwhile, f(r) is just a function of g(r) because of step 3, so the decimals we write down must also terminate or repeat. If they terminate, we just do what you did (lower the last decimal by 1 and add repeating nines). Quote Link to comment Share on other sites More sharing options...

0 jasen Posted November 23, 2015 Author Report Share Posted November 23, 2015 Hidden ContentSomeone still can argue that your reasoning is not a prove. Quote Link to comment Share on other sites More sharing options...

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## jasen

I think I can write every rational number in repeating decimals

1 = 0.99999999........

4 = 3.99999999.......

17.5 = 17.4999999....

and so on.

wheather my statement is right or wrong,

prove your agreement or objection.

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