3 / 3 = 1. But what if I write it as ( 1 + 1 + 1 ) / 3 or 1 / 3 + 1 / 3 + 1 / 3?Now, 1 / 3 is a recurring/repeating/non-ending decimal so if we add these three, i.e. 0.3333... + 0.3333... + 0.3333... we will get infinitesimally close to 1 but not 1Is there a way to show that these decimals do end and will eventually become 1?

Question

3      /    3  =  1. But what if I write it as   (  1  +  1  +  1  )      /    3 or   1      /    3  +  1      /    3  +  1      /    3?Now,   1      /    3 is a recurring/repeating/non-ending decimal so if we add these three, i.e.   0.3333...  +  0.3333...  +  0.3333... we will get infinitesimally close to   1 but not   1Is there a way to show that these decimals do end and will eventually become   1?

kamphundg4 · Accepted Answer

0.            3      ¯       is not in the process of becoming   1      /    3 or anything else. If it is a meaningful expression then it is already equal to   1      /    3 or not equal to   1      /    3.Your Q is not trivial. A logical foundation for       R   was only developed in the 19th century.

Dean Summers · Accepted Answer

The decimals do not end, but that&#039;s not really a problem, since  0.999999999  ⋯  =  1There are several proofs of this, which you can look at yourself.The simplest one is to say that if   x  =  0.99  …, then   10  x  =  9.99  …, and if you subtract the two equations you get   9  x  =  9 which means   x  =  1Another way is to see that  0.99  ⋯  =  0.9  +  0.09  +  0.009  +  ⋯  =    =  9  ⋅  (  0.1  +  0.1  +  0.001  +  ⋯  )  =  9  ⋅      ∑          i      =      1        ∞        10          −      i        =  9  ⋅      1    9    =  1