On May 3, 6:47 pm, WM <mueck...@rz.fh-augsburg.de> wrote: > On 3 Mai, 08:45, Dan <dan.ms.ch...@gmail.com> wrote: > > > 1/9 = 0.1111 ..... (infinity ones) it's ALSO a valid continuous > > magnitude . > > It is a limit, but it has no decimal representation. > > If it had a decimal representation, then it would be in the following > list which contains all decimal representations with the digit 1 > having a finite index. > > 0.1 > 0.11 > 0.111 > ... > > Or can you determine a finite index which is not occupied by a 1? > > Being infinite is a valid mathematical notion, but it is not a means > to distinguish decimal representations. If there is a sequence of 1 > larger than all sequences of the list (not only larger than every), > then let me know the difference by naming the indices. Otherwise stop > asserting unmathematical properties. > > Regards, WM
Every real number ,in decimal representation , is an infinite sequence . But those on your list are followed by an infinity of zeroes : 0.1 ............. 0.110000000 ..... 0.111000000 .....
that we choose not to write , for convenience .If we are to understand the real numbers correctly , we must take this sequence as formally complete . Just as the natural numbers are complete, for every natural number n in existence , make a choice between 0 ,1 .... 9 . This choice will be the nth digit of your chosen real number . Because you can make a choice for 1 , a choice for 2, a choice for 3 , .... the completion of your choice process, the 'summation' of your choices , is the real number you choose . :
r = 0.3 -> 0.31 -> 0.314 -> 0.3141 -> 0.31415 -> ....... = pi /10
Each real number in your list is followed implicitly by an infinite amount of zeroes , just as the antidiagonal is followed explicitly by an infinite amount of ones .
0.10000000..... differs from 0.11111111..... =1/9 at the second digit .
0.11000000..... differs from 0.11111111..... = 1/9 at the third digit .
0.11100000..... differs from 0.11111111..... = 1/9 at the fourth digit .
Real numbers (like pi , or e , or 1/9 ) have their decimal representation formally complete in actual infinity . Natural numbers decimal representation ranges only in potential infinity . If you don't like it, stick to natural numbers. That you can't write the full decimal representation of 1/9 doesn't change its existence . It has a lot of digits . And every . Single.One of them. Is 1 . You're saying black hole singularities (actual infinity) don't exist because we can't test them in the lab. It's effects DO exist (potential infinity) , the bending of light, because we can observe those . But "God" forbid we do an indirect derivation and maybe, just maybe , realize that the Universe is bigger that the inside of our mammalian heads .