# Infinitely smallest number

Without getting side-tracked by arguments of whether 0.9 recurring can ever equal 1, is there an accepted mathematical expression for the difference between 0.9r and 1?
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Commented:
d = 1 - 0.9r
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Commented:
I had to look this one up, and get the gist of current thinking since I have blindly accepted through elementary "proofs" that
``````0.9r == 1.0r
``````
This wiki includes more proofs of this so-called "fact".
https://en.wikipedia.org/wiki/0.999...
While this link tries to prove the above equality, there are also several discussions of the difference between the left hand side and the right hand side of the above equality, which leads to a section on Infinitesimals,
https://en.wikipedia.org/wiki/0.999...#Infinitesimals
which, in turn, leads to Hyperreal number systems.
https://en.wikipedia.org/wiki/Hyperreal_number

According to a quora.com article I read today, but misplaced it (sorry), the hyperreal number system and these so-called infinitesimal numbers are not well-accepted by mathematicians. So, if you believe that one author on quora.com, then your answer is "no". However keep in mind, this is not the first time that a mathematical approach was considered poor math and rejected by the majority of mathematicians, possibly because of lack of rigor. I'm thinking of Fourier whose work has transformed mathematics. Possibly, a new extended math will come out of these hyperreal number systems. Sorry I cannot be of more help. I have TL;DR syndrome at the moment.
First Fourier was an engineer, not a mathematician (my understanding is that he was chief of engineers in Napolean's expedition to Egypt) so that may have had an influence. However, those who rejected his paper were quite right: what he said was not valid. Fourier made two claims: that any "square integrable" function had a Fourier, sine and cosine, series and that any such series, with some conditions on the coefficients, corresponded to such a function. The first is obviously true, just by doing the integration to exhibit the coefficients. The second is false- there existed such series that did NOT converge to such functions. In order for Fourier's method to be valid, both statements had to be true. Yet, engineers went ahead blithely using Fourier's method to solve differential equations, getting solutions that were clearly correct. That was a major reason for developing the "Lesbesque Integral" which was unknown up to that time. Both statements ARE true if you use the Lebesque Integral rather than the Riemann Integral.
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Commented:
In the quora.com article I read, the claim was that for real and complex numbers, the above equality holds true. But that assumes that we all agree upon what a real number system is. If you disagree, then you form your own foundation that deviates from the norm.

How real are real numbers?
https://arxiv.org/pdf/math/0411418.pdf
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Commented:
1 - 0.9r  = 0
1
Author Commented:
Thanks, although I was looking for an expression for the answer rather than the sum, I suppose the sum (1-0.9r) is the closest philosophical expression.
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Commented:
phoffric gives all the info necessary (and a bit more)
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