The Secret Number
strangehorizons.com
strangehorizons.com
Sure, you can multiply 1/3 (== 0.33333_) by 3 and make it look like they're the same, but there's really a tiny little epsilon of something that's getting zeroed out.
It's a conspiracy I tell you! They hide all the cool stuff with infinity and you can't see it any more, then it's like it never existed.
http://en.wikipedia.org/wiki/Archimedean_property
It took thousands of years to accept that the parallel postulate might not be required, and lots of neat geometry emerged. Why not have number systems where infinitesimals are allowed?
There are. A few of them, in fact. Look up the hyperreals.
In mathematics, a definition cannot be 'wrong'; it can lead to an inconsistency, but that isn't the same thing.
The definition of the real numbers (the set of numbers where 0.999... == 1) does not lead to any inconsistency. However, there are other consistent definitions of sets of numbers where (the equivalent of) 0.999... does not equal (the equivalent of) 1. Those sets of numbers have values mathematicians call 'infinitesimals', which do not exist in the set of the real numbers. The hyperreals are one such set of numbers.
More to the point, we aren't just making up numbers here to enjoy infinite decimal expansions; the reason is that you can wrap the radius of a circle around its circumference more than 3 times (or, if you don't like numerals, as many times as letters in 'pie'), but fewer than 4 (not as often as letters in 'easy')—and no changing of units is going to get around that.
Mathematicians are aware of bleem. It's what you get when you remove the induction axiom from Peano arithmetic: A number which is not in the set {0, 1, 2, ...}.
In general the higher up you go in academia the more open people are to "variant realities". Most of our analysis of curved space-time comes to us thanks to mathematicians who thought they were playing with entirely theoretical constructs ("what happens if we remove the parallel postulate?").
See also http://en.wikipedia.org/wiki/Non-standard_model_of_arithmeti...
Generally, I don't think it's fair to characterize this story as something studied by mathematicians. Really, it's an exercise in reasoning about nonsense.
http://en.wikipedia.org/wiki/1_%2B_2_%2B_3_%2B_4_%2B_%E2%80%...
-the Laundry novels (Charlie Stross). Here's one novella: http://www.goldengryphon.com/Stross-Concrete.html
-Luminous/Dark Integers (Greg Egan).
The first story is "Luminous" from 1995, but I don't have a link to a free copy.
The second is "Dark Integers" which you can find here: http://www.asimovs.com/_issue_0805/DarkINtegers.shtml
EDIT: Sorry; on looking closer, it's clearly Gardner's "The No-Sided Professor" (http://en.wikipedia.org/wiki/Fantasia_Mathematica).
The current recommended value of α is 7.2973525698(24)×10−3 = 1/137.035999074(44).
I was absorbed with the story and I could feel the silence of my companion as he traveled behind me gazing at the disappearing stars.
A synopsis for others to decide if they want to read it: Chloe is a beautiful humanoid companion. Deliberately implanted in the programming of her mind is a minor reasoning defect. Whereas an ordinary person could live with it, and perhaps not even be aware of it, Chloe cries and has fits whenever she becomes aware of the flaw in her chain of thought. A psychologist is brought in to figure things out.
By the way, "Sample 27" is the title and not the 27th sample story from the author as I originally thought.
A suggestion to the author: You might want to either not use HTTPS or buy a certificate. I don't think it's a barrier to Hackernews readers but most people won't get past the browser's security exception box.