0.999... Repeating Is Equal To 1, But Something Like It Is Not (Introduction To The Surreal Numbers)
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by Super User, 4 years ago
Technical Points
The surreal numbers are a pretty complicated logical system. One point to make is that the surreal numbers contain the real numbers, and so 0.999.. = 1 since this is true for the real numbers.
However, there is a psychology about 0.999... repeating as a number that is infinitesmally smaller than 1. The surreal number {0, 1/2, 3/4, 7/8,...  1} is infinitesmally smaller than 1 and would satisfy our psychological idea of 0.999.. repeating. I believe {0, 9/10, 99/100, 999/1000,... 1} would also represent the same surreal number; however, I have not been able to verify that so I did not include it in the video.
When factchecking this video, I came across a number of conflicting arguments that seemed valid.
http://math.stackexchange.com/questions/926581/inthesurrealnumbersisitfairtosay09repeatingisnotequalto1
http://forums.xkcd.com/viewtopic.php?f=17&t=76752
http://en.wikipedia.org/wiki/0.999...#Hackenbush
I also did my best to work through John Conway's book that introduces surreal numbers called "On Numbers and Games."
http://en.wikipedia.org/wiki/On_Numbers_and_Games
I welcome thoughtful comments about the surreal numbers.
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