## 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, 3 years ago
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Some of the comments appear to be based solely on reading the title or the thumbnail--so I've changed the title to more accurately reflect the contents. The surreal numbers contain the real numbers, and 0.9 repeating is equal to 1 in both. However, we think about 0.9 repeating conceptually as some number that is infinitely smaller than 1. There is such a number in the surreal numbers, and it is not equal to 1!

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 fact-checking this video, I came across a number of conflicting arguments that seemed valid.

http://math.stackexchange.com/questions/926581/in-the-surreal-numbers-is-it-fair-to-say-0-9-repeating-is-not-equal-to-1
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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• 3 years ago