I find the surreal numbers very interesting. I have tried my best to work through John Conway's On Numbers and Games and teach myself from some excellent online resources.
I have prepared a short video to introduce surreal numbers, but I want to double-check some of my claims and would appreciate some help.
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Is it fair to say that $0.999$ repeating is not equal to $1$ in the surreal numbers? This is the title of my video, so I really want to make sure it's a reasonable statement.
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Can $\{0, \frac12, \frac34, \frac78, \dots \mid 1\} = 1 – \epsilon$ be thought of as $0.999$ repeating?
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Is the number $\{0, \frac{9}{10}, \frac{99}{100}, \frac{999}{1000}, \dots \mid 1\}$ also equal to $1 – \epsilon$?
I am hoping the the answer is "Yes" for each question. If not, please let me know and I will get on the task of majorly revising the video. Thanks!
Best Answer
The surreal numbers contain the real numbers (as well as infinite and infinitesimal numbers).
Both $0.999\dots$ and $1$ are real numbers.
In the real numbers $1 = 0.999\dots$ and so it must also be true in the surreal numbers.
As Bryan correctly points out in his answer, surreal numbers which are not real numbers do not have a decimal expansion. This would appear to undermine the idea of using $0.999\dots$ to represent a surreal number.