Construct an infinite number(?) that has a beginning, an infinite middle, and a end; such as 1000...0001, or 98111...1114 etc. Has this type of number been explored? Under some simple multiplications, 5(1000...0001)=5000...0005, other mathematical operations are not determinable.

1/1000...001, or 1/5200...0008, etc. may have different infinitesimal properties. Can the surreal numbers include these?

π(1000...000) would be sort of like a specific infinite ω and π(1000...000)/(1000...000)=π. Surreal number types as in πω/ω.

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    $\begingroup$ How do you calculate $5(6666...7777)$? $\endgroup$ – vadim123 Aug 28 '13 at 16:30
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    $\begingroup$ You might be interested to learn about "$p$-adic numbers". Ordinary numbers are infinite to the right of the decimal point, for example $\frac{17}7 = 2.4284742\ldots$. $p$-adic numbers are infinite to the left of the decimal point instead; for example $\ldots 9999.2 + 1 = 0.2$, so $\ldots9999.2 = -\frac45$. Something goes wrong if you try to make the numbers infinite in both directions, but I forget offhand what it is. $\endgroup$ – MJD Aug 28 '13 at 16:39
  • $\begingroup$ I am aware of p-adic numbers and have played with them. Some theoretical physicists are also exploring them. $\endgroup$ – 11dim Aug 28 '13 at 16:46

There is a way of writing surreal numbers called "Gonshor's sign expansion". Basically, every Surreal number is a "string" of +s and -s (actually a map from an ordinal to {+,-}). For the finite strings, this matches somewhat closely to tally marks and binary. ""=0, "+"=1, "++"=2, "+++"=3, "-"=-1, "--"=-2, etc. $``+-"=\frac12=.1_2$, $``\underline{++}+-\underline{++-+---+-}''=\underline{10}\,\,. \underline{110100010}\,1_2$, etc.

However, there are infinite ordinals like $\omega$, which give rise to numbers like "+-+-+-+-..."=2/3, but those numbers have no end. Luckily, lots of ordinals do have an end, like $\omega + 3$. Then you get numbers like "+-+-+-+-... +++", which is probably 2/3+3*"+-------...", where "+-------..." is a positive surreal less than "+-"=1/2,"+--"=1/4,"+---"=1/8, etc.

I don't know if this is satisfying to you, but it is a number system where some of the numbers have infinite representations with ends.

  • $\begingroup$ I am aware of the omega + 3, etc. type ordinals. However numbers with a infinite middle form specific infinite ω's, such as π(1000...000), e(1000...000), πe(1000...000), etc.. $\endgroup$ – 11dim Sep 6 '13 at 18:27
  • $\begingroup$ @11dim I'm afraid I am not sure what you are saying here. Can you reword the question? Or if it's not a question, I'm sorry I couldn't be of more help. $\endgroup$ – Mark S. Sep 8 '13 at 4:29
  • $\begingroup$ "Construct an infinite number(?) that has a beginning, an infinite middle, and a end; such as 1000...0001, or 98111...1114 etc." Some mathematical operations are ill defined, others are not. I use a circle of infinite diameter to picture such omega type numbers. My cut on the circle defines a beginning/end demarcation. $\endgroup$ – 11dim Sep 5 '14 at 21:32
  • $\begingroup$ @11dim I still don't know if you're asking a question, or commenting on my answer, or just trying to advertise your ideas in the comments to an answer, which is probably not the best place for them. $\endgroup$ – Mark S. Sep 7 '14 at 4:28

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