1
$\begingroup$

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.

$\endgroup$
3
  • 1
    $\begingroup$ How do you calculate $5(6666...7777)$? $\endgroup$ – vadim123 Aug 28 '13 at 16:30
  • 4
    $\begingroup$ 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
2
$\begingroup$

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.

$\endgroup$
4
  • $\begingroup$ 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.