Construction of an infinite number type and other ideas

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.

• How do you calculate $5(6666...7777)$? – vadim123 Aug 28 '13 at 16:30
• 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. – MJD Aug 28 '13 at 16:39
• I am aware of p-adic numbers and have played with them. Some theoretical physicists are also exploring them. – 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.