#### Non Archimedean version of Zeno's paradox.

Recall the usual version has Achilles chasing a turtle, where the turtle is given a ten cubit head start and Achilles runs ten times faster than the turtle. When Achilles runs 10 cubits, the turtle goes 1 cubit. When Achilles runs 1 cubit, the turtle goes 1/10 of a cubit. It follows that Achilles catches the turtle in 10 + 1 + 1/10 + ... cubits. In modern notation, one can write these numbers as decimals yielding
10 + 1 + .1 + .01 + ... = 11.1111... = 11 1/9.
It follows that Achilles catches the turtle in 11 1/9 cubits.

The non Archimedean case has the turtle trying to catch Achilles who is now given a ten cubit head start.
When the turtle goes 10 cubits, Achilles will have run 100 cubits. When the turtle goes 100 cubits then Achilles will have run 1000 cubits, and so on.
It follows that the turtle goes 10 + 100 + 1000 + .... Ordinarily, this would be considered as meaningless or ``infinity''. However, one can consider this purely formally, in other words, without caring too much about the actual meaning of the numbers and just using algebraic manipulations. So let x = 10 + 100 + 1000+ ..., then
9(x + 1) = 9 + 90 + 900 + 9000 + ... = ... 9999.
Adding 1 to this gives 9( x + 1) + 1

... 9999999

+ 1

= ... 0000000 = 0

where there is never a leading one, as the carries go on forever. It follows that 9(x+1) + 1 = 0, so x = -1 1/9.

Note that this actually gives a correct answer!

See I.M., Gelfand and A. Shen, Algebra Birkhauser, Boston, 1993. Reviewed by R. Askey, Amer. Math. Monthly #102 (1995), 78--81.