**Re: Interesting algebra involving the square root of a summa**
Aha! I will do arithmetic on the numbers of the form 11.....111 (agentredlum discovered such numbers in a very old topic :

viewtopic.php?f=40&t=18441).

the agenterdlum number sqrt(111.....11) can be mapped in the reals by transforming sqrt(11...1) to sqrt(1.111.....) where the place of the decimal doesn't matter. Now, sqrt(1.111....) = sqrt(11(0)/99) where 11(0) means there can be uncountably/countably many zeros after 11 (i.e. 110, 1100,...). we will take 11(0) = 11 for our computational matters. sqrt(11/99) = sqrt(11)/3sqrt(11) = 1/3 = 0.333333.... which is in turn represents the agentredlum number 3333.....333. Hence, no matter where the decimal is situated, the next number after the decimal is 3, and so as the next and next and e.t.c. Hence, the 2013-th digit (irrelevant) after the decimal (irrelevant,too) is always 3.

NOTE : @agentredlum, seems like we find a practical application of your idea

I must admit, you're just brilliant!!

Balarka

.