I was using a calculator one day, and punched in a number, then punched "divided by", then punched another number and got back .01234567890123456789... I thought the calculator was wacko, but the calculator was right.
If the two numbers I put in were relatively prime, what were they?
Express the repeating decimal .01234567890123456789... as an infinite geometric series. We can write .01234567890123456789... = (123456789/10^10) + (123456789/10^20) + (123456789/10^30) + ...
Now, the sum of an infinite geometric is given by the formula S = a / (1-r) , where a is the first term of the series and r is the common ratio that we multiply each term by to get the next. In our series here, we have :
a = 123456789/10^10 and r = 1/10^10 .
Therefore, S = (123456789/10^10) / (1 - 1/10^10)= (123456789/10^10) / (9999999999/10^10)
= (123456789/10^10) * (10^10/ 9999999999)
= 123456789/9999999999
= 13717421/1111111111 .
Hence .01234567890123456789... =
13717421/1111111111 .
Another Solution
