Messing around with the ‘ad infinitum’!

Anybody who has observed the The Ultimate Clock For Nerds properly would know that the 7th hour is designated specially. I know this is pretty elementary level high school stuff but something about this intrigued me. Hence, I am writing on this. Part of the equation had .9 with bar sign. We should remember by now that this is used to symbolize the “infinitely repeating decimals”. The outcome of this all is something like this:

Now let me manifest why it intrigued my mind. How can this be equal to 1. Here’s how it can be proved albeit in a generalized manner:

0.1111……. is 1/9 where … represents “one”’s ‘ad infinitum’.
0.2222……. is 2/9
0.3333……. is 3/9 or 1/3 or 33 1/3%
0.4444……. is 4/9
0.5555……. is 5/9
0.6666……. is 6/9 or 2/9 or 66 2/3%
and so and so on…

until 0.9999……. is 9/9 or 1 whole. the reason for that is because 0.9999 goes on forever….

0.9999999999999999999999999999999999999999999999999999999999999999999999999999
999999999999999999999999999999999999999999999999999999999999999999999999999999
999999999999999999999999999999999999999999999999999999999999999999999999999999
999999999999999999999999999999999999999999999999999999999999999999999999999999
999999999999999999999999999999999999999999999999999999999999999999999999999999
999999999999999999999999999999999999999999999999999999999999999999999999999999
999999999999999999999999999999999999999999999999999999999999999999999999999999
999999999999999999999999999999999999999999999999999999999999999999999999999999
999999999999999999999999999999999999999999999999999999999999999999999999999999
999999999999999999999999999999999999999999999999999999999999999999999999999999
999999999999999999999999999999999999999999999999999999999999999999999999999999
999999999999999999999999999999999999999999999999999999999999999999999999999999
999999999999999999999999999999999999999999999999999999999999999999999999999999
999999999999999999999999999999999999999999999999999999999999999999999999999999
999999999999999999999999999999999999999999999999999999999999999999999999999999
999999999999999999999999999999999999999999999999999999999999999999999999999999
999999999999999999999999999999999999999999999999999999999999999999999999999999
999999999999999999999999999999999999999999999999999999999999999999999999999999
999999999999999999999999999999999999999999999999999999999999999999999999999999
999999999999999999999999999999999999999999999999999999999999999999999999999999
999999999999999999999999999999999999999999999999999999999999999999999999999999
999999999999999999999999999999999999999999999999999999999999999999999999999999
999999999999999999999999999999999999999999999999999999999999999999999999999999
999999999999999999999999999999999999999999999999999999999999999999999999999999
9999999999999999999999999999999999999999999999999999999999999999999999999999……

and so on.

It’s basically one and so it is considered that.

That’s why 0.6666……is 0.66

2/3 that’s 2/3 of 100 is 66.666666666666666666666666666666666666666666666666666666666666666666666…

you get the point.

So 0.99999…. = 9/9 = 1

Taking another route:

let, x = 0.9999…
Therefore, 10x = 9.9999…
Therefore, 10x – x = 9.9999… – 0.9999…
Therefore, 9x = 9
Hence, x = 1

Here’s the same proof albeit in a different & pictorial representation:

By the way, for the more mathematical minded of you, the mathematical equation of this is:

lim(m –> ∞) sum(n = 1)^m (9)/(10^n) = 1

Here’s one more discussion on the same:

http://www.curiousmath.com/index.php?name=PNphpBB2&file=viewtopic&t=48