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:

9s_equals

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:

9s_algebra
9s_thirds
9s_elevenths
9s_sevenths

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

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One response to this post.

  1. Hi, cool post. I have been thinking about this issue,so thanks for writing. I’ll definitely be subscribing to your posts. Keep up the good work

    Reply

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