1 - 0.99999...

i've seen this video posted somewhere in metacafe before...
n i've posted the flaw in that video, so might as well do it here

the flaw is that u assumed that
10 x 0.999...= 9.999...

however the truth is that
10 x 0.999 = 9.999....990(the last digit has to be a zero)

hence,
10x - x = 9.999...990 - 0.9999...
9x = 8.999....991
x = 0.999...

anyway nice try on fooling with kids who aren't fundamentally strong at maths..

Consider this:

define m and n to be real numbers, where m < n.

if m < n then there must exist a real number k such that m < k < n.

now assume 0.999... < 1. there must be some number k such that 0.999... < k < 1. k cannot be defined, which contradicts the original assumption that 0.999... < 1. you can do the same thing assuming 1 < 0.999... (silly) and find that your result contradicts the original assumption. if 0.999... < 1 is false, and 0.999... > 1 is also false, 0.999... must be equal to 1.

I know this isn't a rigorous proof. I've left out the proofs of intuitive assumptions. I hope this helps.

you logic is clearly flawed on your 5th statement because .999... x 9 = 8.999... if not before that one clearly is wrong.

what a bunch of fuckin noobs

If you need another way to prove it, 'equals' can be defined as
if abs(a-b)<m for all m>0 then a=b
suppose you were to choose m = 10^-9, then simply carry the repeat out 10 times and concatenate (cuz rounding would actually be beneficial) and you'll still be fine.
abs (1-.9999999999) = 10^-10
10^-10 < 10^-9
if you want an even tighter tollerance, go with m=10^-100. or m=10^-1000. or the limit as m approaches zero positively. No matter how small m is, you can always take .9bar out further such that 1-.9bar is less than your tollerance

.999... is shorthand for the sum of an infinite series of decimals, which is defined in terms of a limit. anyone can see here that the limit of .999... is 1, since it keeps getting closer and closer to 1 as the decimals go further out. the only room for debate lies in how inadequate your understanding of mathematical notation is.

.999999999... = 1, just like .333333... = 1/3 and 3824.555... = 3824 + 5/9. these are merely different representations of the same numbers.

I know S**t about math. But this just can't be true.

In my opinion, the distance between 9,999(bar) and 10 [at 10x=9,999(bar)] should get ten times infinitesimally bigger than the one between 0,999(bar) and 1 -as it does with rational numbers (0,98 it's 0,02 far from 1; 9,8 it's 0,2 from 10).

Hence, in the example, the ",999(bar)" inside 9,999(bar) is infinitesimally smaller than X; forcing the result of subtracting X from 10x to be logically smaller than 9.

9x=8,999(bar)

x=8,999(bar)/9

x=0,999(bar)

0,999... = 9/10 + 9/100 + ... + 9/10^n (where n approaches infinity)
This series converges (geometrical series, where q=1/10)
So the sum of this series is
9 * [1/(1-1/10) - 1] = 9 * [10/9 - 1] = 9 * 1/9 = 1
So, it's still one.

(I like this better, at least it's smaller)
0,999... = 3 x 0,333... = 3 x 1/3 = 1
:P

999 / 1000 =.999. How can you be a stupid math nerd?