in my last post are some words missing :D but i can edit it.
there a are some strange things happening when infinite numbers are appearing. if you are interested in such things you can google for instance hilberts hotel. or you can ask yourself the tricky question wich numbers are more: the natural or the rational.
#8
i guessing while in school learning this, you were in a class studying math?
and yes that is a proof acceptable in any number theory class.
question is the equation sigma(x - infinity) = (1/2)^x
equal to 1, because that equation can get as close to one as 0.9 repeating.
#stoo
youre convergence argument is wrong
0.9+0.09+0.009+0.0009+...=1 for infinite steps!
it isnt close to 1, it is 1 (for infinite steps of cause)!
@#7
when i add 0.1 9 times i arrive at the reasonable answer of 0.9.
if i try and add the numeric representation of 1/9 ( which is not actually possible) 9 times, i start by trying 0.11 and then 0.111. so when i arrive at the number 0.99999999999999999 or however many places past the decimal you choose, the reality will be that you left out the small fragment of 0.000000000000000001111111111111111111111111
which once accounted for( again not actually possible) give you the necessary difference between 1 and whatever number you got to.
like this
0.1 left out 0.0111111111111111111111111
so when you add
0.111.... to itself
you are not ever adding 1/9 to itself, but something slightly less than 1/9.
and all of the little bits you left out, will be the difference between 1 and your answer.
you can have a proof for mathematicans only when my schoolmath doesnt please you. you know what the definition of the real numbers is? it has to do with an axiom about nested intervals (i dont know the english word. its "Intervallschachtelung" in german). you can reach every real number with nested intervals. for 1 such nested intervals are 0.9+0.09+0.009+0.0009+...we are talking about .
# ah ok now i know what you are meaning. but its mathematical wrong. what you mean is only right if you have no real numbers (not sure about this point) for instance rational numbers.
you can bring the example with a computer. it doesnt has infinitesimal space, obviously. so for it 0.9999... is not 1.
but the world is not a computergame :D and still 0.9999=1.
1/3 = .3333...
3/3 = .9999...
1 = .9999...
this is totally wrong:
"@#7
when i add 0.1 9 times i arrive at the reasonable answer of 0.9.
if i try and add the numeric representation of 1/9 ( which is not actually possible) 9 times, i start by trying 0.11 and then 0.111. so when i arrive at the number 0.99999999999999999 or however many places past the decimal you choose, the reality will be that you left out the small fragment of 0.000000000000000001111111111111111111111111
which once accounted for( again not actually possible) give you the necessary difference between 1 and whatever number you got to.
like this
0.1 left out 0.0111111111111111111111111
so when you add
0.111.... to itself
you are not ever adding 1/9 to itself, but something slightly less than 1/9.
and all of the little bits you left out, will be the difference between 1 and your answer."
you dont let anything out. you have a infinit number of digits. its no problem if you just try to understand what happens when things are going to be infinit.
your mother indeed the world is not a computer, neither is it mathematical theory.
a famous story comes to mind here and it involves Richard Feynman.
Feynman was talking with two theoretical mathematicians around lunchtime. one of them asked Feynman if it was possible to stretch his orange around the sun, given that you can cut the orange into infinitely many pieces and put it back together. Feynman replied no. the mathematicians gathered everyone around Mr. Feynman is going to learn about the Banach–Tarski paradox. at which point Mr Feynman interjected and apologized, he thought they were considering doing this with an actual orange, which cannot be divided smaller than its constituent parts.