there have apparently been circulations of a “mathematical” proof that goes as follows
x=.999[bar] the bar signals recurring numbers.
10x=9.99[bar]
10x -(x)= 9.99[bar] - (.999[bar])
9x=9
x=1
at first you get kind of dumb founded that this seems to be valid… upon further examination this is what i come up with.
when you multiply .999[bar] by ten, those ten “infinitesimal differences from ten” are all represented by a single infinitesimal. what i mean by this is that when you multiply the .999[bar] there is a loss pf precision because the infinitesimal difference from .999[bar] to 1 can always be made more infinitesimal. meaning a 9 can always replace where there would have been an 81.
heres a mental image that does the trick.
imagine a cube that is .999[bar] inches in height. next imagine a wall with inch markers on it from 1 to 10 starting from the ground up.
take the .999[bar] cube and place it next to the 1 inch mark. it is below the 1 line but impossibly close to notice.
take another .999[bar] cube and place it on top of the existing one… logically the difference would be twice as great from the 2 inch mark than the first block is from the 1 inch mark.
add a third and the difference would be even greater.
when you delve into the nature of these differences the only way to describe them is as infinitesimals.
1 divided by infinity…
the problem is that ten infintesimals are represented as one infinitesimal when a number ends like .999[bar]
an obvious misrepresentation… thoughts?