becker (01/28/83)
#N:uiucdcs:28200006:000:1128
uiucdcs!becker Jan 27 17:03:00 1983
I found a really neat proof in a book called "What Are Numbers" by Louis
Auslander; however, my roommate doesn't believe it, either.
notation: b(i) is a subscripted digit. There might be a more posh way
to type this, but I don't know how.
<> means 'is not equal to'.
here goes:
consider the infinite decimal
a = .9999...
we want to find a number
b = b(1)b(2)b(3)b(4)...
such that a = 1 - b
if b(1) <> 0, then a > .99 and b >= .1, so a + b > .99 + .1 (= 1.09).
therefore, b(1) = 0.
if b(2) <> 0, then a > .999 and b >= .01, so a + b > .999 + .01 (=1.009).
therefore, b(2) = 0.
moving along in such a fashion, we come to conclude that b = 0, and, since
we are doing arithmetic with infinite decimals,
1 = .999...
I like this proof alot.
My roommate maintains that 1 - .999... = 9 * 10^infinity, which is sorta
humorous considering that the first proof I showed him involved multiplying
.999... by 10. The only advice I can give is :
Never Room With An Architecture Major.
(no flames please)
cheers,
craigmcewan (01/28/83)
#R:uiucdcs:28200006:uiucdcs:28200007:000:120 uiucdcs!mcewan Jan 28 12:36:00 1983 Actually, 1-.999... = 10^-infinity. If you can't convince your roommate that 10^-infinity=0, I'd give up if I were you.