This excerpt from Danica McKellar's site:
Danica Answers: Hi Tom! Glad to help- you're not the only one who's asked me to repeat this. 🙂
First, we start by labeling:
x= .999999999999..... (repeating infinitely)
Then if you multiply both sides by 10, you get:
10x = 9.99999999999..... (repeating infinitely)
Well,
what if we subtracted the top equation from the bottom one? We'll get
another true statement. Before I do that, let me review why this will
work. Say that a = b , and c = d. Then do you agree that "a-c = b-d" is
also a true statement? After all, a = b , and c = d. So once you agree
with that, let's continue and go ahead and subtract the top statement
from the bottom one. We get, for the left hand side of the equations:
10x -x = 9x
And for the right hand side:
9.99999..... - .99999999...... = 9
And
we conclude that "9x = 9" is also a "true statement." Divide both sides
by 9, and you get x = 1. Since we started with x = .9999999 (repeating
infinitely), we've now shown that the "infinite decimal of .9999
repeating" equals the whole number 1. Yes, I did it much faster on the
TV show. 🙂
What
I didn't talk about on the show is the somewhat philosophical issue
this proof brings to light. Our mathematical system was invented by the
human mind, which assumes that space (and the number line) is
infinitely divisible and that there *is* such thing as a theoretical
"point" that takes up no space, and has no volume or mass. So, this
little proof shows that - according to a math system that assumes such
things - indeed, .999999.... (repeating infinitely) equals 1. There are
those who would say this isn't true, but then they are forgetting that
we are dealing with a "language" that was invented by the human mind,
and also that infinity is a slippery concept. 🙂
[@more@]
The dilemma to this problem is indeed infinity itself.
Don't forget, 10x is actually 10 * 0.99999999.... hence, if we treat infinity as we treat regular numbers which are calculated from one end to another end of the decimal, we have to treat the calculation as such:
10 * 0.9999... would have to have resulted a "0" at the right end of the infinite 9.99999999...."0".
Thus, 10x - x = 9x would not have made 9x = 9. But actually 8.999999999...."1".
I agree with Danica that this is the flaw of human terminology, but I wouldn't give up on the terminology of infinity just so easily.