This excerpt from Danica McKellar's site:<\/p>\n
Danica Answers:<\/strong> Hi Tom! Glad to help- you're not the only one who's asked me to repeat this. \ud83d\ude42 <\/em><\/font><\/font>\n\t\t\t\t\t\t\t\t\t\t\t\t\t<\/p>\n First, we start by labeling: <\/em><\/font><\/font><\/p>\n x= .999999999999..... (repeating infinitely)<\/em><\/font><\/font><\/p>\n Then if you multiply both sides by 10, you get:<\/em><\/font><\/font><\/p>\n 10x = 9.99999999999..... (repeating infinitely)<\/em><\/font><\/font><\/p>\n Well, 10x -x = 9x<\/em><\/font><\/font><\/p>\n And for the right hand side: <\/em><\/font><\/font><\/p>\n 9.99999..... - .99999999...... = 9<\/em><\/font><\/font><\/p>\n And What [@more@]<\/p>\n The dilemma to this problem is indeed infinity itself.<\/p>\n 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:<\/p>\n 10 * 0.9999... would have to have resulted a "0" at the right end of the infinite 9.99999999...."0".<\/p>\n Thus, 10x - x = 9x would not have made 9x = 9. But actually 8.999999999...."1".<\/p>\n 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.<\/p>\n","protected":false},"excerpt":{"rendered":" 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. \ud83d\ude42 First, we start by labeling: x= .999999999999..... (repeating infinitely) Then if you multiply both sides … Continue reading
\nwhat if we subtracted the top equation from the bottom one? We'll get
\nanother true statement. Before I do that, let me review why this will
\nwork. Say that a = b , and c = d. Then do you agree that "a-c = b-d" is
\nalso a true statement? After all, a = b , and c = d. So once you agree
\nwith that, let's continue and go ahead and subtract the top statement
\nfrom the bottom one. We get, for the left hand side of the equations:<\/em><\/font><\/font><\/p>\n
\nwe conclude that "9x = 9" is also a "true statement." Divide both sides
\nby 9, and you get x = 1. Since we started with x = .9999999 (repeating
\ninfinitely), we've now shown that the "infinite decimal of .9999
\nrepeating" equals the whole number 1. Yes, I did it much faster on the
\nTV show. \ud83d\ude42 <\/em><\/font><\/font><\/p>\n
\nI didn't talk about on the show is the somewhat philosophical issue
\nthis proof brings to light. Our mathematical system was invented by the
\nhuman mind, which assumes that space (and the number line) is
\ninfinitely divisible and that there *is* such thing as a theoretical
\n"point" that takes up no space, and has no volume or mass. So, this
\nlittle proof shows that - according to a math system that assumes such
\nthings - indeed, .999999.... (repeating infinitely) equals 1. There are
\nthose who would say this isn't true, but then they are forgetting that
\nwe are dealing with a "language" that was invented by the human mind,
\nand also that infinity is a slippery concept. \ud83d\ude42<\/em><\/font><\/font><\/p>\n