Point nine repeating equals one?Date: 21 Dec 1994 13:16:05 -0500 From: joe redding Subject: Point Nine repeating equals one? Dear Dr Math. I have had a question about this for years but I don't remember finding a satisfactory answer. When I learned how to convert repeating decimals to fractions, we were given the following example: _ _ Let n = .99 so 10n = 9.99 Subtracting the first equation from the second yields: 9n = 9 since the repeating decimals subtract out _ which gives us n = 1, but we know that n = .99 so _ .99 = 1 The problem I have is that I can't logically believe this is true, and I don't see an error with the math, so what am I missing or forgetting to resolve this? (If memory serves, he also said that there are several other ways of proving that .9999... = 1 but I don't remember them) Thanks, Joe Redding Date: Wed, 21 Dec 1994 22:33:21 -0500 From: Stephen Weimar Subject: Re: Point Nine repeating equals one? I think it is true and you did a beautiful job presenting it. > (If memory serves, he also said that there are several other ways of > proving that .9999... = 1 but I don't remember them) If it was not equal to one then there would be a number between it and 1. What number would that be? -- steve ("chief of staff") Date: 22 Dec 1994 14:26:44 -0500 From: Molly Foster Subject: Re: Point Nine repeating equals one? Dear Joe, Hello there! Thanks for writing Dr. Math. You asked an excellent question, and I liked Steve's first response to you, but I thought I might add two things. Another way to think about this is this: Would you agree that 1/3 = .33333...? .3333....is the way to write 1/3 using decimals. If you multiply both sides of the above equation by three you get 1 = .99999...., right? I think the problem you are having, though, is BELIEVING it is true, right? I admit, depending on how you look at it, it can seem false. After all, how can 2 different numbers be equal? The thing is, these 2 numbers AREN'T different. I think saying 1 = .9999... may seem contradictory to us because we aren't realizing that .999.... is a repeating decimal that really does go on forever. Obviously saying 1 = .9 is false, as is saying 1 = .99, 1 = .999, 1 = .9999, etc. But we aren't dealing with finite decimals here. So, you might think of .9999.... as another name for 1, just as .333... is another name for 1/3. What Steve said really should clinch it for you, but I thought I'd just add these thoughts anyway. Hope it helps. Sydney, Dr. "math rocks" Foster Date: 22 Dec 1994 14:38:08 -0500 From: joe redding Subject: Re: Point Nine repeating equals one? Dear Steve and Molly, Thanks so much for your replies. Both were what I was looking for. Molly, thanks for the second method of proving this to be true but does anyone know of the elusive third? By the way, I added Dr. Math to our Web page. I think that it is a great service. Keep up the good work! Thanks, Joe Redding Web site: http://sasd.k12.pa.us/ |
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