2006-07-18 10:45:56 · 10 answers · asked by Anonymous in Science & Mathematics ➔ Mathematics
To all those who say these are not the same: please don't answer questions where you don't know the answer! There are TWO ways of writing the number 1 as a decimal. One is 1.0000.....; the other is .9999....
The problem is to figure out what the symbol .99... means. It *means* the limit of a sequence. Which sequence? The sequence whose first term is .9; whose second term is .99; whose third term is .999; etc. As you go further out in this sequence, is there some number that the terms of the sequence are getting closer and closer to? Yes. That number is what the symbol .999... means. It is also clear that that number is also 1.
2006-07-18 13:51:35 · answer #1 · answered by mathematician 7 · 1⤊ 0⤋
Yes, several proofs were given 1 day ago when the same question was asked:
http://answers.yahoo.com/question/index;...
See also my response to that question.
Furthermore, the people who say it is not the same as 1 are wrong. If 0.99999... has a value, then it is 1. Saying that it is not quite 1 is a conceptual error -- if we were to truncate at some point, i.e., at 0.999999, then that value is not quite 1. But the recurring decimal 0.9999999... is not the same as any of its truncations. Notice that none of its truncations are equal, so if it were the same as one of its truncations, how would you be able to decide which one?
The DEFINITION of 0.9999... is the sum of the series 9/10^1 + 9/10^2+9/10^3+..., an infinite series. Using the theory of such series, the value is 1. People who say 0.99999... is less than 1 would also think that pi=3.14, but it is not. It has infinitely many digits, and if you give me any number n, I could in principal give you the first n digits of pi. The same concept holds for 0.9999...
And btw, thanks for the free 6 points!
2006-07-18 12:48:07 · answer #2 · answered by mathbear77 2 · 0⤊ 0⤋
1 = 1/3 + 1/3 + 1/3 = 0.3333... + 0.3333... + 0.3333... = 0.9999...
OR...
0.9999... = 0.9 + 0.09 + 0.009 + 0.0009 + ...
= 9/10 + 9/100 + 9/1000 + 9/10000 + ...
which is a geometric series. Since the common ratio, r=1/10, is less than one, it converges, and it converges to
a/(1 - r) = (9/10)/(1 - 1/10) = (9/10)/(9/10) = 1.
2006-07-18 10:49:39 · answer #3 · answered by Matt E 2 · 0⤊ 0⤋
It's not EXACTLY equal to one. It is, however, the closest you can possibly get to being 1 without actually being 1.
2006-07-18 10:52:54 · answer #4 · answered by extton 5 · 0⤊ 0⤋
Calculus says that for two numbers to be two separate numbers, there must be a number between them. Since there is no number between .999 repeating and 1, they must be the same number.
2006-07-18 10:51:23 · answer #5 · answered by Nick 4 · 0⤊ 0⤋
It's not exactly equal to one, but "ooh so close!".
However, if you were to round it to the nearest whole number, then it would be an even 1.
2006-07-18 20:27:35 · answer #6 · answered by msoexpert 6 · 0⤊ 0⤋
k here u go
0.3333333 repeated is exactly = 1/3
0.6666666 repeated is exactly = 2/3
0.9999999 repeated is exactly = 3/3 which is = 1
2006-07-18 10:54:50 · answer #7 · answered by Anonymous · 0⤊ 0⤋
Calculus says it converges to 1
2006-07-18 11:10:25 · answer #8 · answered by Anonymous · 0⤊ 0⤋
I believe that Modest Mouse put it best...
"The universe is shaped exactly like the Earth, if you go a straight line, you'll only end up where you were."
2006-07-18 16:33:12 · answer #9 · answered by MobBots 3 · 0⤊ 0⤋
here ya go... (scroll down a bit)
2006-07-18 10:50:25 · answer #10 · answered by Critical Mass 4 · 0⤊ 0⤋