2006-12-27 12:30:34 · 15 answers · asked by yukiphong2006 1 in Science & Mathematics ➔ Mathematics
.99999... represents the infinite series 9/10+9/100+9/1000+... When you sum this infinite series, the answer is 1. This is one example of the well-known fact that the decimal representation of a real number is not unique.
One of the exercises at the beginning of Royden's book Real Analysis discusses this kind of thing.
2006-12-27 12:38:49 · answer #1 · answered by robert 3 · 4⤊ 1⤋
.9 reapeating is equilelent to 3/3. Dealing with fractions, any number overr the same number is equal to 1.
take a pie for instance. it is cut into thirds. 1 slice is .3 repeating and 1/3. if all the slices are there the pie is .9 reapeating and is 3/3 which is 1.
hope I helped!
2006-12-27 14:41:50 · answer #2 · answered by Lisaya 1 · 0⤊ 2⤋
It is EXACTLY equal to 1. There is *no* rounding involved.
x = 0.999999...
10x = 9.999999...
10x - x = 9.999999... - 0.999999...
9x = 9.000000...
= 9
x = 1
The key to understanding this proof is to realize that the 9's go out to infinity for x AND for 10x, so that when they are subtracted, *everything* behind the decimal point cancels. This is the essence of repetition to infinity.
lulu's example actually proves that 1=0.999999...!
If she accepts that 1/3=0.333333... and 2/3=0.666666..., then when we add the two fractions we get 3/3=1=0.999999.... Done!
2006-12-27 13:24:19 · answer #3 · answered by Jerry P 6 · 4⤊ 1⤋
This is a repeating decimal.
Let S = .9999999...(1)
10 S = 9.9999999...(2)
Subtract (1) from (2):
9S = 9
which leads to S = 1
2006-12-27 12:39:33 · answer #4 · answered by sahsjing 7 · 2⤊ 1⤋
We'll base this proof around 3/3.
Any number divided by itself is 1, so 3/3 is 1.
3/3 = 1
1/3 = .3333..., so 1/3 x 3 is .9999... (.3333... x 3)
3/3 = .9999...
Theorem: if a = b, and b = c, then a = c
So if 3/3 = 1 and 3/3 = .9999..., then 1 = .9999...
2006-12-27 15:53:46 · answer #5 · answered by Trip 3 · 0⤊ 1⤋
there is an infinite series justification, but here is a rather simple proof that makes it easier to believe, esp. if you have not learned about infinite series:
let x = 0.9999...
10x = 9.9999...
10x - x = 9.9999... - 0.9999...
9x = 9
x = 1.
2006-12-27 12:38:51 · answer #6 · answered by pinkpearls 3 · 2⤊ 1⤋
Lets look at the fraction 2/3 and 3/3.
2/3 is equal to 0.6666666666...
1/3 is equal to 0.3333333333...
3/3 is equal to 2/3 plus 1/3 so 3/3 is 0.99999999.. or 1.
But this is imposible so 3 divided by 3=1
But 1= 0.99+0.01
1=0.9999999999+0.0000000001
and because there is an extra 0.01/0.01/0.001 etc. it is an imposible chance that 3/3=0.999
so 0.9999... is not equal to 1.
2006-12-27 13:07:45 · answer #7 · answered by lulu 3 · 0⤊ 7⤋
To anyone who would say that 0.9999... is not exactly equal to 1, I would ask this:
Since there is always a number between two unequal numbers, can you name me a number that is between 0.9999... and 1? If you show me such a number, I will show you that it is equal to 0.9999... so you can't really find one.
2006-12-27 13:40:39 · answer #8 · answered by Anonymous · 5⤊ 1⤋
Suppose:
x=.99999999999999999999999...
then:
10x=9.999999999999999999999999...
-x -x
9x=9
x=1
Therefore, .9999999999999999999999999999...=1
2006-12-27 14:13:42 · answer #9 · answered by abcde12345 4 · 2⤊ 1⤋
.9 = 1 - .1
.99 = 1 - .01
.999 = 1 - .001
As more digits is added to the string, we say the result approaches 1 as a limit. It never quite gets there, but gets ever nearer.
2006-12-27 12:36:08 · answer #10 · answered by Anonymous · 1⤊ 5⤋