1/3 is .333.. 2/3 is .666 why isn't 3/3 .999..?

Question

If 1/3=.33333... and 2/3=.66666666... and .6666666..+.33333333333..=.99999999.... shouldn't 3/3=.999999999?

Answer 1

It is
if x = .9999999....
then 10x = 9.9999999...
subtracting,
10x - x = 9x = 9.9999999... - .9999999.... = 9
9x =9 so x = 1

Answer 2

It is.

3/3 = .999999... = 1

This can be proven using infinite series. 0.99999... can be expressed as the geometric series .9 + .09 + .009 + .0009 + ... = .9 + .9*(.1) + .9*(.1)^2 + .9(.1)^3 + ... + .9*(.1)^n + ... = the sum of .9*(.1)^n from n=0 to n=infinity (anyone know how to type sigma notation in internet explorer?). This series is convergent, and using a formula that I won't derive (you can easily find a derivation on the internet if you're interested, it's a simple proof) the sum of an infinite convergent geometric series, S, is given by S = a / (1-r) where a is the first term of the series (.9) and r is the common ratio (.1 - note that the (n+1)th term of the series is the nth term times .1 [hence the name common ratio]). Thus the sum is S = .9 / (1-.1) = .9/.9 = 1. Therefore, .9999999... = 1.

Answer 3

The KEY is to think of a pie.

1) With 1/3, you only have 1 slice out of a 3 slice pie. A lot less than the entire pie!

2) With 2/3, you have 2 slices out of a 3 slice pie, double the amount in #1. Less than the entire pie, but more than #1.

3) With 3/3, you have all 3 slices of that 3 slice pie. Why isn't is triple #1? Because you've them all!! There's nothing left over.

But let's say you had 2.9 slices, meaning you ate 2 full slices and nearly all of the 3rd. You still have some left over, although it's a small piece, but anything less than the whole has a value less than 1.

If you've got them all and there's nothing remaining, then the value is 1.

Answer 4

Because it is. 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...

Answer 5

1/3 = 0.333333
2/3 = 0.666667
3/3 = 1/3 + 2/3 = 1

Answer 6

Mathematics says that .99999999999999999 repeating to infinity and 1 are actually the same number. To be separate numbers, there must be a number between them. Since there is no number between .999999999 repeating and 1, they are the same.

Answer 7

You got the decimal for 2/3's wrong, that's why.

First up, keep the same number of decimal places, let's say five.

1/3 = .33333
2/3 = .66667 (notice the last digit gets rounded up)

Ergo,
.33333 + .66667 = 1.00000

Answer 8

It is. .999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999

is equal to 1 eventually. Eventually. There are an infinate amount of 9's, so eventually they'l reach 1.

(inf)

SIGMA 9/(10^j) = 1

j = 1

Is the proper notation.

Answer 9

2 / 3 = 0.666666667

Answer 10

3 goes into 3 perfectly. It's a whole of a whole