Does .9999999999999999999999999 = 1 ?

Question

Thats .9999999 recurring. Does it equal 1?

Answer 1

0.999999.... is absolutely equal to one and I'm surprised by just how mny people still stubbornly refuse to accept it. Here is another proof.

Define S as follows

S = 9∑10^-k from k=1 to ∞
S = 0.9 + 0.09 + 0.009 + 0.0009 + ...
S = 0.999...

This is the same 0.999... as above.

10S = 10*9∑10^-k from k=1 to ∞
10S = 9∑10*(10^-k) from k=1 to ∞
10S = 9∑10^(-k+1) from k=1 to ∞

Here we let -j=-k+1 so j= k-1

10S = 9∑10^-j from j=0 to ∞
10S = 9[10^0 + ∑10^-j from j=1 to ∞]
10S = 9*1 + 9∑10^-j from j=1 to ∞
10S = 9 + S
9S = 9

S = 1

0.999... = 1 is an inescapable result based on the definition of the real numbers which is based on the Archimeadean principle which states that there is no infinitely large or infinitely small number.

If you disagree with this principle then you should be able to find a number z such that there is no number x where

1<1+x<1+z

In other words if you disagree with the Archimedean principle you are saying there exists a smallest real number z where the above holds (dx from the inifinitesimal calculus is not a real number which is why you don't see it used outside of a differential or integral equation).

Answer 2

A lot has already been said, so I'm not sure that I can add much, but in the hopes that you have an open and scientific mind, I would like to point out to you the following observations I've made.

(1) The people who claim that 0.999... is exactly equal to 1 have a proof (often several proofs) for it; the ones who claim that it isn't, don't, and speak vaguely of "infinity", "infinitesimal" or "limit" without really elaborating on what it's supposed to mean in this context.

(2) The people who claim that they are not equal get easily upset and irritated over this. This irrational insistence on their being right is borderline religious. They are more likely to defend this belief by claiming that a math professor or some other "reliable" source told them this. That is, of course, not a proof.

When it comes to mathematics, you MUST draw your conclusions from proofs that follow a train of logical reasoning. Try not to get confused by laymen's hearsay. And, to be honest, there are so many proofs for this particular equation, most of which are so simple and easy that even schoolchildren can follow it.

Answer 3

Yes, of course it equals one if you understand a few items, as follows:

> the three dots mean that the numbers continue on to infinity [which is a *very* long way to continue], and
> 1/3 = 0.333333... , and
> 1/3 +1/3 + 1/3 = 0.99999999999... ,and
> 1/3 +1/3 + 1/3 = 1 *exactly*, SO
> 0.999999999... = 1.
QED


Wiki quote:

Multiple decimal representations
Main article: Proof that 0.999... equals 1
Some real numbers have two infinite decimal representations. For example, the number 1 may be equally represented by 1.00000... as by 0.99999... (where for the sake of brevity the infinite sequences of digits 0 and 9, respectively, have been replaced by "..."). Conventionally, the version with zero digits is preferred; by omitting the infinite sequence of zero digits, removing any final zero digits and a possible final decimal point, a normalized finite decimal representation is obtained.
end quote

(so chin, amiram, tsr 21, and I are correct and the others are not)

Answer 4

Proof by contradiction. Assume that .999... < 1.

Then there is a number x between .999... and 1, i.e.
.999... < x < 1

What is the decimal representation of x?
- The unit digit must be 0, otherwise it wouldn't be smaller than 1.
- The first decimal must be 9, otherwise it would be smaller than .999...
- The second decimal must be 9, otherwise it would be smaller than .999...
- Ditto the third, fourth etc. decimals. ALL of the decimals must be 9's.

In other words... x = .999... This contradicts our previous conclusion that .999... < x < 1. Therefore we must reject our initial assumption, and

0.999... = 1. QED.

Not "very close to". Not "tends towards". Not "infinitesimally close".

Equal. Identity. Same thing.

Answer 5

Yes it IS definitely equal to 1. When I was younger I thought that It's not equal but, over a year ago a mathematician in a mathematics club i had been going, told as that it is. At first I said that it's just not possible, that 0.999999... was very close but not equal to 1. Then she (the mathematician) gave us a very simple proof which made me accept it, because I respect a mathematical prove when it's right (I'm saying this cuz I know guys that would not accept it, even if I gave them the proof).
So, there's the proof:
x=0.999...
0.999.../0.099...=10
Therefore, 0.099=x/10
0.999...-0.099... =x-(x/10)=0.9x=0.9
(0.9x)/9=0.1x=0.9/9=0.1
Therefore, 0.1x=0.1
If you multiply both sides by 10 it comes out that
x=1

Answer 6

it may be.
1/3 = 0.333333333333333333333
0.999999999999999=0.3333333333333333333 X 3
=1/3 X 3
= 1

Answer 7

Almost all of the above answers are incorrect.

Sorry guys.

Yes, they are equal. Not just close, but exactly equal.

I'm not sure a proof will convince anyone, but anyway, here goes. There are several variations - all are trivial.
1/3 = .3333333333......
3* 1/3 = 1 = .99999999999.....

.

Answer 8

It will always be minute less than 1 but if you round it off then yes, it is equivalent to 1.

Answer 9

when u change .9999999999999 to a fraction it becomes 9/9=1...donno y it just does

Answer 10

Do you find it difficult to imagine...

1/9= 0.1111111111111111111111 endless repeating

Now multiply both sides by 9

9/9= 0.99999999999999999999999...endless repeating

You know that 9/9=1

1= 0.9999999999999999999999999... endless repeating

Any problem?


Regards!