# 0.999... = 1

Heh...I have no idea. Apparently when Dragon says "0.999..." he means something different from the rest of the world, but I don't really understand what his unique definition is.

Klippymitch's comments make me think that he is somehow imagining that the "..." is sort of an active process, with the series of 9s appearing one after another, with new ones appearing at every moment. So, although it might be arbitrarily long, it will never actually equal 1 because you would have to wait an infinitely long time for the numbers to all appear. Of course, that's not what "..." means. In 0.999... the 9s are already at an infinite length, so there isn't any sort of time issue. Of course, maybe I'm completely misunderstanding him.

and for absane.

You cant just jump to infinite. Numbers follow a sequence of 123456789 you cant jump or skip through the sequence only with logical thought processes can this happen. For when we give objects numbers. 3 red mm's + 2 red mm's = 5 red mm's

You cant just jump to infinite. Numbers follow a sequence of 123456789 you cant jump or skip through the sequence only with logical thought processes can this happen. For when we give objects numbers. 3 red mm's + 2 red mm's = 5 red mm's

When am I "jumping to infinite?"

As for the rest of what you said.... I don't see the connection between that and everything I said.

When am I "jumping to infinite?"

As for the rest of what you said.... I don't see the connection between that and everything I said.

I can't see how you can't see the connection.

I can't see how you can't see the connection.

Well I understand what you mean. For example we have number $$x_{0}x_{1}...x_{n}$$ in base ten such that $$x_{0},x_{1},...,x_{n} \in \{0, 1, 2, ..., 8, 9\}$$

But what's the connection? Fill in the blanks... I can't read minds over the Internet.

As far as your statement that there exists no number between 0.999... and 1, there must be a number next to 0.999... what number is this? How about the number next to that one?

If you can answer those questions, I will detect "the pattern" and I will devise a counting function. I will explain everything else at that time.

Err.. that, and as you have been said somewhere else, 0.999... is equal to 1.
So is that mean 1 - 0.9999... = 0 or 0.0000.....

I suppose you will say 0, because there is no number between 0 and 0.0000...

Sorrrrry if that irritates you, I'm not a mathematician, yet I have a bit interest in it.

and btw,.. when we want to sum up a number which contain 9 behind a decimal to get a result which contain 0 as in:

0.9
0.1
--- +
1.0

or

0.1009
0.0001
---------+
0.1010

arent we always have to add this with something contain 1?
So, shouldnt it be: 1 - 0.9999..... = 0.00000...........1 ?

then which one is equal to 0? 0.0000.... or 0.00000.....1 ?

p.s: ok, ok I'll read from 1st page, maybe will find something there. I was expecting a volunteer to give a summary for that

Last edited:
.999...

.999...
=

123456789
01233456789
00123456789
000123456789
0000123456789
00000123456789
....

That's the best I can explain it.
WTF is this? Nasor.... help?! Draqon.. you seem to know what he's talking about. Explain.
Oh it's perfectly simple really. 1 = 123456789, so 0 = 123456788 and 0 = 1. This in turn implies that all reals are equal to each other. There are no bigger or smaller numbers. All numbers are equally big and small. I think this is in line with draqon's philosophies.

Err.. that, and as you have been said somewhere else, 0.999... is equal to 1.
So is that mean 1 - 0.9999... = 0 or 0.0000.....

Yes.

I suppose you will say 0, because there is no number between 0 and 0.0000...

0 = 0.000...

Sorrrrry if that irritates you, I'm not a mathematician, yet I have a bit interest in it.

You'll only start to irritate me if you are arrogant enough to think you know more about mathematics

0.00000...........1 ?

"0.000...1" is self-contradictory. The "..." means that the zeros repeat forever. If they repeat forever, there's no way for there to be a 1 on the end.

"0.000...1" is self-contradictory. The "..." means that the zeros repeat forever. If they repeat forever, there's no way for there to be a 1 on the end.

Ahh.. she made a correction. Thanks.

Thanks, too, from me.

Ok I understand now that this number is exist:
0.9999999............ which is equal to 1

so I assume that:

0.8888888............ is equal to 0.89

and 0.898888...... is equal to 0.899

So the result of:
0.89888....... - 0.8888....... = 0.899 - 0.89 = 0.009

Is it correct?? It is strange for me because how can you substract something
unlimited by another unlimited thing and then get a limited thing

so I assume that:

0.8888888............ is equal to 0.89
No. The general "rule" goes something like this: if you have an infinite sequence of 9's in the decimal representation of a number, you can remove it and increase the next digit to the left by 1. For example:

0.8899999...​

There's an infinite sequence of 9's after the 8, so you can remove it and add 1 to the 8. So:

0.8899999... = 0.89

It also works inside a sequence of 9's:

0.99999999...​

If we try to add 1 to the 9, it becomes 10 and we have to carry the 1 to the next digit on the left. This goes on until we run out of 9's

This only works for repeating 9's. Other sequences of repeating digits can be expressed as fractions, eg:

$$0.888 \ldots = \frac{8}{9}$$​

Ok, thanks, but that has not answer my question yet... or

Well 0.89888... - 0.88888... = 0.01, if that's what you wanted to know. The rules you learned in primary school for adding and subtracting numbers digit-by-digit still apply.

Is it not 0.009? Surely 0.009 is not equal to 0.01

I assume that:

0.8888888............ is equal to 0.89

and 0.898888...... is equal to 0.899

So the result of:
0.89888....... - 0.8888....... = 0.899 - 0.89 = 0.009

Is it correct?? It is strange for me because how can you substract something
unlimited by another unlimited thing and then get a limited thing

Is it not 0.009? Surely 0.009 is not equal to 0.01

0.00999... = 0.01.

As P said, all your normal rules of addition and subtraction work. The difficulty you are having is trying to deal with addition and subtracting with an infinite number of digits. As far as I understand, no one truly knows how to subtract decimal numbers with an infinite number of digits (assuming you don't rewrite it into another forum or make assumptions). Of course, many assumptions are reasonable like with: 0.333... - 0.123123123... = 0.210210210... because the pattern appears to repeat. But if you have sqrt(2) - sqrt(3) = 1.41421356... - 1.73205081... = ? How do we do this? Clearly, we can't start from the very "right hand side" because there is no right hand side. Starting from the left hand side and we have to keep making corrections when caring numbers over to towards the right (what I mean is if we have 73-68... we can see 7-6 = 1 and 3 - 8 = -5. But, we can't have -5 in the answer so we take "10" from "1" and make that 0 and add 10 to -5. The answer is 5.)

Sorry if I confused you.

so I assume that:

0.8888888............ is equal to 0.89
and 0.898888...... is equal to 0.899

Both of these are wrong. (Others have said the same but not explicitly). Everything that follows hinges on these mistakes.

The correct interpretation is 0.898... - 0.888... = 0.01.

As both numbers involve repeating decimal fractions they are both rational numbers. You may have taught how to convert these to rationals: 9*0.89888... = 10*0.898888... - 0.898888... = 8.9888... - 0.898888... = 8.09. Thus 0.89888... = 8.09/9 = 809/900. Similarly, 0.888... = 8/9. Subtracting, 809/900-8/9 = 9/900 = 1/100 = 0.01.

It is strange for me because how can you substract something
unlimited by another unlimited thing and then get a limited thing
Why is this confusing? 1/3 + 1/6 = 0.333... + 0.166... = 0.499... = 0.5 = 1/2.