# 0.999... = 1

Discussion in 'Physics & Math' started by Andrej64, Jun 8, 2006.

1. ### KlippymitchThinkerRegistered Senior Member

Messages:
699

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

3. ### AbsaneRocket SurgeonValued Senior Member

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8,989
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.

5. ### KlippymitchThinkerRegistered Senior Member

Messages:
699
I can't see how you can't see the connection.

7. ### AbsaneRocket SurgeonValued Senior Member

Messages:
8,989
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.

8. ### SciencelovahRegistered Senior Member

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4,349
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.

9. ### SciencelovahRegistered Senior Member

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4,349
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: Oct 12, 2007
10. ### przyksquishyValued Senior Member

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3,171
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.

11. ### AbsaneRocket SurgeonValued Senior Member

Messages:
8,989
Yes.

0 = 0.000...

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

12. ### NasorValued Senior Member

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6,221
"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.

13. ### AbsaneRocket SurgeonValued Senior Member

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8,989
Ahh.. she made a correction. Thanks.

14. ### SciencelovahRegistered Senior Member

Messages:
4,349
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

15. ### przyksquishyValued Senior Member

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3,171
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:

$0.888 \ldots = \frac{8}{9}$​

16. ### SciencelovahRegistered Senior Member

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4,349
Ok, thanks, but that has not answer my question yet... or

17. ### przyksquishyValued Senior Member

Messages:
3,171
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.

18. ### SciencelovahRegistered Senior Member

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4,349
Is it not 0.009? Surely 0.009 is not equal to 0.01

19. ### AbsaneRocket SurgeonValued Senior Member

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8,989
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.

20. ### D HSome other guyValued Senior Member

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2,257
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.

Why is this confusing? 1/3 + 1/6 = 0.333... + 0.166... = 0.499... = 0.5 = 1/2.