# Does 0.9 repeated = 1?

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• 10-24-2008, 09:47 PM
Xei
...interesting maths? :/

Neither of those statements are untrue. 1 = .99... = 1 = 1, if you like... if I understand what I think you're trying to say correctly, then I don't think you've quite mastered the equals sign.

.99... is effectively an alternate way of writing 1.
• 10-24-2008, 09:51 PM
Sornaensis
I'm only in Algebra II...

What I mean is:

.333... is 1/3 in decimal form. If you multiply it by 3, you get 1, not .999..., right?

And are there types of decimals like this?

2.99...3

Where 3 follows an infinite amount of repeating nines. At the end of infinity.
• 10-24-2008, 09:55 PM
Xei
.333 = 1/3
.999 = 1

.333 x 3 = .999;
.333 x 3 = 1;
1/3 x 3 = .999;
1/3 x 3 = 1

There are no decimals like the ones you gave above. Technically I don't see why you couldn't express them mathematically, but it would be pointless, as the numeral at the end would be equal to 0, and so your decimal above = 2.99... = 3.
• 10-24-2008, 09:59 PM
Sornaensis
I get that. I'm not stupid.

2 = 5 is an incorrect statement. If you come to that for an equation, the equation has no solution in algebra.

Do you understand what I was asking now?
• 10-24-2008, 10:21 PM
Xei
There are no equations which would lead to that conclusion.

2 = 5 is not like the above because 2 is in fact not 5.
• 10-24-2008, 10:25 PM
Sornaensis
Yes, I just tried.
• 10-24-2008, 10:33 PM
[SomeGuy]
0 .1 .2 .3 .4 .5 .6 .7 .8 .9 .9999999~9 1
THEREFORE it is not even close to equal 1. 0.9999~9 is a different number. Ever hear of the density property?
• 10-24-2008, 10:39 PM
Sornaensis
There is no way of expressing .9... as a fraction other than 1/1.
• 10-24-2008, 10:42 PM
[SomeGuy]
No, 1/1 is a whole...1.
• 10-24-2008, 10:43 PM
Xei
Quote:

Originally Posted by xXSomeGuyXx
0 .1 .2 .3 .4 .5 .6 .7 .8 .9 .9999999~9 1
THEREFORE it is not even close to equal 1. 0.9999~9 is a different number. Ever hear of the density property?

wtf
• 10-24-2008, 10:46 PM
[SomeGuy]
my number line failed sorry
• 10-24-2008, 10:47 PM
Xei
But .999... is exactly 1.
• 10-24-2008, 10:48 PM
[SomeGuy]
No, .9999999~9 is .999999999~9. 1 is 1.
• 10-24-2008, 10:49 PM
Sornaensis
Quote:

Originally Posted by xXSomeGuyXx
No, 1/1 is a whole...1.

So then express .999... as a fraction.

It is a rational number, so this should be simple, right?

.999... = 1.
• 10-24-2008, 10:50 PM
drewmandan
Quote:

Originally Posted by xXSomeGuyXx
No, .9999999~9 is .999999999~9. 1 is 1.

We're not talking about 0.9...9. We are talking about an infinite number of 9's. What you wrote has a termination.
• 10-24-2008, 10:51 PM
arby
The big problem I have with these proofs is that you are applying rules meant for rational numbers on irrational ones.

For example: 10*0.99999999... = 9.9999999999... would normally move the 9 out of the last place value and so in a rational number you get

10*0.999 = 9.99

When we do the next step, that 9 we would moved right gets subtracted and we're left with 8.991 Of course, the trick is to say "theres infinate numbers of nines, lets ignore that" but in reality, it should end up being something like

9x = 8.999999999(infinite times)99991 and thus x = 0.9999999... still

You can't apply those sorts of rules to irrational values. You'd have to leave it as un-evaluated like:

10x = 10*0.99999999...
• 10-24-2008, 10:56 PM
drewmandan
Quote:

Originally Posted by arby
The big problem I have with these proofs is that you are applying rules meant for rational numbers on irrational ones.

For example: 10*0.99999999... = 9.9999999999... would normally move the 9 out of the last place value and so in a rational number you get

10*0.999 = 9.99

When we do the next step, that 9 we would moved right gets subtracted and we're left with 8.991 Of course, the trick is to say "theres infinate numbers of nines, lets ignore that" but in reality, it should end up being something like

9x = 8.999999999(infinite times)99991 and thus x = 0.9999999... still

You can't apply those sorts of rules to irrational values. You'd have to leave it as un-evaluated like:

10x = 10*0.99999999...

The big problem I have with people arguing that 0.9~ =/= 1 is that their math education is so lacking that they don't even know what a fucking rational number is.
• 10-24-2008, 10:57 PM
Sornaensis
0.99... isn't irrational...

I don't know what you were doing. This, maybe?

10x = 9.9...
x = 0.9...

Subtract.

9x = 9

x = 1
• 10-24-2008, 10:59 PM
Xei
arby; Well, you don't normally find decimals in proper maths.

However if you define them properly as I did above with sigma notation, there is absolutely nothing wrong with the proof.

.999... is not irrational, by the way. Even if it was irrational, you can perform all the usual operations on it.
• 10-24-2008, 11:01 PM
[SomeGuy]
I know. It's like calling me Grod. Grod is Grod. I'm SomeGuy.

A number is a number. If you round up, (which means you suck at maths) then it equals 1.

If you think .99999~99 equals 1, you're high.
• 10-24-2008, 11:04 PM
Xei
But it's just been proved for you in two different and valid ways.
• 10-24-2008, 11:04 PM
Sornaensis
Then what is the fractional equivalent of .999...?
• 10-24-2008, 11:06 PM
drewmandan
Quote:

Originally Posted by xXSomeGuyXx

If you think .99999~99 equals 1, you're high.

No one said that number equals one. We said 0.9~ equals one. See, the way you wrote your version of the number, it's obvious that you think the 9's "end". They don't. As soon as you understand that, you will see that it clearly is 1.

Also, here's something to consider.

1/3 in base 10 is 0.3~
1/3 in base 6 is 0.2000~ (in other words, just 0.2. Do the long division if you don't believe me)

Then, in base 6, if you do 3 * 0.2, you get 1.

This would suggest that there's no fundamental difference between numbers with an infinite number of non-zero repeating decimals and an infinite number of zeros. Therefore, it would seem that the "1/3 proof" is, in fact, quite valid.
• 10-24-2008, 11:13 PM
[SomeGuy]
I also have 6 ways to prove that it, in fact, is not 1.

1. They have different points on a number line.
2. .999~ in a fraction is .999~/1
3. .999~ + .111~ =1
4. Density property ;)
5. Rounding up doesn't count
6. The lines y=1 and y=.999~ are on different positions.
• 10-24-2008, 11:14 PM
[SomeGuy]
Quote:

Originally Posted by drewmandan
No one said that number equals one. We said 0.9~ equals one. See, the way you wrote your version of the number, it's obvious that you think the 9's "end". They don't. As soon as you understand that, you will see that it clearly is 1.

Also, here's something to consider.

1/3 in base 10 is 0.3~
1/3 in base 6 is 0.2000~ (in other words, just 0.2. Do the long division if you don't believe me)

Then, in base 6, if you do 3 * 0.2, you get 1.

This would suggest that there's no fundamental difference between numbers with an infinite number of non-zero repeating decimals and an infinite number of zeros. Therefore, it would seem that the "1/3 proof" is, in fact, quite valid.

Nooo, i understand that repeating decimals don't end. I can stop writing it that way.

But if you took the time to actually plot them on a number line, they would have seperate positions.
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