0.99 repeating = 1

[Denex]

0.33 = 1/3.
3(times)1/3 = 3/3.
3/3 = 1.
0.33(times)3 = 0.99.
3/3 = 0.99.
0.99 = 1.

You may also think that 9/9 = 1, which is true.
Now, think about it in a slightly more complex matter.
1/9 = 0.11 repeating. 8/9 = 0.88 repeating. 0.11 repeating(plus)0.88 repeating = 0.99 repeating.
9/9 = 0.99 repeating = 1.

Discuss.

[denacioust]

3 x .3333 repeating equals one.

[Denex]

Multiplying 3 by each of the infinity decimal digits of 3 will get repeating 9s.

[denacioust]

Yeah but since they infintely repeat generally they're rounded off. I know where you're getting it all we were discussing it in Maths the last day.
I'm sure that to some extent it is flawed, I just don't want to get into it.

[The Brown Cow]

Multiplying 3 by each of the infinity decimal digits of 3 will get repeating 9s.
Right, sorta. That also happens to be 1.

.333333 repeating is 1/3

Now, we all know:
1/3 x 3 = 1

So it follows that .3333333 repeating x 3 = 1

The .9999999 that you get is infinitely close to 1.

[LizardRob]

Of course .999 rounded off is 1, you need to be able to prove it with no rounding.

And without rounding, 1/3 can have no decimal value.

[Tanktunker]

Discuss.
What's to discuss?

[Denex]

Nope.

.333333 repeating is 1/3

Now, we all know:
1/3 x 3 = 1

So it follows that .3333333 repeating x 3 = 1

The .9999999 that you might think you get is infinitely close to 1.
.99 repeating is equivalent to 1, which is why 1/3(times)3 also equals 1.
It's not infinitely close, it IS 1.

[Logic]

No, they are not equivalent. The whole reason it is 0.3 repeating infinitely is because there is no numerical way of expressing 1/3 without using a fractal value. Saying that they are the same value is ambiguous because it forces you to choose an arbitrary number of decimal places to keep.

0.9 repeating is still not equal to one. It is less than one by an infinitely small value which can be expressed by the function 10^-x when x is positive infinity. I can't write this in more clear mathematical terms on this site so you'll have to read and interpret.

[Denex]

So then how would you represent the number fitting between 1 and 0.99 repeating?
There is no 0.0(0)1. Only the last digit in the decimal can be infinitely repeated.
You can't represent 1 - 0.99 repeating, to put it in short.

[LizardRob]

(1-0.999) would be one way of representing it.

Equations are often used to represent numbers which would be otherwise illogical to write. For instance, Standard Form ( N*10^y ).

[Denex]

(1-0.999) would be one way of representing it.

Equations are often used to represent numbers which would be otherwise illogical to write. For instance, Standard Form ( N*10^y ).
An equation requires the usage of an equal sign, so I don't know what you're going on about.

[LizardRob]

Just replace the word Equation with whatever then. Sum? I'm not sure.

Say you wanted to show 800000000000000000000000000000 * 2 = 1600000000000000000000000000000, you could convert it to Standard Form as 8*10^29 *2 = 1.6*10^30

So if you wanted to show the answer to 1 - 0.999, you could simply write it as (1-0.999). We were taught that this is an acceptable answer on any test paper.

[Denex]

So if you wanted to show the answer to 1 - 0.999, you could simply write it as (1-0.999). We were taught that this is an acceptable answer on any test paper.
So if you got the question 5-3 on a test, you could just write (5-3) as an answer? Seems like that's what you're saying.

Besides, that is off the point.

[denacioust]

The word you're looking for is expression.

Anyway, there is no number between 1-.999...repeating. It is as close to 1 as is possible. It's like if you add 1/2 + 1/4 + 1/8 + 1/16 etc. you will never ever reach one. You will probably come as close as .9999 repeating but it will never reach 1.

Anyway, .9999 repeating cannot be expressed as a farction. Just as 1/3 cannot accurately be expressed as a decimal, .3333 repeating just being an approximation.]


So if you got the question 5-3 on a test, you could just write (5-3) as an answer? Seems like that's what you're saying.

Besides, that is off the point.

You know exactly what he means. If an answer to a question was 1+ root 5 you could write it as that rather than calculating it.

[Denex]

The word you're looking for is expression.

Anyway, there is no number between 1-.999...repeating. It is as close to 1 as is possible. It's like if you add 1/2 + 1/4 + 1/8 + 1/16 etc. you will never ever reach one. You will probably come as close as .9999 repeating but it will never reach 1.

Anyway, .9999 repeating cannot be expressed as a farction. Just as 1/3 cannot accurately be expressed as a decimal, .3333 repeating just being an approximation.]
.999 repeating represented as a fraction is 9/9.
Once again, 1/9 is .111 repeating, and 8/9 is .888 repeating, which is .999 repeating when added together.
And in that case, 1/3 is actually .33 repeating. What could it be an approximation of? You know that it makes mathematical sense.

[LizardRob]

9 = 9, so 9/9 = 1 not 0.999

Again, you're using Rounding to make your answers. 1/9 ~= 0.111, 8/9 ~=0.888, so 9/9 ~= 0.999

[Denex]

9 = 9, so 9/9 = 1 not 0.999

Again, you're using Rounding to make your answers. 1/9 ~= 0.111, 8/9 ~=0.888, so 9/9 ~= 0.999
There is NO rounding. I believe you are making the mistake that a repeating decimal has an end of some sort- it doesn't.

If you add .11 with INFINITE 1's at the end with 0.88 with INFINITE 8's at the end, you end up with .99 with INFINITE 9's at the end.

And by the way- your first statement proves my point. 9 DOES equal 9, and either 1 or 0.99 repeating can be used to represent 1.0. It depends on HOW it is equated.

[LizardRob]

(1-0.999), no matter how small, is still a number which has an important role in this idea of 1 = 0.999

You're adding (1-0.999) to the answer to make it come to a whole number. That is rounding.

[Denex]

(1-0.999), no matter how small, is still a number which has an important role in this idea of 1 = 0.999

You're adding (1-0.999) to the answer to make it come to a whole number. That is rounding.
So how is (1-0.999) being added?
Let's go over this clearly.
1/9= .11 INFINITELY REPEATING.
8/9= .88 INFINITELY REPEATING.
0.11 repeating + 0.88 repeating = 1/9 + 8/9.
1/9 + 8/9= ONE.
Therefore, 0.11 repeating + 0.88 repeating ALSO equals ONE.
Where is the rounding?

[LizardRob]

For 0.999 to equal 1, (1-0.999) needs to be added. That's just logic.

This idea is really debatable, I can definately see how the idea works and it's very clever.

[Freddy]

You can't represent 1/3 in decimal form. The best you can do is round, and rounding wont bring you an exact answer.
Using 0.33 or 0.333 or 0.333333333333 or whatever wont work. You can't use an equals sign (=) with a decimal for that.

This is why a teacher always prefers you put answers as fractions. Fractions are always exact.

By using an equals sign your whole equation was wrong.

[DarkReality]

1/3 is an irrational number, meaning there is, by definition, no exact decimal representation of it. You can define 1/3 as 0.3333~ but it's like dividing by 0. You can define 1/0 as infinity, but it'll only work in a very small frame.

Your (copied, pasted and effing old, I might add) calculation is an example of a definition used out of context. When comparing irrational numbers to rational numbers, you can't assume that there's a decimal representation of said irrational numbers.

And yes, it has jack to do with actual rounding. That's exactly what it pretends to prove: 0.999~ = 1 without rounding. But it's not.

[Denex]

For 0.999 to equal 1, (1-0.999) needs to be added. That's just logic.
You seem to be still using that argument. I have more points.

For example, let's say x=0.99 repeating
so 10x=9.99 repeating.
10x-x=9x.
9x=9x.
Using simple algebra, x equates to 1.

Your (copied, pasted and effing old, I might add) calculation is an example of a definition used out of context. When comparing irrational numbers to rational numbers, you can't assume that there's a decimal representation of said irrational numbers.
Using an argument that others have used doesn't count as an immediate copy/paste.
And have I used any irrational numbers in my entire argument? I might have overlooked it, but I don't immediately recall doing so.

You can't represent 1/3 in decimal form. The best you can do is round, and rounding wont bring you an exact answer.
Using 0.33 or 0.333 or 0.333333333333 or whatever wont work. You can't use an equals sign (=) with a decimal for that.
By saying that it is 0.33 repeating, I can use an equal sign, because it is equivalent in that sense to 1/3.

[Freddy]

You seem to be still using that argument. I have more points.

For example, let's say x=0.99 repeating
so 10x=9.99 repeating.
10x-x=9x.
9x=9x.
Using simple algebra, x equates to 1.

x can be what ever the hell it wants to be. The coefficient is the same on both sides. You didn't prove anything there.

[Denex]

x can be what ever the hell it wants to be. The coefficient is the same on both sides. You didn't prove anything there.
Very well. Let's say, instead, that 9.9 repeating = 10x and 0.9 repeating = 1y.
Subtract the two, and you get 9 = 10x-y.

[LizardRob]

For example, let's say x=0.99 repeating
so 10x=9.99 repeating.
10x-x=9x.
9x=9x.
Using simple algebra, x equates to 1.


10(0.99)=9.99
10(0.99)-(0.99)=9(0.99).
9(0.99)=9(0.99)=9
---------------
9.99=9.99
9=8.999 <- Here's where that pesky (1-0.999) went
8.999=8.999

[denacioust]

Using an argument that others have used doesn't count as an immediate copy/paste.
And have I used any irrational numbers in my entire argument? I might have overlooked it, but I don't immediately recall doing so.



1/3 is an irrational number. Well, the decimal form of it is.

[Denex]

1/3 is an irrational number. Well, the decimal form of it is.
A repeating decimal isn't irrational. http://img186.imageshack.us/img186/3183/37090534lr1.png represents 1/3 as a decimal.

[The Brown Cow]

http://en.wikipedia.org/wiki/0.999...
http://polymathematics.typepad.com/polymath/2006/06/no_im_sorry_it_.html
http://mathforum.org/library/drmath/view/57081.html

I could go on. Just Google it. Try to find a good source saying anything contrary.

[denacioust]

A repeating decimal isn't irrational. http://img186.imageshack.us/img186/3183/37090534lr1.png represents 1/3 as a decimal.


God you are stubborn.

An irrational number is one which cannot be expressed simply as a fraction.
Although there are various definitions of it, including it being a number which does not repeat.
Anyway, regardless, it cannot be expressed as a fraction. It is not the decimal calue of 1/3 as proven in your original post.

[Denex]

9.999 = 10x
0.999 = y

Wait what?
(9.999 = 10x) - (0.999 = y)

I don't think that's worded right.
Scratch that.

If 10x = 9.99 repeating, subtract .99 repeating, and you get 9x = 9. With an coefficient on only one side, x must equal 1. But wait, didn't the original x equal .99 repeating?
That must mean 1 = .99 repeating.

Anyway, regardless, it cannot be expressed as a fraction. It is not the decimal calue of 1/3 as proven in your original post.But why would that be the case? Divide 1 by 3, and you get .33 repeating.
.33 repeating must also equal 1/3 as well.

[LizardRob]

You can't use the idea of 0.999 (which implies recurring by the way) = 1 to PROVE the idea of 0.999 = 1.

That's like defining a word using the word itself.

[Denex]

You can't use the idea of 0.999 (which implies recurring by the way) = 1 to PROVE the idea of 0.999 = 1.

That's like defining a word using the word itself.
And where did I do that? Tell me, and I'll fix it.

[LizardRob]

10(0.99)=9.99
10(0.99)-(0.99)=9(0.99).
9(0.99)=9(0.99)=9
---------------
9.99=9.99
9=8.999 <- Here's where that pesky (1-0.999) went
8.999=8.999

There. That is assuming that 8.999 = 9, which is what you're trying to proove.

[Logic]

Scratch that.

If 10x = 9.99 repeating, subtract .99 repeating, and you get 9x = 9.

Where the hell did you learn algebra? YOU CANNOT SUBTRACT CONSTANTS FROM VARIABLES. Do you understand this basic concept?

10x = 9.99
10x - 0.99 = 9

That is the simplest algebraic form you can have when you perform that subtraction operation.

[Denex]

Where the hell did you learn algebra? YOU CANNOT SUBTRACT CONSTANTS FROM VARIABLES. Do you understand this basic concept?

10x = 9.99
10x - 0.99 = 9

That is the simplest algebraic form you can have when you perform that subtraction operation.
Excuse the typo.
This is what I meant:
http://img90.imageshack.us/img90/6011/9salgebraps4.jpg
There. That is assuming that 8.999 = 9, which is what you're trying to proove.
I said scratch that equation.

[Logic]

It's not possible to subtract infinitely long decimals. If you present the argument that the end-behavior of the decimal is uncertain which validates your argument, you must also realize that it disproves other parts.

If the values repeat INFINITELY, arithmetic cannot be performed short of introducing limits and getting approximations of the values.

[DarkReality]

Linguistic mistake. I didn't mean irrational number, I meant a number that cannot be displayed as a finite decimal. Although Logic's argument brings it to the point much better.

[Denex]

It's not possible to subtract infinitely long decimals. If you present the argument that the end-behavior of the decimal is uncertain which validates your argument, you must also realize that it disproves other parts.

If the values repeat INFINITELY, arithmetic cannot be performed short of introducing limits and getting approximations of the values.
Subtracting .99 repeating from 9.99 repeating simply means that the .99 repeating is negated, leaving you with a simple 9.0.

[DarkReality]

It sadly doesn't, as that would imply that they have the same "length". You can't subtract infinity from infinity. It seems logical, but it leads to the mistake 0.99~=1 as well as other similar wrong answers that can be gotten by the same logic.

[Denex]

It sadly doesn't, as that would imply that they have the same "length". You can't subtract infinity from infinity. It seems logical, but it leads to the mistake 0.99~=1 as well as other similar wrong answers that can be gotten by the same logic.
Are you saying you can't subtract 1/3 from 2/3? Which, I remind you, is the same as .33~ and .66~, respectively.

[denacioust]

Are you saying you can't subtract 1/3 from 2/3? Which, I remind you, is the same as .33~ and .66~, respectively.]


You can't add or substract two rough estimations. That'd be very pointless. You can add or subtract them as fractions and make a rough estimation of that's decimal value.

[Denex]

]


You can't add or substract two rough estimations. That'd be very pointless. You can add or subtract them as fractions and make a rough estimation of that's decimal value.
Yes, but in the end, 1/3 is still .33 repeating, and 2/3 is still .66 repeating. It makes no difference whether you subtract/add them in decimal or fractal form; you are still doing it in such a way that it is an infinitely repeating decimal.

[denacioust]

Yes, but in the end, 1/3 is still .33 repeating, and 2/3 is still .66 repeating. It makes no difference whether you subtract/add them in decimal or fractal form; you are still doing it in such a way that it is an infinitely repeating decimal.

1/3 is not .333 repeating. That's a decimal approximation. Not an exact value.

[Denex]

1/3 is not .333 repeating. That's a decimal approximation. Not an exact value.
Please explain.

[denacioust]

Please explain.

Well if you multiply .333 repeating by 3 you get .999 repeating.
If you multiply 1/3 by 3 you get one.
Therefore they are not equal.
I assume you know this since it's pretty much the foundation of your thread.

[Denex]

Well if you multiply .333 repeating by 3 you get .999 repeating.
If you multiply 1/3 by 3 you get one.
Therefore they are not equal.
I assume you know this since it's pretty much the foundation of your thread.
The foundation of my thread is that 0.99 repeating, which is what .33 repeating (times) 3 equals, is equivalent to 1. So it makes perfect sense (to me, that is, that .333 repeating is what 1/3 is. Besides, most people accept that as the value of 1/3.)

[denacioust]

The foundation of my thread is that 0.99 repeating, which is what .33 repeating (times) 3 equals, is equivalent to 1.

Well then 1/3 * 3 = 1.

Then you've just effectively destroyed your own argument.

[Denex]

No. What I'm saying is that there are TWO values that can represent 1.0.

[DarkReality]

That would pretty much destroy fundamental mathematics at it's core. It would mean a number can have two values, which means that no mathematical operator would have a unique answer, which would put us in a state of perpetual indecision and make mathematics about as worthless as poo on a stick.

So no.

1/3 is NOT 0.3333~. The idea of an infinitely repeating decimal is complete bollocks for exactly the reason you're claiming to discover: 1/3*3=1. 0.333~*3=0.999~. Hence they CAN'T be the same thing as there is one and only one decimal notation for a specific (real) number. As I said earlier, it's like the idea that 1/0 is infinity. It makes perfect sense and in most cases, it's absolutely OK to assume that. It's ok because you're omitting several steps that would lead to the same result (transform to 1/a, find the limes for a->0, etc). But dividing by 0 is still not allowed, despite you technically doing so. You can technically notate 1/3 as 0.333~, but the only reason you would ever do that is if you're about to round it off.

[Blastedt]

Is it possible to write out the difference between .99 repeating and 1? No. Thus, for all intents and purposes, it's true. However, it is technically false, as there is an infinitely small difference.

[Denex]

That would pretty much destroy fundamental mathematics at it's core. It would mean a number can have two values, which means that no mathematical operator would have a unique answer, which would put us in a state of perpetual indecision and make mathematics about as worthless as poo on a stick.

So no.

1/3 is NOT 0.3333~. The idea of an infinitely repeating decimal is complete bollocks for exactly the reason you're claiming to discover: 1/3*3=1. 0.333~*3=0.999~. Hence they CAN'T be the same thing as there is one and only one decimal notation for a specific (real) number. As I said earlier, it's like the idea that 1/0 is infinity. It makes perfect sense and in most cases, it's absolutely OK to assume that. It's ok because you're omitting several steps that would lead to the same result (transform to 1/a, find the limes for a->0, etc). But dividing by 0 is still not allowed, despite you technically doing so. You can technically notate 1/3 as 0.333~, but the only reason you would ever do that is if you're about to round it off.
So let's say that hypothetically, 1/3 is not 0.333 repeating. So then what number, multiplied by 3, fits completely into 1.0?

[The Brown Cow]

Sure, just ignore my post...
http://en.wikipedia.org/wiki/0.999...
http://polymathematics.typepad.com/polymath/2006/06/no_im_sorry_it_.html
http://mathforum.org/library/drmath/view/57081.html

I could go on. Just Google it. Try to find a good source saying anything contrary.

Honestly, I challenge you to find any even remotely reliable source that says that 0.9 repeating is not equal to 1.

The equality has long been accepted by professional mathematicians and taught in textbooks.

Students of mathematics often reject the equality of 0.999… and 1, for reasons ranging from their disparate appearance to deep misgivings over the limit concept and disagreements over the nature of infinitesimals. There are many common contributing factors to the confusion:
Students are often "mentally committed to the notion that a number can be represented in one and only one way by a decimal." Seeing two manifestly different decimals representing the same number appears to be a paradox, which is amplified by the appearance of the seemingly well-understood number 1.[1]
Some students interpret "0.999…" (or similar notation) as a large but finite string of 9s, possibly with a variable, unspecified length. If they accept an infinite string of nines, they may still expect a last 9 "at infinity".[2]
Intuition and ambiguous teaching lead students to think of the limit of a sequence as a kind of infinite process rather than a fixed value, since a sequence need not reach its limit. Where students accept the difference between a sequence of numbers and its limit, they might read "0.999…" as meaning the sequence rather than its limit.[3]
Some students regard 0.999… as having a fixed value which is less than 1 by an infinitely small amount.
Some students believe that the value of a convergent series is an approximation, not the actual value.

[DarkReality]

So let's say that hypothetically, 1/3 is not 0.333 repeating. So then what number, multiplied by 3, fits completely into 1.0?

Not hypothetically. 1/3 isn't 0.333~. You essentially proved it with the first post.

That number would be 1/3. It doesn't have to have a decimal notation in order to be a number. PI is, after all, also a number. As is euler's constant. Both those numbers are impossible to correctly notate with decimals, or anything but their respective symbols.

[Vagrant]

Let's take a different approach to this whole problem.

You take a full human. You scratch off a cell. Is the guy now 99.99% repeating human, but not a full human?

That's my whole problem with this discussion: It's fruitless, and ultimately, retarded. The reason that 3/3 = 1 is because 1/3 is NOT equal to .333 repeating. That is used merely as an approximation to a value which we cannot hope to ever reach in this lifetime or the next.

[stickdude237]

I couldn't be bothered to read all 6 pages of this, but really this is the stupidest thing I have ever heard. 1/3 doesn't equal, 0.333333333333333333 it equals .3 repeated for infinity, no matter how far you go it's still rounding, so no.

[The Brown Cow]

You take a full human. You scratch off a cell. Is the guy now 99.99% repeating human, but not a full human?
Humans have a finite number of cells in their bodies at a given moment, so that percentage will have a finite number of decimal places.

I know it's hard to accept, but 0.9 repeating is equal to 1. That's an accepted mathematical truth. Ask Google or ask a math teacher and I promise you'll get the same answer.

[Denex]

Let's take a different approach to this whole problem.

You take a full human. You scratch off a cell. Is the guy now 99.99% repeating human, but not a full human?

That's my whole problem with this discussion: It's fruitless, and ultimately, retarded. The reason that 3/3 = 1 is because 1/3 is NOT equal to .333 repeating. That is used merely as an approximation to a value which we cannot hope to ever reach in this lifetime or the next.

We don't have an infinite number of cells. If you remove a cell from the human body- (and let's assume that there is no cellular reproduction involved in this case) you will be 99.99999999999999999999999999999999999999 percent of a human- that's right, there will be that last decimal 9. 99.99 with 9 trillion nines at the end is not the same as 99.99 repeating.

You say 1/3 is not the same as .33. But if you do long division of 1/3, you will get .3 initially. Then another .03 at the end. And another 0.003. And another 0.0003. There is subsequentially no reason to believe that the decimals will start to vary, because you can do it 5 hundred billion times and still end up with another 3 as the last decimal place, and therefore the best possible thing to do is to simply infinitely repeat the 3. You just simply can not fill any decimal of 3 into the exact value of 1.0. It's not difficult to wrap your head around, it's simple logic.

If you still don't believe it- you can ask a mathematics professor/teacher (who should be more of a professional at this than I am) or search the internet for a reliable source.

[Vagrant]

Humans have a finite number of cells in their bodies at a given moment, so that percentage will have a finite number of decimal places.

I'm well aware of this.

My point was that at some point, the decimal points become irrelevant. You can go on for a googol if you please, But each decimal becomes less and less significant.

We don't have an infinite number of cells. If you remove a cell from the human body- (and let's assume that there is no cellular reproduction involved in this case) you will be 99.99999999999999999999999999999999999999 percent of a human- that's right, there will be that last decimal 9. 99.99 with 9 trillion nines at the end is not the same as 99.99 repeating.http://en.wikipedia.org/wiki/E_%28mathematical_constant%29

In terms of functionality, .99 repeating forever is 1. It behaves like 1. The very final decimal point (which is never reached) becomes irrelevant in just about any equation you can concoct. The person I thought of is not any less human -- the reason I said "scratch off a cell" is because a single cell is insignificant in the scope of things, and it will eventually be replaced anyways.

You say 1/3 is not the same as .33. But if you do long division of 1/3, you will get .3 initially. Then another .03 at the end. And another 0.003. And another 0.0003. There is subsequentially no reason to believe that the decimals will start to vary, because you can do it 5 hundred billion times and still end up with another 3 as the last decimal place, and therefore the best possible thing to do is to simply infinitely repeat the 3. You just simply can not fill any decimal of 3 into the exact value of 1.0.We still don't have the exact numbers for pi or e, yet we still use them.

http://en.wikipedia.org/wiki/E_%28mathematical_constant%29
http://en.wikipedia.org/wiki/Pi

limx->3 (x/3) = 1

Tell me that limit is not true. The limit as x approaches 3 of f(x)= x/3 is 1.

[Woolfenstien]

1 = c4ca4238a0b923820dcc509a6f75849b
0.(9) = bded8f90879792d29cfdcfe39d761194

Therefore the two values are not equal.

Imagine this. You're in France, however you are infinitely close to Belgium. Are you in Belgium?

Change that around a bit, and you have: You are a number less than 1, but you are infinitely close to 1. Are you 1?

I am aware that 0.(9) might as well be one.

[The Brown Cow]

This is becoming tiring... I know it's hard to accept - I have trouble with it too. But it doesn't matter what clever little logic puzzle you use; 0.9... is 1. It's simply the truth.

Go look around the internet. You'll find a bunch of people asking the same questions, and getting the same answers.

The mind (yes, even mine) instinctively rebels at this conclusion. We readily concede that .999~ gets infinitely close to 1--to put it in mathematical terms, 1 is the sum of the converging infinite series .9 + .09 + .009 + . . . But, we protest, .999~ never quite reaches that limit. If at any step we halt the progression to infinity to take a sum, we find that we remain separated from 1 by some infinitesimal amount.

But that's just the point, the mathematicians say. When a decimal repeats ad infinitum, you never stop.

The lower primate in us still resists, saying: .999~ doesn't really represent a number, then, but a process. To find a number we have to halt the process, at which point the .999~ = 1 thing falls apart.

Nonsense. The fraction 1/3 is an ordinary number, and .333~ is the same ordinary number; an infinite series of 3s simply happens to be the only way to express said number given the limitations of decimals. Granted, decimals let us express the quantity 1 without difficulty, but the process of infinite repetition produces the same result; .999~ is merely another way of saying 1. Likewise, pi is an ordinary number; it's just a quirk of the real number system that we have to express it as 3.14159 etc (without ever repeating or stopping). Rational numbers, which by definition can be expressed as fractions, translate to repeating or terminating decimals; irrational numbers (like pi) never repeat or terminate in their decimal form.

If you're still having trouble, consider another example involving a converging infinite series: Zeno's paradox, proposed by the Greek philosopher Zeno in the fifth century BC. Suppose Achilles and a tortoise have a footrace. Achilles is ten times faster than the tortoise, but the tortoise has a ten-meter head start. In the time Achilles runs those ten meters, the tortoise crawls one meter. In the time Achilles runs that one meter, the tortoise plods another .1 meter. In the time Achilles runs that .1 meter, the tortoise lumbers ahead .01 meter. You get the picture. We seem to be reasoning ourselves to the conclusion that Achilles can never pass the tortoise.

But common sense says he does, and common sense is right. The expression 10 + 1 + .1 + .01 . . . is a converging infinite series whose sum is 11.111~ (or, to express it as a mixed number, 11 1/9). Common sense also tells us that Achilles does not merely approach this limit (as Zeno's paradox would have us believe), but reaches and then passes it--i.e., that Achilles overtakes the tortoise at 11 1/9 meters. We thus see (I hope) that there's nothing magical and unattainable about limits, and so no barrier to grasping that .999~ = 1.

Doesn't that enhance your quality of life? Of course it does. Not that body paint doesn't have its place, but there's just no substitute for the pleasures of an infinite series.

It come up often on forums, and you always wind up with a massive debate amongst people who really don't know what they're talking about. Hell, Blizzard even issued a (fake) press release (http://www.blizzard.com/press/040401.shtml) to shut people up on their forums.

[Logic]

There is subsequentially no reason to believe that the decimals will start to vary, because you can do it 5 hundred billion times and still end up with another 3 as the last decimal place

You just disproved your own argument.

3 * .3 = 0.9
3 * .33 = 0.99
3 * .333 = 0.999

If it never varies, guess what? It never equals one. Go contradict yourself some more.

@ TBC: The proof is not valid for reasons previously stated in this thread (namely that you cannot perform arithmetic on irrational numbers unless they are represented as fractions or symbols).

[Denex]

You just disproved your own argument.

3 * .3 = 0.9
3 * .33 = 0.99
3 * .333 = 0.999

If it never varies, guess what? It never equals one. Go contradict yourself some more.

@ TBC: The proof is not valid for reasons previously stated in this thread (namely that you cannot perform arithmetic on irrational numbers unless they are represented as fractions or symbols).
Err, my point was that the infinite 9s on the end of the decimal makes it equivalent to 1.0. I didn't say .33 repeating multipled by 3 equals the integer form of 1.0, instead as .99 repeating which is, essentially, 1.

0.3 repeating can be represented as 1/3. It doesn't matter whether you represent it as 0.3 repeating or 1/3, it is the same thing. Surely it is easier to represent it in that way- but if you multiply, say, 1/3 by 3, essentially, you are still multiplying .33 repeating by 3.

[The Brown Cow]

@ TBC: The proof is not valid for reasons previously stated in this thread (namely that you cannot perform arithmetic on irrational numbers unless they are represented as fractions or symbols).
Who the hell decided that? 0.3... is a perfectly valid way of expressing 1/3. They're the same number; it doesn't matter how you write it.

The Wikipedia article gets into some seriously advanced proofs which, quite frankly, go right over my head. But if you want to read about how you can do it with Dedekind cuts or Cauchy sequences (thereby avoiding writing those pesky long decimals), it's all there.

There are proofs of this dating back to the 1700's, guys. I'm pretty sure the mathematicians have it right.

[LizardRob]

Is it possible to write out the difference between .99 repeating and 1? No. Thus, for all intents and purposes, it's true. However, it is technically false, as there is an infinitely small difference.

(1-0.999...) as I said earlier in the thread. That's an expression (thanks for reminding me of the word, Den) to show the smallest number above one.

Although you could probably argue that (1-0.999...) = 0. It really just depends on your opinion.

Edit: Also, I don't know what you're taught, but everywhere I know here teaches that 0.333... is the only way to deal with 1/3 in decimal form, but it's certainly not exactly 1/3.

[Logic]

Who the hell decided that? 0.3... is a perfectly valid way of expressing 1/3.

The whole point of this thread is to argue about whether that is correct or not. -_-

I will look at the proofs on Wikipedia though.

EDIT: YOUR OWN SOURCES SAY THAT YOU ARE WRONG:



Section about Dedekind Cuts...
Clearly 0.9* = 1 + 0¯, so 0¯ is a sort of negative infinitesimal. On the other hand, you can't solve the equation 0.9* + X = 1 because, in cut D, the sum of a traditional real with any real is a traditional real.

Section about Open Problems...
Although we can introduce negative decimal fractions, negative numbers in general present a serious problem because we don't have cancellation in cut D. We can't simply write them as additive inverses of positive numbers. Moreover, we have no interpretation for the number -3.14159265... because this represents a process of approximation from above, -3.14159 being greater than -3.14159265..., whereas in cut D all real numbers are approximated from below. Of course we could just introduce symbols like -3.14159265..., but it's not clear how to get a satisfactory coherent system that incorporates them.

Because of this, multiplication of arbitrary real numbers is also a serious problem, if for no other reason than that we don't know how to multiply -1 by 3.14159265.... Even in the traditional approach, multiplication is awkward. The elegant treatment of addition is replaced by an ugly division into cases: one defines how to multiply positive numbers, and extends to negative numbers according to the usual rules

[Denex]

(1-0.999...) as I said earlier in the thread. That's an expression (thanks for reminding me of the word, Den) to show the smallest number above one.

Although you could probably argue that (1-0.999...) = 0. It really just depends on your opinion.

Edit: Also, I don't know what you're taught, but everywhere I know here teaches that 0.333... is the only way to deal with 1/3 in decimal form, but it's certainly not exactly 1/3.
There is no number between .99 repeating and 1.

[Tanktunker]

There is no number between .99 repeating and 1.
There is, it's the smallest positive number possible (inf^-1), or something like that.

[Logic]

There is no number between .99 repeating and 1.

I suppose you're blind to reading your own sources - LOOK AT MY ABOVE POST:

"Clearly 0.9* = 1 + 0¯, so 0¯ is a sort of negative infinitesimal. On the other hand, you can't solve the equation 0.9* + X = 1 because, in cut D, the sum of a traditional real with any real is a traditional real."

Guess who's wrong and whose proofs are all worthless!

[Zapurdead]

Paradox, #####!

[Denex]

There is, it's the smallest positive number possible (inf^-1), or something like that.
Yes, but it can't be represented numerically or even with a symbol.

[LizardRob]

Therefore it is not a number! I see that logic, genius.

[Denex]

I suppose you're blind to reading your own sources - LOOK AT MY ABOVE POST:

"Clearly 0.9* = 1 + 0¯, so 0¯ is a sort of negative infinitesimal. On the other hand, you can't solve the equation 0.9* + X = 1 because, in cut D, the sum of a traditional real with any real is a traditional real."

Guess who's wrong and whose proofs are all worthless!
Have I cited any sources?
And can you tell me exactly how that disprove anything?

[Logic]

If you haven't, then your argument is baseless and has been disproven already by all of the posts against it. Otherwise, all of your sources have been disproven and are worthless as well.

My last sentence in that post still stands. Also, "it can't be represented with a symbol" - you really must have vision problems:

"0¯ is a sort of negative infinitesimal"

Do you see the 0¯? Do you? I just want to make sure because that clearly would constitute a symbolic notation of the number.

[Denex]

Notices how it says "On the other hand, you can't solve the equation 0.9* + X = 1" afterwards. So x can't equal 0¯?
And here are some sources if you insist:
http://mathforum.org/dr.math/faq/faq.0.9999.html
http://www.cut-the-knot.org/arithmetic/999999.shtml
http://www.newton.dep.anl.gov/askasci/math99/math99167.htm
http://descmath.com/diag/nines.html
http://mathforum.org/library/drmath/view/55746.html
Those are to name a few.

[Logic]

Do you read?

http://descmath.com/diag/nines.html

Your own source here entirely disproves EVERYTHING you have said. Jesus Christ.

[a1b23]

0.99999999 /= 1. It's as simple as that. With the addition of a decimal you get closer to 1 but never reach one. You are infinitely close to 1 (limits/asymptotes).

[Denex]

Do you read?

http://descmath.com/diag/nines.html

Your own source here entirely disproves EVERYTHING you have said. Jesus Christ.
Read the conclusion.
"Repeating nines and the unit whole are essentially the same quantity."


0.99999999 /= 1. It's as simple as that. With the addition of a decimal you get closer to 1 but never reach one. You are infinitely close to 1 (limits/asymptotes).Common misconceptions: http://en.wikipedia.org/wiki/0.99...
Students are often "mentally committed to the notion that a number can be represented in one and only one way by a decimal." Seeing two manifestly different decimals representing the same number appears to be a paradox (http://en.wikipedia.org/wiki/Paradox), which is amplified by the appearance of the seemingly well-understood number 1.[1] (http://en.wikipedia.org/wiki/0.99...#_note-0)
Some students interpret "0.999…" (or similar notation) as a large but finite string of 9s, possibly with a variable, unspecified length. If they accept an infinite string of nines, they may still expect a last 9 "at infinity".[2] (http://en.wikipedia.org/wiki/0.99...#_note-1)
Intuition and ambiguous teaching lead students to think of the limit of a sequence as a kind of infinite process rather than a fixed value, since a sequence need not reach its limit. Where students accept the difference between a sequence of numbers and its limit, they might read "0.999…" as meaning the sequence rather than its limit.[3] (http://en.wikipedia.org/wiki/0.99...#_note-2)
Some students regard 0.999… as having a fixed value which is less than 1 by an infinitely small amount.
Some students believe that the value of a convergent series (http://en.wikipedia.org/wiki/Convergent_series) is an approximation, not the actual value

[Logic]

Read the conclusion.
"Repeating nines and the unit whole are essentially the same quantity."

My God. I read the document; that is the author's summation and opinion. Maybe if you read the mathematical facts of the document you would see that this is an opinion and not true.

Somehow you managed to overlook EVERY SINGLE point that he/she makes against this being true and instead plucked the fruitless summation out at the end of the text. Brilliant.

[Tanktunker]

Yes, but it can't be represented numerically or even with a symbol.
inf^-1
durrr.

Do you read?

http://descmath.com/diag/nines.html

Your own source here entirely disproves EVERYTHING you have said. Jesus Christ.
That's because he went to google, and assumed that since his opinion, and his opinion alone, is the correct one, all of the links would support him.
Don't be fooled into thinking he read any of his own sources.

[a1b23]

only 1=1 .99999999/=1 .999999<1

[Denex]

My God. I read the document; that is the author's summation and opinion. Maybe if you read the mathematical facts of the document you would see that this is an opinion and not true.

Somehow you managed to overlook EVERY SINGLE point that he/she makes against this being true and instead plucked the fruitless summation out at the end of the text. Brilliant.
Please give me several points in which he says .99 repeating does not equal 1, or in fact, anything that disproves me.

That's because he went to google, and assumed that since his opinion, and his opinion alone, is the correct one, all of the links would support him.
Don't be fooled into thinking he read any of his own sources.What a generalization. I will not take that as a personal insult (however much it seems to be), but how about you actually point out some areas in which any of these sources disproves me?

[Logic]

Please give me several points in which he says .99 repeating does not equal 1.

Ready?

Note, that if c is equal to the first n digits of pi, then 1 - (pi - c) contains a string of (n-1) repeating nines. With each additional digit in our calculation of pi, we extend our series of nines and get closer to the real value of pi. However, a million digit decimal is still not equal to pi. If we subtract the difference of the absolute value of pi from a finite decimal representation of pi, we will get a decimal of repeating 9s. 1 - (pi - 3.14159265358979323846...) = 0.99999999999999999999... Expanding pi indefinitely still leaves us with a mysterious logical entity between pi and the decimal.

Saying that 0.99999... equals 1 is the same thing as saying that converging sequences actually reach the point of convergence. Although the quantities are essentially the same, absolute equality requires a major metaphysical leap.

The first thing I hope you notice is that there is essentially nothing between the two numbers. On the other hand, it is impossible to actually do the subtraction. We write decimals from left to right, but do subtraction from right to left.

Infinite decimals are unbound on the right. There is not a furthest right hand digit. That means we cannot do standard subtraction.

Yet there is always a remainder. Subtracting 0.99999... from 1.0000... leaves 0.00000... plus a remainder that has been infinitely dimished.

Repeating nines seem to offer the same problem as multiplying and dividing by zero 0 * 5 = 0 * 4. Dividing both sides of the equation by 0 we get 5 = 4. There really is not a moral precept against dividing by zero. It is just that information gets lost when we perform division by zero...causing problems in our reasoning process.

You are likely to come across math texts claiming that: if three thirds equals a whole, and repeating threes add up to repeating nines; Then repeating nines equals a whole. The flaw in this proof is with the stipulation that repeating threes equal a third. The same mysterious logical entity that stands between between repeating nines and a whol stands between repeating threes and a third. After then nth iteration of the expanding a third into decimal form, I still have a remainder 1/(3*10(n+1)). A third equals repeating threes plus the logical entity. Adding three thirds equals repeating nines plus the strange logical entity.

The demonstatration that three repeating threes add up to a repeating nine adds to my belief that repeating threes might be different from the absolute value a third.

There are some strange things going on with fractions. Let's say I had a fraction of the form a/b where b is has factors other than 2 or 5. a is a number between 1 and b. Such a fraction would be between 0 and 1.

The digits in the decimal expansion of a/b and (b-a)/b will add up to repeating nines. Here are a few samples:

1/7 = 0.142857... (142857 repeats)
6/7 = 0.857142...

2/7 = 0.285714...
5/7 = 0.714285...

3/7 = 0.428571...
4/7 = 0.571428...

1/6 = 0.166666... (6 repeats)
5/6 = 0.833333... (3 repeats)

5/11 = 0.454545... (45 repeats)
6/11 = 0.545454... (54 repeats)

1/11 = 0.0909090... (90 repeats)
10/11= 0.9090909... (09 repeats)

I find it interesting that the infinite decimals of a/b and (b-a)/b will always add up to repeating nines.

arlier in this brain fart, I mentioned that the (1 - 1/2n) converges at a different rate than (1 - 1/3n). Altough both numbers produce an endless string of repeating nines, I am willing to accept that they still are different numbers. The fact that the infinite decimal expansions of a/b + (b-a)/b always creates an string of repeatings nines in one step makes me wonder if the repeating nines produced by adding 0.333... and 0.666... is absolutely equal to the repeating nines produced by adding 0.090909... and 0.909090...

In Everything and More, David Foster Wallace presents an interesting trick to convince the world that repeating nines "equal" the unit whole. He starts with x = 0.9999... He then subtracts x from 10*x as follows:

9.99999...
0.99999...
===========
9.00000...

10 - 1 = 9. Hence 9x = 9.000... implying that x=1.0. We start with x = 0.999... and conclude that x equals 1.0. In the essay above we noted that a digit to digit comparison from left to right does not prove that numbers are actually the same thing. This leaves the possibility that (10 * 0.9999... - 9) is not actually equal to 0.9999... despite the fact that we can match the digits from left to right.


Did I miss anything?

[Denex]

Ready?



Did I miss anything?
Am I missing anything, that is what I am wondering.

Several things.

Firstly, my sources don't have to BACK ME UP completely (although, this does, to some extent, it provides both views). I may want to look at COUNTER-SOURCES for the very reason.

What I am saying- essentially- is that the author of this article is not being entirely biased. He is NOT saying that 0.99 is NOT 1.0. NOR is he making the claim that they are in fact different. Therefore, that does not mean I still can't use it as a source.

Secondly, just as it is unprovable that .99 repeating = 1, the contrary is also unprovable.

[The Brown Cow]

That's because he went to google, and assumed that since his opinion, and his opinion alone, is the correct one, all of the links would support him.
Don't be fooled into thinking he read any of his own sources.
It's my opinion too. And the opinion of pretty much every mathematician out there. Sure, you can find a paper or two written about how it can't be conclusively proven, but what does that prove? Anyone can write for the internet.

Considering the sheer volume of sources saying that the symbols are equal, I would have to label it as an accepted fact. And it's certainly not the outlandish, radical idea that people seem to be trying to label it as.

[Logic]

Consider this: if I can define the difference between 0.9* and 1, how can you say that it does not exist? Just think about it. It certainly does exist and is defined by:

10^-x where x is negative infinity

[Vagrant]

Consider this: if I can define the difference between 0.9* and 1, how can you say that it does not exist? Just think about it. It certainly does exist and is defined by:

10^-x where x is negative infinity

You just said 10^infinity.

You mean 10^-infinity, I assume though.

Let me put it this way: Since 0.(9) goes far beyond 10^100 digits, the entire argument is pointless. Even if there is some infinitely small difference between 1-0.(9), you've gone beyond all real and useable numbers into something that is, essentially, pointless.

only 1=1 .99999999/=1 .999999<1Limx->3 (x/3) = 1. Prove that wrong.

[Denex]

"As part of Ed Dubinsky's "APOS theory" of mathematical learning, Dubinsky and his collaborators (2005) propose that students who conceive of 0.999… as a finite, indeterminate string with an infinitely small distance from 1 have "not yet constructed a complete process conception of the infinite decimal". Other students who have a complete process conception of 0.999… may not yet be able to "encapsulate" that process into an "object conception", like the object conception they have of 1, and so they view the process 0.999… and the object 1 as incompatible..."

From Wikipedia. http://en.wikipedia.org/wiki/0.999...

[Vagrant]

"As part of Ed Dubinsky's "APOS theory" of mathematical learning, Dubinsky and his collaborators (2005) propose that students who conceive of 0.999… as a finite, indeterminate string with an infinitely small distance from 1 have "not yet constructed a complete process conception of the infinite decimal". Other students who have a complete process conception of 0.999… may not yet be able to "encapsulate" that process into an "object conception", like the object conception they have of 1, and so they view the process 0.999… and the object 1 as incompatible..."

From Wikipedia. http://en.wikipedia.org/wiki/0.999...

I understand the concept fully. Stop throwing quotes at me. Trust me, I've had epiphanies about infinity. When you visualize it, it is mind staggering.

What I'm trying to say is that the whole debate is pointless after a certain point. There is nothing measured or measurable in this universe that goes up to a googol.

[DarkReality]

We've only established, at this point, that no one can even understand Wikipedia articles and that there is obviously not one correct solution, as various algorithms and methods will lead to varying results.

[Zapurdead]

It is generally accepted, and theoretically, 0.9999... is 1, but numerically, 1- 0.999... is 0.1....

[Denex]

1- 0.999... is 0.1....
No it's not.

[Logic]

Shut up and let the thread die. We're both wrong or right.

[Denex]

I was going to let the thread die until Zapurdead's post.
1 - 0.99 repeating does not equal 0.11 repeating; nor is that really up for debate.

[denacioust]

Yeah, it'd be .00000...(infinite)1

Anyway, leave this, it's a stupid thread.