Why does an infinite number of .3's not equal 1/3?

[ice109]

Since nobody took the burden to give the famous example for these debates,

<br /> 3\frac{1}{3} = 3(0.333333\ldots) \Longrightarrow 1 = 0.999999\ldots<br />

Just to stir up the soup.

and of course the right side is a correct equality.

 

[lurflurf]

How then is it possible for an infinite number of .3's to equal 1/3, or indeed any real number? Wouldn't an infinite number of positive values invariably equal infinity?

Also, by definition, adding an extra 3 to a finite string of 3s would never result in 1/3, so why would repeating a failed process an infinite number of times result in a success?

.3333333333...
is not a failed brocess it is a way of writing 1/3
in particular if x and y are real numbers with
|x-y|<p
where p is any positive real number
then
x=y

 

[zgozvrm]

I don't understand why ANYONE is confused as to why 0.33333... is exactly equal to 1/3!

If you divide 3 into 1 (using long division), you get: 0, remainder 1. But if you keep carrying out the division, you get 0.33333...

<br /> \begin{array}{rc@{}c}<br /> &amp; \multicolumn{2}{l}{\, \, \, \00.33333\dotsb} \vspace*{0.12cm} \\<br /> \cline{2-3}<br /> \multicolumn{1}{r}{3 \hspace*{-4.8pt}} &amp; \multicolumn{1}{l}{ \hspace*{-5.6pt} \Big) \hspace*{4.6pt} 1.00000} \\<br /> &amp; \multicolumn{2}{l}{\, \, \,0}<br /> \vspace{1mm}<br /> \\<br /> \cline{2-3}<br /> &amp; \multicolumn{2}{l}{\, \, \,1\phantom{.}0} \\<br /> &amp; \multicolumn{2}{l}{\, \, \, \phantom{1.}9}<br /> \\<br /> \cline{2-3}<br /> &amp; \multicolumn{2}{l}{\, \, \, \phantom{1.}10} \\<br /> &amp; \multicolumn{2}{l}{\, \, \, \phantom{1.0}9}<br /> \\<br /> \cline{2-3}<br /> &amp; \multicolumn{2}{l}{\, \, \, \phantom{1.0}10} \\<br /> &amp; \multicolumn{2}{l}{\, \, \, \phantom{1.00}9}<br /> \\<br /> \cline{2-3}<br /> &amp; \multicolumn{2}{l}{\, \, \, \phantom{1.00}10} \\<br /> &amp; \multicolumn{2}{l}{\, \, \, \phantom{1.000}9}<br /> \\<br /> \cline{2-3}<br /> &amp; \multicolumn{2}{l}{\, \, \, \phantom{1.0000}etc.}<br /> \end{array}<br />As you can see, the cycle continues and the quotient is never exactly resolved since the 3's repeat forever.

 

[Gib Z]

I don't understand why ANYONE is confused as to why 0.33333... is exactly equal to 1/3!

Probably because people like YOU say things like this:

As you can see, the cycle continues and the quotient is never exactly resolved since the 3's repeat forever.

That is precisely the incorrect logic that stumbles people on this.

I would like to say everything would be resolved if people just understood what defines the decimal notation, but even then a lot of people seem to think its impossible to sum those in "finite amounts of time" .

 

[zgozvrm]

I'm sorry to say, that my logic is not incorrect when I state that the "quotient is never exactly resolved."

You can never stop this division: You cannot say that 1 million "3's" will exactly equal 1/3, nor can you say that 1 million and 1 "3's" equals 1/3, etc.

I think everyone can agree that each successive division appends an additional "3" to the sequence and gives a decimal representation of 1/3 that is closer than the previous.

My point is that you can never stop the division; the division is never satisfied to an decimal number that can be quantified an any way other than stating that the "3's" continue forever (there is always a remainder of 1). Therefore, it is never exactly resolved (as a decimal number).

People may stumble on this, but that has nothing to do with "my logic."

 

[trambolin]

Do you occasionaly revive this topic?

 

[CRGreathouse]

You can never stop this division: You cannot say that 1 million "3's" will exactly equal 1/3, nor can you say that 1 million and 1 "3's" equals 1/3, etc.

Do you think Gib Z was suggesting you could?

 

[zgozvrm]

Do you think Gib Z was suggesting you could?

No. I was only justifying my statement that "the quotient is never exactly resolved" since Gib Z seemed to have a problem with "people like me" saying things like that...

 

[fluidistic]

Since nobody took the burden to give the famous example for these debates,

LaTeX Code: <BR>3\\frac{1}{3} = 3(0.333333\\ldots) \\Longrightarrow 1 = 0.999999\\ldots<BR>

Just to stir up the soup.

OMG, please don't!

Ahahahahah! Sorry, this was hilarous.

 

[Algr]

zgozvrm,

What is odd is you start out saying one thing, and then give an example that proves exactly the opposite. There is no place in math where an unfinished equation can prove anything. It seems to me that if a math problem can't be completed, then it has no answer. You can't simply assume that because the running total seems to be approaching 1/3 that that it must arrive at it.

 

[Algr]

Divided parts? Who said anything about divided parts?

If you want me to take your answer seriously, you might not want to pretend you don't understand the question.

 

[zgozvrm]

zgozvrm,

What is odd is you start out saying one thing, and then give an example that proves exactly the opposite. There is no place in math where an unfinished equation can prove anything. It seems to me that if a math problem can't be completed, then it has no answer. You can't simply assume that because the running total seems to be approaching 1/3 that that it must arrive at it.

Wow! I'm not sure where you are getting this.

1) I did NOT "start out saying one thing, and then give an example that proves exactly the opposite." There seemed to be some doubt as to whether 0.333333... was exactly equal to 1/3 or not. I simply proved that it was by showing the inverse: since 1 divided by 3 = 0.3333333... then 0.3333333... MUST equal 1/3! PERIOD!

2) I never said the equation was "unfinished," only that you can never finish the long division, as it continues forever.

3) I never said that since "the running total seems to be approaching 1/3 that that it must arrive at it." What I said was, the more 3's you append to the sequence, the closer you get to 1/3 (which cannot be argued), but that you cannot ever stop appending 3's. Once you stop appending 3's, you will have come to some finite number of 3's (and therefore, an EXACT decimal number) and this value will NOT exactly equal 1/3. The value ONLY equals 1/3 exactly, if the series of 3's continues forever.

By the way, the same holds true for 1/9 = 0.11111..., 2/9 = 0.222222..., or for that matter 2369/9999 = 0.2369236923692369...

 

[statdad]

"I simply proved that it was by showing the inverse" - no, you didn't. Your work indicates that the result is true, but it doesn't qualify as a proof.

 

[Sorry!]

I guess this leads to the famous conclusion that 1/3 x 3 = 0.9999999...

Because everyone knows that 1/3 = 0.333333... so therefore 0.333333... x 3 = 0.999999...
No matter how many extra threes you add to the end of the decimal it will always be 3x3 which gives you 9. It might approach one... but it never is one come on this is elementary stuff. :D :D :D :D.

 

[zgozvrm]

"I simply proved that it was by showing the inverse" - no, you didn't. Your work indicates that the result is true, but it doesn't qualify as a proof.

So you're saying that showing that x/y = z doesn't prove that z = x/y!

This fact cannot be disproved.


If you have an expression that looks complicated to you, such as

\frac{XY}{X+Y} = \frac{1}{\frac{1}{X} + \frac{1}{Y}}

It may be easier to show the inverse is true:

\frac{1}{\frac{1}{X} + \frac{1}{Y}} = \frac{XY}{X+Y}

\(\frac{1}{1/X + 1/Y} \times \frac{X}{X} = \frac{X}{X(1/X + 1/Y)} = \frac{X}{1 + X/Y} = \frac{X}{1 + X/Y} \times \frac{Y}{Y} = \frac{XY}{Y(1 + X/Y)} = \frac{XY}{Y + X} = \frac{XY}{X+Y}\)

How does this NOT qualify as a proof? It may not be a theorem or a postulate. We all know that if A=B, then B=A. If there is any doubt as to whether A=B is true, but we can show that B=A, we have PROVEN that A=B.

This is basic stuff. If you're going to argue this further, you obviously just like to stir up crap!

 

[D H]

So you're saying that showing that x/y = z doesn't prove that z = x/y!

No, statdad was saying that your informal demonstration does not qualify as proof.

 

[alxm]

2) I never said the equation was "unfinished," only that you can never finish the long division, as it continues forever.

Which is what you're getting criticized for. You're saying that as if it meant something, which it doesn't. So what if the division 'continues forever'? That doesn't mean you can't know the result. It doesn't mean the answer isn't a 'real' number. (I can't ever write all the decimals in Pi. So?). The statement doesn't have any relevance or significance, but is exactly what confuses people.

Once you stop appending 3's, you will have come to some finite number of 3's (and therefore, an EXACT decimal number) and this value will NOT exactly equal 1/3. The value ONLY equals 1/3 exactly, if the series of 3's continues forever.

No, infinitely, not 'forever'. Stop being anthropocentric. It's not about what you can practically add up in an amount of time. It's an infinite series, and 'infinite' means infinite in math. Not 'a really big number'. And an summation mark means a sum, not a command to 'add up these numbers'. Just because there's an infinite number of terms in a summation doesn't mean that its value isn't finite, or exactly calculable.

By the way, the same holds true for 1/9 = 0.11111..., 2/9 = 0.222222..., or for that matter 2369/9999 = 0.2369236923692369...

By the way, any number, whether it has recurring decimals or not, can be written as an infinite series in an infinite number of ways, e.g. 1 = \sum^\infty_{n=1}\frac{1}{2n}

The value of that sum is exactly 1. The fact that a human being manually summing up the numbers would never 'reach' 1 doesn't enter into it. You're not making anything clearer by saying that, you're bringing up exactly what gets people mixed up.

 

[zgozvrm]

No, statdad was saying that your informal demonstration does not qualify as proof.

I disagree: I showed that 1 divided by 3 was equal to 0.33333... therefore proving that 0.33333... was equal to 1/3. Besides, how is my "demonstration" informal? I showed by the simplest means (no calculus necessary) that 1/3 is equal to 0.333333... This is merely long division (is long division now considered to be informal?) Also, I referred to a fundamental law of mathematics: If X=Y then Y=X (is THAT considered to be informal?)

If person "A" doesn't understand how "X" can be equal to "Y", and I show to them the "Y" is equal to "X" then I have proven to person "A" that "X" is in fact equal to "Y"

Do I need to re-prove every basic mathematical law and theorem in order to make my point (I don't think so). I think it is a clear fact that 1/3 = 0.33333... and 0.33333... = 1/3. No matter how formal or informal you think that I've shown this to be true, my justification is valid, and therefore it is a proof.


If I put a million dollars in a box, then seal the box. I could tell somebody that there is a million dollars in that box, but they may not believe me. If I then open the box, show them the money and count it out, then I have proven to them that there was a million dollars in the box.

You guys are all over-thinking this. Algr didn't seem to believe something was true, I showed him a way to see that it WAS in fact true, therefore, I proved it to him.

End of story.

 

[arildno]

Division shows you that:

a) There is NO decimal sequence of finite length, say 0.333, that equals 1/3
b) You have NOT shown that the sequence of numbers, 0.3, 0.33, 0.333 converges to anything that could be called "a number".
c) Therefore, due to the crucial lack of b), you have NOT proven that 1/3=0.333... since you haven't proved that the right-hand side is anything meaningful.

 

[zgozvrm]

So what if the division 'continues forever'? That doesn't mean you can't know the result. It doesn't mean the answer isn't a 'real' number. (I can't ever write all the decimals in Pi. So?). The statement doesn't have any relevance or significance, but is exactly what confuses people.

Obviously, the fact that the division continues forever, indefinitely, or infinitely is ABSOLUTELY relevant! If the "3's" stop repeating at some point, then the value will not be exactly 1/3.

I never said that I/we don't know the result, only that it couldn't be written as an definite decimal number (only as a infinitely repeating decimal number, or as a fraction).

No, infinitely, not 'forever'.

Infinitely, forever? Pish-posh! If people don't understand that a number series that repeats infinitely means that it goes on forever, that's their (your) problem.

It's not about what you can practically add up in an amount of time. It's an infinite series, and 'infinite' means infinite in math. Not 'a really big number'. And an summation mark means a sum, not a command to 'add up these numbers'.

I never once mentioned adding or summation, nor have I used the summation mark in any of my posts.

(It) doesn't mean that its value isn't finite, or exactly calculable.

I never said that the value wasn't finite. In fact, it is finite and calculable, it's 1/3. I merely stated that you couldn't come up with an exact (finite, if you will) decimal representation of 1/3, since the "3's" continue infinitely (forever). On the other hand, 1/4 CAN be given as an exact decimal (0.25).

 

[Mark44]

By the way, any number, whether it has recurring decimals or not, can be written as an infinite series in an infinite number of ways, e.g. 1 = \sum^\infty_{n=1}\frac{1}{2n}

The value of that sum is exactly 1. The fact that a human being manually summing up the numbers would never 'reach' 1 doesn't enter into it. You're not making anything clearer by saying that, you're bringing up exactly what gets people mixed up.

Just for the sake of accuracy, your LaTeX script was almost correct. Here is what I'm sure you meant.
1 = \sum^\infty_{n=1}\frac{1}{2^n}

The sum as previously written does not add to 1. In fact, the sequence of partial sums can be shown to be increasing without bound.

 

[Hurkyl]

I merely stated that you couldn't come up with an exact (finite, if you will) decimal representation of 1/3

If you mean finite, then say finite. Words like "exact" and "definite" suggest a meaning that is flat out wrong here.

 

[arildno]

Furthermore, zgozvrm:

You cannot utilize the fact that division SEEMS to yield 0.3333... as an "answer" to ague for that 0.3333... IS a "number".

You might simply be misapplying the process called "division" on an illegitimate object, invoking thereby the well-known GIGO principle.

 

[zgozvrm]

Furthermore, zgozvrm:

You cannot utilize the fact that division SEEMS to yield 0.3333... as an "answer" to ague for that 0.3333... IS a "number".

You might simply be misapplying the process called "division" on an illegitimate object, invoking thereby the well-known GIGO principle.

You guys are killing me!

I never stated that the division SEEMS to yield 0.3333... Rather, I stated that the division DOES yield 0.33333... which is clearly evident by doing the long division.

And, since when is dividing 1 by 3 "illegitimate"?

It is obvious that 1 divided by 3 equals 0.33333... (where "..." represents infinitely repeating "3's"). Therefore, since 1/3 = 0.33333..., then 0.333333... = 1/3.

Let it go.

 

[arildno]

You guys are killing me!

I never stated that the division SEEMS to yield 0.3333... Rather, I stated that the division DOES yield 0.33333...

Indeed you did!
And how do you know that is something meaningful??

which is clearly evident by doing the long division.

And, since when is dividing 1 by 3 "illegitimate"?

How do you know it is legitimate, and indeed, applicable to the particular case 1/3?

 

[zgozvrm]

Indeed you did!
And how do you know that is something meaningful??



How do you know it is legitimate, and indeed, applicable to the particular case 1/3?

Are you for real?

Do the division yourself and see.

<< insult deleted by Mentors >>

 

[arildno]

Are you for real?

Indeed I am.

The point is that you haven't the faintest clue about what a proper mathematical proof consists of.

 

[zgozvrm]

This is not a general proof, this is a specific one.

I'm still not sure why anyone with at least an elementary school level of math cannot understand this:

The original question (as far as I can remember) was basically, "How can 0.333333... be equal to 1/3?"


To paraphrase my answer:

1) "1/3" is a fraction that can be represented by dividing 3 into 1.

2) The result of this division is "0.33333..." where "..." represents
a never-ending (or infinite) series of "3's" (a repeating decimal)

3) If it is true that A=B, then it MUST be true that B=A

4) Therefore, since 1/3 = 0.33333..., then 0.3333... MUST equal 1/3


These 4 statements are TRUE and cannot be disputed. So, therefore, I have proven that 0.333333... = 1/3.

Period.
The end.
Fini.
Au Revoir.
Auf Wiedersehen.

 

[arildno]

1) "1/3" is a fraction that can be represented by dividing 3 into 1.

Not obvious at all.

 

[zgozvrm]

Now, put down the spoon and stop stirring...