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

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AI Thread Summary
The discussion revolves around the mathematical concept of representing the repeating decimal 0.333... as the fraction 1/3. Participants argue about the implications of infinite sequences, limits, and the nature of infinity in mathematics. Key points include the assertion that 0.333... can be derived from the sum of the series 3/(10^i), which converges to 1/3, and the distinction between finite and infinite processes in mathematics. The conversation also touches on philosophical implications of infinity and the completeness property of real numbers, emphasizing that 0.333... equals 1/3 under standard mathematical definitions. Ultimately, the discussion highlights the complexities of understanding infinite decimals and their representation in real numbers.
  • #51
alxm said:
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.
 
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  • #52
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.
 
  • #53
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.
 
  • #54
arildno said:
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.
 
  • #55
zgozvrm said:
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?
 
  • #56
arildno said:
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 >>
 
Last edited by a moderator:
  • #57
zgozvrm said:
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.
 
  • #58
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.
 
  • #59
1) "1/3" is a fraction that can be represented by dividing 3 into 1.
Not obvious at all.
 
  • #60
Now, put down the spoon and stop stirring...
 
  • #61
zgozvrm said:
This discussion is beyond converting fractions to decimals and vice-versa, so if that is beyond your level of math, then so is this discussion.

Maybe you could prove to us all why the division algorithm necessarily works?

Why it cannot produce garbage as long as the divisor is non-zero?
 
  • #62
zgozvrm said:
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.[/b]

I think you are UNDER-thinking things, Zgozvrm. You can't assume that a proof is correct simply because it gives you the answer you want. That is circular logic.

Edit:
In your latest proof, I don't have a problem with step 1, but in step 2, there is no final result of the division. You simply DECLARE the result to be ".333..." as one would declare a variable. That doesn't prove anything about what it means for a decimal to repeat infinitely.
 
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  • #63
I think you are UNDER-thinking things, Zgozvrm. You can't assume that a proof is correct simply because it gives you the answer you want. That is circular logic
Indeed, Algr!

"Plug&chug"-mentalities confuse their ability to churn something out of a machine with what is required as proof.
 
  • #64
Algr said:
Edit:
In your latest proof, I don't have a problem with step 1, but in step 2, there is no final result of the division. The "..." simply represents a failure to complete a process that can never be completed.

Why is representability (or lack of such) obvious?

I just stopped at the first hurdle, so I haven't reached the other three yet. :smile:
 
  • #65
Wow! Apparently you guys don't believe the basic laws of math nor that division is a valid "algorithm" with non-zero numbers.

I shouldn't have to re-invent the wheel to make a point. If I told you that 2+5=7, we all know this is true, and I shouldn't have to prove it to anyone in a discussion that is beyond that level of math.

The assumption was made all along that people know how to divide and "convert" a fraction to a decimal.

I cannot help either of you if you are not willing to accept or understand the basic laws of math, nor am I willing to try. That is way beyond the scope of this discussion.


Any further posts along these lines will not be answered and/or acknowledged by me.
 
  • #66
zgozvrm said:
Wow! Apparently you guys don't believe the basic laws of math nor that division is a valid "algorithm" with non-zero numbers.
No, we don't.

Rather, we strive to construct consistent, mathematical systems, rather than rely upon "received wisdom" as some sort of oracle.
 

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