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

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SUMMARY

The discussion centers on the mathematical concept that the repeating decimal 0.333... is equal to the fraction 1/3. Participants clarify that this equality arises from the definition of decimal representation and the properties of limits in real numbers. The sequence of partial sums, represented as \sum^{\infty}_{i=1}\frac{3}{(10)^i}, converges to 1/3, demonstrating that an infinite series can yield a finite value. The conversation also addresses misconceptions about infinity and the nature of real numbers, emphasizing that 0.333... is a valid representation of a real number.

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  • #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?
 
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  • #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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