Prove 3/3 ≠ 1: Math Puzzle Challenge

  • Context: High School 
  • Thread starter Thread starter krypto
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Discussion Overview

The discussion revolves around the mathematical assertion that 3/3 is equal to 1, with participants exploring the implications of decimal representations, particularly the recurring decimal 0.999... and its relationship to 1. The scope includes mathematical reasoning, conceptual clarification, and debate over the nature of infinite series and convergence.

Discussion Character

  • Debate/contested
  • Mathematical reasoning
  • Conceptual clarification

Main Points Raised

  • Some participants assert that 3/3 equals 1, but argue that 0.999... is an alternative representation of 1, leading to the conclusion that 3/3 does not equal 1.
  • Others counter that 0.999... is indeed equal to 1, citing mathematical principles and the concept of Cauchy sequences.
  • One participant emphasizes that an infinite series converges to its limit, suggesting that 0.999... equals 1 due to the nature of convergence in mathematics.
  • There are claims that the terms "convergence" and "member" apply to sequences and not to numbers themselves, indicating a distinction in how numbers are defined.
  • Some participants express skepticism about the equality of 0.999... and 1, suggesting that proofs supporting this claim contain contradictions.
  • Discussions include the idea that different decimal representations can lead to confusion, with references to how numbers are constructed and defined in mathematics.

Areas of Agreement / Disagreement

Participants generally disagree on the equality of 0.999... and 1, with multiple competing views presented. Some assert the equality while others reject it, leading to an unresolved debate.

Contextual Notes

Limitations include varying interpretations of mathematical definitions, the nature of infinite series, and the implications of decimal representations. The discussion reflects differing levels of understanding and acceptance of mathematical concepts.

krypto
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Everyone knows that 3/3 is equal to 1...
but 1/3 is equal to 0.3333333 recurring...
this means that 2/3 is 0.6666666 recurring...
and therefore 3/3 is 0.99999999 recurring...
So therefore 3/3 does not equal 1!
Id like to try and see you prove all this wrong! :devil:
 
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krypto said:
Everyone knows that 3/3 is equal to 1...
but 1/3 is equal to 0.3333333 recurring...
this means that 2/3 is 0.6666666 recurring...
and therefore 3/3 is 0.99999999 recurring...
So therefore 3/3 does not equal 1!
Id like to try and see you prove all this wrong! :devil:

Your first mistake is on line 5, where you assume there is a contradiction. There's nothing wrong with 0.9999... = 1.
 
This is a classic dilema. In fact what you've shown that 1 actually has two decimal representations. ie 0.9999...=1. The reals are actually just Cauchy sequences of rational numbers modded out by the equivilancy that two sequences are equivilant if the interpolated sequence is still Cauchy. So the claim is the sequences [tex](\sum_{j=1}^k 9\times 10^-j )_{k=1}^\infty[/tex] is equivilant to the sequence [tex](1)_{k=1}^\infry[/tex]. It is not hard to show the interpolated sequence is Cauchy. And clearly the first one is 0.999... and the second is 1.
 
shhh lol 0.999 recurring isn't 1, just look at it simply
 
krypto said:
shhh lol 0.999 recurring isn't 1, just look at it simply

I assume "like at it simply" really means "look at it without bothering to learn how decimal notation works". :rolleyes:
 
Do not make duplicate posts.
 
krypto said:
shhh lol 0.999 recurring isn't 1, just look at it simply

You mean that the two don't look the same? Well, sure. 1+1=2, but the string "1+1" isn't the same as the string "2". They just happen to have the same numeric value.
 
krypto said:
Everyone knows that 3/3 is equal to 1...
but 1/3 is equal to 0.3333333 recurring...
this means that 2/3 is 0.6666666 recurring...
and therefore 3/3 is 0.99999999 recurring...
So therefore 3/3 does not equal 1!
Id like to try and see you prove all this wrong! :devil:

This has been discussed so many times here. Hmn, do you know what the mathematical keyword called "difference" means? It means subtraction. So tell me, what is the difference from 1.0 and 0.999...~ ? :eek:


Hmn, you have two options. Either 0 or .000...~1
The second answer is wrong. .999... = 1
This concept does not violate common sense at all. Its a mathematical thing. Were you aware that 5/10 is the same as 1/2? Hmn, does that violate your logic too?

Settled :smile:
 
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  • #10
strictly speaking 1/3 is not equal 0.33333...
what do we have here is an infinite series 0.3, 0.33, 0.333, 0.33333...
this series converges to 1/3.
At the same time there is an infinite number of another series, i.e.
0.4,0.2, 0.34,0.32,0.334,0.332... which converges to the same number.

So 0.9999... is actualy a representation of series 0.9, 0.99, 0.999... which converges
to the 1. So, roughly speaking, 1=0.99999... if we keep in mind that on the left side we have not the number, but a set of numbers.

In the same way we can write: 1=1.1, 0.9, 1.01, 0.91, 1.001,0.991... But
 
  • #11
shyboy said:
strictly speaking 1/3 is not equal 0.33333...
what do we have here is an infinite series 0.3, 0.33, 0.333, 0.33333...
this series converges to 1/3.
At the same time there is an infinite number of another series, i.e.
0.4,0.2, 0.34,0.32,0.334,0.332... which converges to the same number.

So 0.9999... is actualy a representation of series 0.9, 0.99, 0.999... which converges
to the 1. So, roughly speaking, 1=0.99999... if we keep in mind that on the left side we have not the number, but a set of numbers.

In the same way we can write: 1=1.1, 0.9, 1.01, 0.91, 1.001,0.991... But

Man I get sick of these threads but here goes anyway. Shyboy an infinite series does NOT approach it's limit point it EQUALS it. A finite series of length n approaches the limit as n approaches infinity but a convergant infinite series is (let me repeat) EQUAL to the limit.
 
  • #12
Man I get sick of these threads but here goes anyway. Shyboy an infinite series does NOT approach it's limit point it EQUALS it. A finite series of length n approaches the limit as n approaches infinity but a convergant infinite series is (let me repeat) EQUAL to the limit.

you may be right, but why all of these mathematicians use the term convergence?
 
  • #13
shyboy said:
you may be right, but why all of these mathematicians use the term convergence?

Why wouldn't they use the word convergence? Convergence isn't a reserved word that you're only allowed to apply to sequences.
 
  • #14
shyboy:
We say that the sequence of finite partial sums converges to some limit.
An infinite series is typically defined as that limit (i.e, another name for it).
 
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  • #15
shyboy said:
strictly speaking 1/3 is not equal 0.33333...
what do we have here is an infinite series 0.3, 0.33, 0.333, 0.33333...
this series converges to 1/3.
At the same time there is an infinite number of another series, i.e.
0.4,0.2, 0.34,0.32,0.334,0.332... which converges to the same number.

So 0.9999... is actualy a representation of series 0.9, 0.99, 0.999... which converges
to the 1. So, roughly speaking, 1=0.99999... if we keep in mind that on the left side we have not the number, but a set of numbers.

In the same way we can write: 1=1.1, 0.9, 1.01, 0.91, 1.001,0.991... But

Right, infinite decimal expansions are defined (I can't stress this too much as it really is only a matter of how soemone choose to define something), as the value that the sequence of partial sums of the related decimal series converges to.

To be fair infinite sequenecs are sequences of numbers not numbers themselves, but if the are convergent thye do have a partocualr number associated with them (i.e. the value they converge to).
 
  • #16
It's all a matter of definition really. As I said before a standard way to build the reals is just the completion of the rationals under the standard norm. If you think of the rationals this way then every number is just a congruency class of Cauchy sequences. Even non-infinite decimal representations. Even numbers in radical form. Perceived this way there is no [tex]\pi[/tex] beyond the Cauchy sequences that represents [tex]\pi[/tex]. Or rather I should say "Perceived this way there is no [tex]\pi[/tex] beyond the Cauchy sequences that are represented by the symbol [tex]\pi[/tex]."
 
  • #17
I guess that a congruency class of Cauchy sequences is not what you can find in a high school texbook, but the question itself is within high-school range.
 
  • #18
Another important thing to keep in mind is that [itex]0.\bar{9}[/itex] is a number. Words like "convergence" or "member" don't apply to numbers.

We might build numbers, or represent numbers, with objects for which convergence and membership do mean something, but those words only apply to those objects, not to [itex]0.\bar{9}[/itex] as a number.
 
  • #19
since when is 1/3=.333333...its the same as
.99999...=1, which i don't believe is true...
I don't care what kinda proofs might even show that it is, it just shows that something contradicts in those proofs...
 
  • #20
You can disbelieve it all you want. Why you would come to a math forum if you reject mathematics?
 
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  • #21
Another case of jcsd's theorum: If a bulletin board exist for any length of time, someone will claim that 0.999.. is not equal to 1 on it.

Not that no-one's ever claimed that 0.999.. isn't equal to 1 on PF before...
 
  • #22
moose said:
I don't care what kinda proofs might even show that it is, it just shows that something contradicts in those proofs...

Don't discard the possibility that you just don't understand the proofs and explanations offered.
 
  • #23
I don't believe today is Friday- but they made me come to work anyway!

(Apparently, it's part of the mental health plan.)
 
  • #24
moose said:
since when is 1/3=.333333...its the same as
.99999...=1, which i don't believe is true...
I don't care what kinda proofs might even show that it is, it just shows that something contradicts in those proofs...

Can you back up your theory at all?

hmn, let's see my theory...

1 - .999... = 0;

Ah, there is no mathematical difference, now do you see?
 
  • #25
The series [tex]\sum_{i=-1}^{-\infty} 9\times 10^i[/tex] converges to 1.

Most people accept that [tex]\frac{1}{3} = 0.333...[/tex], but not [tex]1 = 0.999...[/tex], for some reason.
 
  • #26
What boggles me most are the people who refuse to accept [itex]0.\bar{9}=1[/itex], yet fully accept 1/2 = 5/10 without even blinking.
 
  • #27
Hurkyl said:
What boggles me most are the people who refuse to accept [itex]0.\bar{9}=1[/itex], yet fully accept 1/2 = 5/10 without even blinking.
yeah cos 5/10 is a half like 1/2 lol 0.999 isn't 1!
 
  • #28
Of course 0.999 isn't 1. 0.999... is though.
 
  • #29
if uit was 1 you would write 1 not 0.999...
 
  • #30
Just like if it was 1/2, you'd write it as 1/2 and not 5/10?
 

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