Prove 3/3 ≠ 1: Math Puzzle Challenge

[krypto]

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!

 

[CRGreathouse]

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!

Your first mistake is on line 5, where you assume there is a contradiction. There's nothing wrong with 0.9999... = 1.

 

[snoble]

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 (\sum_{j=1}^k 9\times 10^-j )_{k=1}^\infty is equivilant to the sequence (1)_{k=1}^\infry. It is not hard to show the interpolated sequence is Cauchy. And clearly the first one is 0.999... and the second is 1.

 

[LeBrad]

https://www.physicsforums.com/showthread.php?t=65181&highlight=.9999+equals

Search for more on the subject.

 

[krypto]

shhh lol 0.999 recurring isn't 1, just look at it simply

 

[master_coda]

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".

 

[Hurkyl]

Do not make duplicate posts.

 

[CRGreathouse]

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.

 

[eNathan]

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!

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...~ ?


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

 

[shyboy]

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

 

[uart]

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.

 

[shyboy]

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?

 

[master_coda]

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.

 

[arildno]

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).

 

[jcsd]

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).

 

[snoble]

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 \pi beyond the Cauchy sequences that represents \pi. Or rather I should say "Perceived this way there is no \pi beyond the Cauchy sequences that are represented by the symbol \pi."

 

[shyboy]

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.

 

[Hurkyl]

Another important thing to keep in mind is that 0.\bar{9} 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 0.\bar{9} as a number.

 

[moose]

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...

 

[Hurkyl]

You can disbelieve it all you want. Why you would come to a math forum if you reject mathematics?

 

[jcsd]

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...

 

[shmoe]

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.

 

[HallsofIvy]

I don't believe today is Friday- but they made me come to work anyway!

(Apparently, it's part of the mental health plan.)

 

[eNathan]

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?

 

[Icebreaker]

The series \sum_{i=-1}^{-\infty} 9\times 10^i converges to 1.

Most people accept that \frac{1}{3} = 0.333..., but not 1 = 0.999..., for some reason.

 

[Hurkyl]

What boggles me most are the people who refuse to accept 0.\bar{9}=1, yet fully accept 1/2 = 5/10 without even blinking.

 

[krypto]

What boggles me most are the people who refuse to accept 0.\bar{9}=1, 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!

 

[Lonewolf]

Of course 0.999 isn't 1. 0.999... is though.

 

[krypto]

if uit was 1 you would write 1 not 0.999...

 

[Hurkyl]

Just like if it was 1/2, you'd write it as 1/2 and not 5/10?

 

[Lonewolf]

So, if I chose another symbol for 1, its numerical value would change?

 

[krypto]

nope cos it still means 1 even if u named it bob

 

[Lonewolf]

Precisely. So writing 0.999... shouldn't falsify 0.999... = 1, right?

 

[gregmead]

A recurring decimal can be written like:

S = A*r^n

0.999... = 0.9*0.1^n

Sum to infinity of a converging series: \frac {A}{1-r}

So, \frac {0.9}{1-0.1}=\frac {0.9}{0.9} = 1

 

[The Rev]

It's times like this that we need to consider switching to a duodecimal system. In duodecimal 1/3 = 0.4

Much easier.

\sigma

The Rev

 

[Hurkyl]

Well, since nobody seems to be learning anything, I'll close it up.