Simple mathematical problem (?)

[DanishDynamite]

And after this argumentation I'd need a drink.

I'm already on my second. Skål!

 

[LuxFerum]

I don't have any idea why you keep on bringing that "jump" forward or even what you really mean by it.

k=0;
sum=0;
while sum!=2 do
(
sum=sum+(1/2)^k;
k=k++;
)
end

Thats is our endless sum

k=0;
while k!=1 do
(
)
end

And this is an infinity loop.


Put anyone of then in a perfect computer and the computer will never get out of the loop.

Do you understant that or not?

No, you use the properties of the bloody infinite series. Not your bloody imagination.

Exactly, you have to use that because you know that that is not good enough.
An infinite series will lead you to an aproximation, not to the exactly point.
That is when you say that the series result will tend to answer.
Then you use the properties of the infinite series to find this answer that the sum is tending for.(and will never reach, as we can see with the computer).

Now. let me make it clear. If you still hold that 0.99... is not precisely equal to 1, then you are contradicting every single mathematician who has written about infinite series since after they have been invented.

Now there are exactly two choices:

1. Every single mathematician who has ever worked with real numbers and infinite series is wrong.

2. You are wrong.

There is no third alternative.

I don't have a problem with being wrong.
I have a problem with disseminating ideias that I do not agree.
Therefore 1!=0,999999...

 

[LuxFerum]

but that is the way it is.

That is the way you accept it.
You came here to discuss, or to point the finger and say "you are wrong"?

 

[geni]

Ok can someone explain to me that if 1=0.9999.. why they behave differently if I square them an infinit number of times?

(too early for me to start drinking here)

 

[DanishDynamite]

That is the way you accept it.
You came here to discuss, or to point the finger and say "you are wrong"?

There is nothing to discuss, unless you wish to change the axioms of number theory.

 

[DanishDynamite]

Ok can someone explain to me that if 1=0.9999.. why they behave differently if I square them an infinit number of times?

(too early for me to start drinking here)

Kindly show that they do. Rigorously.

 

[slimshady2357]

Ok can someone explain to me that if 1=0.9999.. why they behave differently if I square them an infinit number of times?

(too early for me to start drinking here)

They don't.

Can you explain why you think they do?

Adam

 

[geni]

Kindly show that they do. Rigorously.

This may take some time
so
1)1 squared an infinit number of times =1 ( if this is not true I think I am going to start drinking)

2)Any number less than 1 squared and infinit number of times tends to zero
3)1-an infinitesimal = less than 1 or 0.9999...

I think the problem is in part 3

 

[slimshady2357]

k=0;
sum=0;
while sum!=2 do
(
sum=sum+(1/2)^k;
k=k++;
)
end

Thats is our endless sum

k=0;
while k!=1 do
(
)
end

And this is an infinity loop.


Put anyone of then in a perfect computer and the computer will never get out of the loop.

Do you understant that or not?

You are still confusing a process with something that is outside of time.

When you have run an infinite amount of time, the computer will stop and 0.999..... will be equal to 1

As I said to Zep, you tell me when an infinite amount of time has passed and I'll show that I'm right

Seriously, why do you think such things should be able to be represented by real world examples?

Please, please, go back and read Cabbage's excellent post as to how numbers are defined.

Otherwise, as DD said, you're talking about changing standard definitions within mathematics.

Also, if you're still stuck, perhaps you could read up on Hilbert and his finite math work. Maybe you could take up from where they are now (Hilbert's followers).

Adam

 

[LW]

I have a problem with disseminating ideias that I do not agree.

Well, I have to agree that it certainly is your problem and no one else's if you don't agree with how mathematics works and how it has worked for at least 300 years.

 

[boooeee]

Originally Posted by LuxFerum
An infinite series will lead you to an aproximation, not to the exactly point.
That is when you say that the series result will tend to answer.

Po - TAY - to

Originally Posted by DanishDynamite
There are equal signs all the way through. Try to understand that the expression "0.999..." is simply another way of writing the expression "0.9 + 0.09 + 0.009 + ..." These expressions are EQUAL. And the latter expression is EQUAL to 1 as the sum of a geometric series shows. EQUAL. There is no "tends" involved.

Po - TAH - to

Okay, this thread has devolved into a semantic argument.

The Po - TAH - to camp says 0.999(recurring) = 1.

The Po - TAY - to camp says it only tends to 1, but does not equal 1.

I'm firmly with the Po - TAH - to's here. For every conceivable intent and purpose, 0.999(recurring) = 1. Adding qualifiers like "it only tends to one", or "it never reaches one" only obfuscates the issue.

Zep, LuxFerum, and the rest of the Po - TAY - to's: If you stand by your assertion that 0.999(recurring) does not equal one, then you must also deny the following mathematical statements:

1 + (1/2) + (1/2)^2 + (1/2)^3 + ..... = 2
1 + (1/2)^2 + (1/3)^2 + (1/4)^2 + .... = (pi^2)/6

And so on and so on....

 

[geni]

Are you saying that 0.0000...1=0

( it all used to be so simple sop)

 

[xouper]

While I was compising this long reply, more stuff got posted. So I'll just post this and address the subsequent points in another reply.

Zep: "Xouper, perhaps a question to make this clear in my mind. How far away from 1 must a number be for it NOT to be 1? What is that delta that "makes the difference"?"

angard: Hi, I think he will answer more than 0.

That's exactly what I would answer. If the delta is greater than 0, then the number is no longer 1.

Colloden: I’m still unconvinced that 0.9999etc. is 1. After four pages I’m none the wiser.

May I suggest that you would be wise to consider the advice from math experts and dismiss the nonsense spouted by LuxFerum, Zep, and Suggestology.

Suggestologist: Well then you may as well get .888888888..... as a result of your division. Since 1 into 10 is 8 remainder 2. And then 1 into 20 is 8 remainder 12. And then 1 into 120 is 8 remainder 112. Etc.

You would need to prove that 0.888.... is a valid limit of using your method. Since that is not possible with a diverging remainder, your proposed result is meaningless. Also, 0.888... is exactly equal to 8/9, so once again, your nonsense is shown for what it is.

Walter Wayne: a/9 = 0.aaa... for any a which is a single digit

Suggestologist: It doesn't matter what your equations seem to show. They are obviously not generalizable because a counterexample exists.

Walt is quit correct. You have shown no counterexample. There is no counter example. Walt's explanation is airtight. (It also applies in any base, not just base ten.)

LuxFerum:

1-0,9         =0,1
1-0,99        =0,01
1-0,999       =0,001
1-0,9999      =0,0001
1-0,99999     =0,00001
1-0,999999    =0,000001

when exactly the result will be zero??
never!!!!

Quite true. But this is not the meaning of the notation 0.999.... in which all the nines are already there. The notation 0.999... is not a process where you tack on another nine and never quite get them all tacked on. You will get nowhere with mathematicians if you try to redefine the meaning of the notation 0.999... as you are trying to do here.

This is the same fundamental error made by Zeno. If you try to sum the infinte series one term at a time, you will fail. However, if you use the geometric series theorem (which has been rigorously proven and is airtight), you can get an exact answer. If you do not believe this, then you need to study some more math and bow out of the argument here.

Suggestologist: Yeah, and the number "infinity" is not a process? Anytime you deal with infinity of anything, decimals or steps in a limit, or whatever -- it IS a process that theoretically does not stop.

Wrong, wrong, and wrong. Please stick to the standard math definitions please and quit trying to redefine them. One exception, approaching a limit (using a smaller and smaller epsilon-delta) is a process. The limit itself is not. And the number 0.999... is not a process, it is an exact rational number.

LuxFerum: 0,9999... is not 1
0,3333... is not 1/3
no matter how close you get.
The difference tend to zero.
The limit is the point where they tend to go, but they really never get there.

You are wrong.

1/3 is exactly equal to 0.333...

That is the definition of the notation used here. There is no limit or process inherent in this defintion.

If you have a computer with and infinity precision and infinity power of calculation, and you put him to add "0.9 + 0.09 + 0.009 + ..." and only stop when the result is 1,he will never come out. this sum will never reach the "1" point.

Your observation is correct, but this is not how the number 0.999.... is defined. So your objection is moot.

It is like the runner and the turtle problem, if you stick with the infinty sum, the runner will never transpass the turtle, But with a different analyse you can see that he will.

Now we are getting somewhere. You recognize the flaw in Zeno's Paradox. In the same way that the runner does in fact catch the turtle, 0.999.... is exactly equal to 1.

Perhaps the reason you do not see this is because you are trying to use a flawed method of analysis (using Zeno's process).

LuxFerum: the equation is right, your interpretation of it is wrong, imho.

The equation is right and so is the interpretation by LW. Your humble opinion is misinformed.

LW: Tell me, if I shoot an arrow at you, will it reach you in a finite time? [Supposing that I first travel near to you].

LuxFerum: yes it will. and that is where you missed the point.

No. That is exactly the point. You are the one missing the point.

that is when you change the analise from and endless sum to a linear movement.
The endless sum is a good tool but incomplete.
you need that jump from one analise to another to make it pratical.
When you can see where that sum is going, that is the time when you leave the loop.
And that is why the computer will never leave the loop.
The computer will not make that assumption.
And will never reach the end of that sum.

How many times do we need to tell you that the process you are trying to use is not valid for the question at hand? Just because a computer cannot ever get there has no bearing on the definition of the notation 0.999...

As Cabbage and LW have already pointed out, your computer algorithm is not how the number 0.999... is defined. Obviously you are not getting this point. How else can we make this clear to you?

DanishDynamite: For those still in doubt that 0.999... = 1, allow me to cut and paste xouper's proof from the link provided by boooeee on page 1 of this thread:

Thank you for repeating that here. I am still waiting for one of the challengers to show any flaw in that proof.

DanishDynamite: I give up. Mathematically, when considering real numbers, 1 and 0.999... are equal as has been proven. You can wax philosophical about whether this seems reasonable or not, but that is the way it is. "You can lead a horse to water, but you can't make it drink".

Good point.

I have only one piece of advice to anyone lurking this thread and is trying to figure out who to believe. Go ask a mathematician you trust or study some mumber theory from a reputable book. Don't listen to the nonsense being posted here by LuxFerum, Zep, and Suggestologist.


Edited to fix a typo. No content was changed.

 

[LW]

This may take some time
so
1)1 squared an infinit number of times =1 ( if this is not true I think I am going to start drinking)

2)Any number less than 1 squared and infinit number of times tends to zero
3)1-an infinitesimal = less than 1 or 0.9999...

I think the problem is in part 3

Now, this will be my last post on this thread today. I have the feeling that I'll be working to the last bus in anyway and wasting time in trying to teach basic mathematics will not increase the change that I will catch an earlier buss.

Anyway 1^2 = 1. (here we agree)

If you square the infinite series

sum ( 0.9 * (1/10)^k), 0 <= k < infinity,

you get:

sum( 0.81 * (k+1) * x^k), 0 <= k < infinity.

(I skipped here an intermediate step since it is not possible to have clear notation for it without sigmas).

You can take the 0.81 out to get:

0.81 * sum((k +1) * x^k)

Next, divide this into two cases:

A = 0,81 * (sum(k * x^k) + sum(x^k))

Going to the trusty R&W we find the formula.

sum(k^m x^k) = (1-x)^(-m-1) * sum( a_{j,m} x^j) where 1 <= j <= m

Here, m = 1 so this simplifies to: [forgot to add: a_{1,1} = 1 for all such series]

(1-x)^(-2) * x

Plugging this back to A above we get:

A = 0.81 ( 100/81 * 1/10 + (10 / 9))
= 0.81 * (10 / 81 + 90 / 81)
= 0.81 * 100 / 81 = 81 / 81 = 1

So, (0.99..)^2 = 1.

 

[DanishDynamite]

This may take some time
so
1)1 squared an infinit number of times =1 ( if this is not true I think I am going to start drinking)

2)Any number less than 1 squared and infinit number of times tends to zero
3)1-an infinitesimal = less than 1 or 0.9999...

I think the problem is in part 3

The problem is somewhere between 2) and 3). You assume that 0.999... is less than 1.

 

[slimshady2357]

Po - TAY - to

Po - TAH - to

Okay, this thread has devolved into a semantic argument.

The Po - TAH - to camp says 0.999(recurring) = 1.

The Po - TAY - to camp says it only tends to 1, but does not equal 1.

I'm firmly with the Po - TAH - to's here. For every conceivable intent and purpose, 0.999(recurring) = 1. Adding qualifiers like "it only tends to one", or "it never reaches one" only obfuscates the issue.

Zep, LuxFerum, and the rest of the Po - TAY - to's: If you stand by your assertion that 0.999(recurring) does not equal one, then you must also deny the following mathematical statements:

1 + (1/2) + (1/2)^2 + (1/2)^3 + ..... = 2
1 + (1/2)^2 + (1/3)^2 + (1/4)^2 + .... = (pi^2)/6

And so on and so on....

Yes, as xouper pointed out in the other thread (DD quoted it here) and as LW has shown in this thread again, they would have to throw out all sums of infinite series... which would lead to a denial of all kinds of things....

Adam

 

[xouper]

LuxFerum: That is the way you accept it. You came here to discuss, or to point the finger and say "you are wrong"?

DanishDynamite: There is nothing to discuss, unless you wish to change the axioms of number theory.

Agreed.

I have to ask the obvious question here to LuxFerum, Zep and Suggestologist. What is your objective for participating in this thread? Are you just here to post your mistaken notions, or are you interested in learning why you are wrong?

 

[geni]

Now, this will be my last post on this thread today. I have the feeling that I'll be working to the last bus in anyway and wasting time in trying to teach basic mathematics will not increase the change that I will catch an earlier buss.

Anyway 1^2 = 1. (here we agree)

If you square the infinite series

sum ( 0.9 * (1/10)^k), 0 <= k < infinity,

you get:

sum( 0.81 * (k+1) * x^k), 0 <= k < infinity.

(I skipped here an intermediate step since it is not possible to have clear notation for it without sigmas).

You can take the 0.81 out to get:

0.81 * sum((k +1) * x^k)

Next, divide this into two cases:

A 0,81 * (sum(k * x^k) + sum(x^k))

Going to the trusty R&W we find the formula.

sum(k^m x^k) = (1-x)^(-m-1) * sum( a_{j,m} x^j) where 1 <= j <= m

Here, m = 1 so this simplifies to:

(1-x)^(-2) * x

Plugging this back to A above we get:

A = 0.81 ( 100/81 * 1/10 + (10 / 9))
= 0.81 * (10 / 81 + 90 / 81)
= 0.81 * 100 / 81 = 81 / 81 = 1

So, (0.99..)^2 = 1.

Ok I belive you (The bits I understand cheak out and the bits that I don't seem logical)

This leavs me with two queations:
1)does 0.0000.....1=0?
2)and how close to one can a number get? (ie does the proof hold true for 0.9999....8)

 

[xouper]

boooeee: Zep, LuxFerum, and the rest of the Po - TAY - to's: If you stand by your assertion that 0.999(recurring) does not equal one, then you must also deny the following mathematical statements:

1 + (1/2) + (1/2)^2 + (1/2)^3 + ..... = 2
1 + (1/2)^2 + (1/3)^2 + (1/4)^2 + .... = (pi^2)/6

And so on and so on....

slimshady2357: Yes, as xouper pointed out in the other thread (DD quoted it here) and as LW has shown in this thread again, they would have to throw out all sums of infinite series... which would lead to a denial of all kinds of things....

Agreed. Well said, boooeee and slim...

 

[xouper]

geni: This leavs me with two queations:
1)does 0.0000.....1=0?
2)and how close to one can a number get? (ie does the proof hold true for 0.9999....8)

Neither of those (0.000...1 and 0.999...8) are real numbers according to any of the accepted definitions of real numbers, so it is not meaningful to ask if a real number is equal to one of those things you asked about.