Simple mathematical problem (?)

[DanishDynamite]

xouper:

Agreed.

If only you could be as reasonable in gun control debates.

 

[Mendor]

Are you saying that 0.0000...1=0

( it all used to be so simple sop)

0.0000...1 is meaningless, because the ... means an infinite number of zeroes, and you can't put a 1 on the end of an infinite number of zeroes. So what you actually have is 0.0000... . Which does equal 0.

 

[geni]

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.

I did wonder if the answer was along thoes lines.
Since thoes were the only two cracks left that I could see I think the proof holds. ( I think I'm going to have to change my sig again)
Purely out of interest whay are they not real? Under the definitions I was taught a A level they would be.

 

[xouper]

DanishDynamite: xouper: If only you could be as reasonable in gun control debates.

I'm doing the best I can considering that we do not agree on the initial axioms in the gun control issue.

 

[T'ai Chi]

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!!!!

What don't you understand about "for any e>0" ?

 

[xouper]

xouper: Neither of those (0.000...1 and 0.999...8) are real numbers according to any of the accepted definitions of real numbers,

geni: Purely out of interest whay are they not real?

That's not a bad question. Here's a hint.

Is the following a real number?

(That's a power of ten with infinity as the exponent.)

I put the rest of the answer in the spoiler box in case you wanted to puzzle it out first using the above hint.

<table cellpadding="4" cellspacing="1" border="0" bgcolor="#666699" align="center" width="80%"><tr><td bgcolor="#666699"><font size="1" face="Arial, Helvetica, sans-serif" color="#FFFFFF">Spoiler:</font></td></tr><tr><td bgcolor="white"><font size="2" face="Arial, Helvetica, sans-serif" color="white">
The correct answer is no, that's not a real number.

Since infinity is not a real number, niether is any number raised to the power of infinity.

Therefore the reciprocal, one divided by that "number", is also not a real number.

Therefore 0.000...1 is not a real number.

Does that help?

</font></td></tr></table>

 

[angard]

"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
"

Hmm, I have, like some others before me in this thread, poured a ( In this case a whiskey. ) glass.

I'm not a math guru. I like numbers for what they are, numbers. Some say numbers are subjective. I say no, numbers are what they are, letters.( People are subjective.) Letters for an alphabet. An alphabet to speak a language that is the same for all who agree to the rules which is itself defined by (very basic) axioms.
If you know the language, it can't be missinterpretated.( Counter to english/swedish and such)

1+1=2 or what? 1+1=1? Its all in the definition of the rules.

To the point: The quote above. Its all good. Unfortunatly, there is no number 1 in 1-0.9999... where ... is 9:s in infinity(The 1 appears at the last instance of 9, which of there is none.).

People TEND (overly used word in this math tread.) to think that its about the NUMBER off nines in the 0.9999999999....that defines the "exactness" of the number. You are right in asuming just that, if not its an infinite number.


This whole thread reminds me of when I was in (I guesstimate) around my third school year(Third grade?). My friend and I got into an argument regarding 1/3 vs 0.3333333...I can't remember exactly, but I think it was about some answer to a math test,, where I had answered in decimal and he in pure 1/x. I said that 0.3333 into infinity must be the same as your 1/3 and he said no. Mine is more "exact" than yours. The thing I agreed to is that his number(notation) is more "clean" than mine, but otherwise no different. I still hold to that.

In closing, sorry for the bad english. Its part of i'm being swedish, though most beingintoxinated. I
like numbers, what can I say.

ps. Yes. A 33cl coke is 33cl. Not 1/3 of a liter.


EDIT: Yes, I understand your "resistance" to why everyone is telling you that 33cl coke is 1/3 of a liter, because they are not telling you that.

EDIT2: yes, I'm up to 10 posts. I hope youre not holding that to my disadvantage. I TEND to post only when I read something that I can/must reply to, hehe. Actually I agree with Suggestologist regarding education and teachers teach only whats "appropriate" for your lvl. I would have aprecciated if I had learned the true basics about derivates( Is it called derivates?) instead of some textbook "shortcuts".

 

[Zep]

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?

Both.

Herewith a formal theorem for the sum of a geometric series, from Calculus and Analytical Geometry, George B Thomas Jr, MIT, Addison-Wesley Publishing, 1975 - p624:

Theorem:

If |r| < 1, then the geometric series

a + ar + ar^2 + ... ar^(n-1) + ...

converges to the sum a/(1-r). If |r| >= 1, the series diverges unless a = 0. If a = 0, the series converges to the sum 0.

My highlighting.

The wording, please note, is "converges" and "diverges," and is specifically NOT "is equal to".

So may we now cease with the personal attacks on our respective mathematical knowledges? I may be old and rusty, but I did graduate too.

 

[slimshady2357]

Both.

Herewith a formal theorem for the sum of a geometric series, from Calculus and Analytical Geometry, George B Thomas Jr, MIT, Addison-Wesley Publishing, 1975 - p624:

My highlighting.

The wording, please note, is "converges" and "diverges," and is specifically NOT "is equal to".

So may we now cease with the personal attacks on our respective mathematical knowledges? I may be old and rusty, but I did graduate too.

Have you read Cabbage's post near the end of the 4th page? The long one that talks about how decimal notation is defined?

0.999... is defined to be EQUAL TO the limit of it's expansion.

0.999... is actually equal to the limit of the series. It's EQUAL to the value the series converges to, that is, 1.

What do you think of Cabbage's post? Where do you disagree with it?

Just asking, no ill will or insulting tone intending

Adam

 

[xouper]

Zep:

Theorem:

If |r| < 1, then the geometric series

a + ar + ar^2 + ... ar^(n-1) + ...

converges to the sum a/(1-r). If |r| >= 1, the series diverges unless a = 0. If a = 0, the series converges to the sum 0.

My highlighting.

The wording, please note, is "converges" and "diverges," and is specifically NOT "is equal to".

Sorry, but you have misinterpreted what the book says. The terms "converging" and "diverging" describe the behavior of the series, not the sum. The theorem says the sum is exactly equal to the series if it is a converging series. I just looked in one of my old calc texts (1995) and it specifically says a coverging geometric series equals the sum a/(1-r).

The whole point of the theorem is to equate an exact sum with a converging geometric series. In your textbook, you should find specific examples where the series is set equal to the sum, using the theorem as justification for that equality. Right?

The point is, this theorem is used to prove that 0.999... is exactly equal to one. The series represented by the notation 0.999... is a convergent series and this characteristic is what makes the theorem applicable.

I apologoze for calling you names. My frustration at seeing the overwhelming amount of crap and bad information in this thread ticked me off, but that's no excuse. I apologize for behaving uncivilly and I will try not to do it again (if I do, please call me on it ). And that apology extends to eveyone here.

 

[xouper]

Let me try another angle.

I know of no mathematician who thinks 0.999... is not exactly equal to one. Why do you suppose that is? Have they all lost their marbles?

I know it is not valid to use an argumentum ad populum, but I offer it only as a hint that just maybe those of you who think 0.999.... is not equal to one, might wanna take a look at where you may be going wrong.

When most everyone else says the world is round and offers sound evidence of it, and you are still saying it's flat, perhaps it's time to rethink your position.

 

[Zep]

Have you read Cabbage's post near the end of the 4th page? The long one that talks about how decimal notation is defined?

0.999... is defined to be EQUAL TO the limit of it's expansion.

0.999... is actually equal to the limit of the series. It's EQUAL to the value the series converges to, that is, 1.

What do you think of Cabbage's post? Where do you disagree with it?

Just asking, no ill will or insulting tone intending

Adam

I would respectfully disagree with Cabbage, and submit the following quote from the same reference, ibid, p621 (Chapter 18 - Infinite Series):

Limit of a sequence

A sequence {(n, an)} may have a different value an for each different value n. But it may also happen that as n increases, the different an's tend to cluster around some fixed number L. If there is such a number L, with the property that |L - an| is arbitrarily small for all sufficiently large values of the index n, we say that an converges to L as limit, and write

lim an = L
n-> infinity

By the phrase "|L-an| is arbitrarily small for all sufficiently large values of n," we mean that to any positive number E there corresponds an index N such that

|L - an| < E for all n > N

That is, all terms after the Nth lie within the distance E from L. If no such limit exists, then we say that the given sequence diverges.

So this would suggest to me that the limit of the sequence 0.9 + 0.09 +... would indeed converge on 1, and the formula derived previously is used to calculate what this limit value is. However, it also indicates that this is a convergence towards a limit value, not an equality.

 

[DangerousBeliefs]

Just a quick note to say that I went in search of a better proof that 1 and .999(recurring) are the same.

Found it.

Moving on with life...... (suggest you do the same)

 

[ceptimus]

I'm happy with the definition. There are two ways of writing any number that can be written (conventionally) using a finite number of digits. If we take seventeen-and-an-eighth as an example, we can write either:

17.125 followed by an infinite string of zeros, which are by convention omitted. or

17.124 followed by an infinite string of nines.

Irrational and transcendental numbers can't be written down accurately at all. All the counting numbers (integers) can be written in two ways. Zero however, can only be represented in one way.

Remember, decimal notation is just a human invention. It is an imperfect way of representing numbers. Why all the fuss?

 

[xouper]

Zep: So this would suggest to me that the limit of the sequence 0.9 + 0.09 +... would indeed converge on 1, and the formula derived previously is used to calculate what this limit value is. However, it also indicates that this is a convergence towards a limit value, not an equality.

Please do not confuse the sequence with the series.

The infinite sequence R = {.9, .99, .999, .999, ...} converges on a limit.

The infinite series S = .9 + .09 + .009 + .0009 + ... is exactly equal to one.

In other words, the series S is the limit of the sequence R.

lim R = S.

1 = 0.999...

 

[Suggestologist]

For starters, "infinity" is not a number. There are actually an infinite number of different orders of infinity.

Which one are you using to take a limit to "infinity"? Is it omega, or aleph_naught?

 

[UKBoy1977]

Everyone read Earthborn's posts again she(?) actually knows what she is talking about!

The original proof (in the first post) is perfectly convincing to me (ignoring the typo in it)

If you think 0.9999.... is not the same as 1 then I am assuming you believe it to be less than 1. In which case if it is less than 1 then you must be able to find another number between 0.9999... and 1. If anyone can provide such a number I would be interested to see it. If not we will have to accept they are one and the same.

 

[aerosolben]

For those who think 0.9~ != 1:

The problem seems to be that you want to treat infinity as a really big number. This is not the case. Infinity is in a whole different ballpark from really big numbers.

re: Zep's string cutting example. You cannot cut a string into 90%/10% pieces an infinite number of times. This is impossible. You cannot do it, you cannot construct a machine to do it or even simulate doing it. You must accept that infinity is not a number, but a concept. You must take the limit of the series for the use of infinity to have any meaning whatsoever in real numbers.

On the surreal "iota" being an infinitely small number. This is dumb. An iota does not exist, except perhaps in some bizarre number system. If you think it does, I give you:

iota/2.

Eat my shorts.

 

[xouper]

Suggestologist: Which one are you using to take a limit to "infinity"? Is it omega, or aleph_naught?

Aleph_naught, since the cardinality of the infinite sequence { .9, .99, .999, ... } is the same as the cardinality of the natural numbers.

 

[Zep]

Please do not confuse the sequence with the series.

The infinite sequence R = {.9, .99, .999, .999, ...} converges on a limit.

The infinite series S = .9 + .09 + .009 + .0009 + ... is exactly equal to one.

In other words, the series S is the limit of the sequence R.

lim R = S.

1 = 0.999...

I haven't confused them, but I will simply quote more of my textbook that says,

"If sn [partial sum] converges to a limit S as n -> infinity: lim(n->inf) sn = S, then we say that the series converges and that its sum is S".

"the behaviour of the [infinite] series, convergence or divergence, is the behaviour of its sequence of partial sums".

Anyway, I think we will just have to agree to disagree. Yes?