Do you think 0.9999999~ =1 , that is infinite 9s.

[rahuldandekar]

Mister Earl, you seem to have the same misconception as many other "doubters": namely, that the notation 0.99~ (or 0.99... or 0.99R) represents a process. It doesn't. It represents a number, one with an infinite number of 9s to the right of the decimal.


Why don't people get this even after 50 pages?? The topic has not "wandered" much even after 50 pages, we're still there, trying to explain the same thing over and over again.

Reminds me of :wall:.

[Moose]

Why don't people get this even after 50 pages??

38, but it's because people are intrinsically lazy. Unfortunately, we're becomming an instant gratification society. Not wanting to take the time to do the appropriate research needed to justify our instincts.

Unfortunately, a person's first instinct is that 0.999~ cannot possibly be equal to 1, for the various misconceptions we've seen so often in this thread.

The fact of it is, 0.999~ is, and indeed must be equal to 1. The math bears it out, physics bears it out. Engineering bears it out. So does Computer Science. I can say that because engineering works. We can build bridges. We can make computer chips. We can launch rockets.

Limit theory works. And one minor but fundamental result of limit theory is that 0.999~ must be equal to 1. And it is.

I again suggest that anybody still wanting to dispute this equality should read the first seven pages of this thread. Then try to come up with a number that comes between 0.999~ and 1.

Short of that, it's all been said.

[hhEb09'1]

38, but it's because people are intrinsically lazy. Posters can set the size of their pages, so the number of pages in the thread varies.

I have a question, especially for anyone who is entertaining the idea of 0.999...8 sorta things: Does 0 = 0.000~ ?

I mean, what if it ends in a one?

:) :) :)

[rahuldandekar]

Show me a number that lies between 0 and 0.0000.....1 .
;)

[worzel]

Unfortunately, a person's first instinct is that 0.999~ cannot possibly be equal to 1, for the various misconceptions we've seen so often in this thread.

The fact of it is, 0.999~ is, and indeed must be equal to 1. The math bears it out, physics bears it out. Engineering bears it out. So does Computer Science. I can say that because engineering works. We can build bridges. We can make computer chips. We can launch rockets.
What I find odd about the inequality believers is that they treat numbers as if they can argue about them independantly of their definitions as if they have a reality of their own. A few people have said things like "well if you define it that way then yeah, it's true by definition, but I still say it's not true." What can that mean? Everything in maths that is true is true by definition (although it may not be trivial to demonstrate), unlike science there is no reality check.

[Moose]

Posters can set the size of their pages, so the number of pages in the thread varies.

Good point, I forgot I'd lengthened my page from the default. 50 it is. :)

[Moose]

What I find odd about the inequality believers is that they treat numbers as if they can argue about them independantly of their definitions as if they have a reality of their own. A few people have said things like "well if you define it that way then yeah, it's true by definition, but I still say it's not true." What can that mean? Everything in maths that is true is true by definition (although it may not be trivial to demonstrate), unlike science there is no reality check.

I think your observation is right. I think this is yet another symptom of us being continuously bombarded with the idea that reality is subjective. It would help explain why our younger generations seem so enamored with conspiracy fantasies.

[Mister Earl]

If you're so certain that 0.99~ is unequal to 1, you have to do only one thing to convince us: specify a number n such that 0.99~ < n < 1. If you can do this, you will have revolutionized mathematics (or at least redefined some fundamental elements of mathematics).

Ok, I'll take on that challange :D

1- 0.9R = X
X=0.0R1

There are an infinite number of zeroes before you achieve the final 1 required to make 0.9R into 1.

That's my take on it.

[worzel]

Ok, I'll take on that challange :D

1- 0.9R = X
X=0.0R1

There are an infinite number of zeroes before you achieve the final 1 required to make 0.9R into 1.

That's my take on it.
You've just move the question. If your 0.0R1 is not exactly equal to 0 then there must be an infinite amount of numbers between 0 and 0.0R1.

Can you give even 1?

[rahuldandekar]

Ok, I'll take on that challange :D

1- 0.9R = X
X=0.0R1

There are an infinite number of zeroes before you achieve the final 1 required to make 0.9R into 1.

That's my take on it.

Give a number. Dont trust the difference, express the number between 0.999~ and 1 in decimal form. If you can, we accept defeat ;).

[Moose]

1- 0.9R = X
X=0.0R1

There are an infinite number of zeroes before you achieve the final 1 required to make 0.9R into 1.

Can't be. Either your R represents an infinite repetition, (in which case it 0.0R1 cannot possibly be considered as a finite number with a terminating 1, as you're suggesting,) or else 0.9R can't be an infinite repetition. You can't have it both ways, and you can't move the goalposts.

[Disinfo Agent]

Ok, I'll take on that challange :D

1- 0.9R = X
X=0.0R1

There are an infinite number of zeroes before you achieve the final 1 required to make 0.9R into 1.

That's my take on it.Here's something else that "doubters" of the equality 0.999~=1 systematically forget: "0.0R1" is not a decimal expansion. Decimal expansions of real numbers have no final digit after the infinite others.

I want to be the least technical possible: the problem with that last digit is "How much does it contribute to the expansion?" We know the first digit after a decimal point countributes 0.1 to the full number. The second digit after the point contributes 0.01, and so on... the digit at the n-th place contributes to the 0.1n-order term of the expansion.

So, what does that last digit contribute with? Logically, it should be 0 = 0.1infinity...

[Mister Earl]

Give a number. Dont trust the difference, express the number between 0.999~ and 1 in decimal form. If you can, we accept defeat

My whole take on it is that rounding 0.9R to 1 is just a way of rounding up and eliminating infinity from the problem. 0.9R isn't equal to 1 because it isn't 1, it's 0.9R. That's how the decimal system works.

Let's say you have two giant vats. Both are filled with identical red paint. They are equal in this sense. Now say you put a drop of yellow paint into one tank. They are no longer equal. It doesn't matter how big or small that drop is in relation to the other vat. The tanks are no longer equal.

No matter how far along you tack on 9's to 0.9R, you approach but never achieve 1. I haven't seen a proof here yet that convinces me.

[Mister Earl]

Here's something else that "doubters" of the equality 0.999~=1 systematically forget: "0.0R1" is not a decimal expansion. Decimal expansions of real numbers have no final digit after the infinite others.

I could just as easily say that you folks who believe that 0.9R = 1 are forgetting the basics of the decimal system. If the first digit to the left of the period is a zero, then it is impossible that the number is a one.

[Disinfo Agent]

I could just as easily say that you folks who believe that 0.9R = 1 are forgetting the basics of the decimal system. If the first digit to the left of the period is a zero, then it is impossible that the number is a one.Says who?

P.S. Please reread my previous post, as I've added a paragraph to it which makes my argument clearer.

[Mister Earl]

I want to be the least technical possible: the problem with that last digit is "How much does it contribute to the expansion?" We know the first digit after a decimal point countributes 0.1 to the full number. The second digit after the point contributes 0.01, and so on... the digit at the n-th place contributes to the 0.1n-order term of the expansion.

Since calculating infinity is impossible, I'd consider 0.0R1 to be 0.(however many zeroes you do calculate to)1. That added to 0.9R you calculated out to the same length, would finally give you a 1.

[Disinfo Agent]

Since calculating infinity is impossible, I'd consider 0.0R1 to be 0.(however many zeroes you do calculate to)1.We want infinite zeroes in there (see the thread title).

How much does the "1" contribute to the total in that case?

[Moose]

No matter how far along you tack on 9's to 0.9R, you approach but never achieve 1.

That's where you're going wrong. The 9s are already there.

See, you're arguing that 0.99 != 0.999. Absolutely. If you compare a short or rounded number, then of course they're different.

But 0.999~ isn't a process. It's a number. A full complete number. All the nines are there already. They have to be in order to be expressable. And when all infinite nines are there, the lot is equal to 1. This is something that is only true because there are infinite nines. If that infinity isn't present, then yes, 0.[finite number of nines] must be less than 1.

But it's not hard in that instance to find a number that lies between 0.[finite number of nines] and 1. There are an infinite number of them. But we're talking about 0.[infinite number of nines] and 1. And they're the same.

Have you read the first seven pages of this thread yet?

[Moose]

Since calculating infinity is impossible

Says who?

[Mister Earl]

Sure have. And while it is impossible to express a number between 0.9R and 1, that doesn't mean they're equal. I haven't seen anything that convinces me of that.

[Moose]

Sure have. And while it is impossible to express a number between 0.9R and 1, that doesn't mean they're equal.

Yeah. It does. It's a property of the real number system.

Between any two discrete numbers in the real number system, you necessarily have to have an infinite number of numbers that lie between them. The only numbers you can't do this with are numbers that are equal to each other. And that's specifically because they're equal to each other.

So as you seem to have admitted, it's impossible to express a number that lies between 0.999~ and 1. But that necessarily means they must be equal.

[Disinfo Agent]

I haven't seen anything that convinces me of that.Here's the simplest proof (http://www.bautforum.com/showthread.php?p=379619#post379619) we came up with, back when we were discussing this the last time around. Read it and let us know what you think about it.

[Moose]

That next bit (from the post you linked to oneward) where you guys are walking Candy through the 10x proof, step by step, is absolute gold. Mister Earl should definitely read on in that thread.

[Mister Earl]

*1.00000......
-0.99999.....
----------------
0.000000......
and 0.000... = 0 therefore 0.999... = 1


That math isn't proof at all. 1.0000~ = 1, but "-0.99999....." != 0.9R.

The solution to the problem above would be that 1 = 0.9R + 0.0R1. I know that isn't mathmatical norm, but it's the only way to express that number.

# EDIT #
Reading over that 10x proof now.

[Moose]

That math isn't proof at all. 1.0000~ = 1, but "-0.99999....." != 0.9R.

Sure it is. How do you figure?


Reading over that 10x proof now.

Good.

[Mister Earl]

The culmination of that 10x proof is,

Therefore, the answer would be 9.999... or 10.
Since when does 9.9R = 10? It's the same argument as 1=0.9R without the proof.

[Moose]

Since when does 9.9R = 10?

Since always.

The proof is that you wouldn't be able to do those manipulations if they were different. They'd break down and diverge. And it would be pretty obvious very quickly. Pretty much like the rest of mathematics.

[Mister Earl]

All I see there is trying to prove one unproveable statement by submitting another unproveable statement. 0.9R != 1 by the same amount that 9.9R != 10. It's substituting one imperfect formula with another.

[Mister Earl]

I'm sorry guys, I'm not trying to be anal or thickheaded here, or closed to new ideas. I'm just a stickler for proof in cases such as this. If someone tells me 1 and 0.9R are equal, then to me that's like saying an orange is just a different color apple. Sure, it's possible, but I'm going to want irrefutable proof.

[Moose]

I'm pretty sure I see where you're getting messed up, Mister Earl, but just to be sure, and to try and illustrate it for your benefit, could you tell me what the final digit of 0.999~ is?

[Mister Earl]

There isn't one. But I still do not see how that makes it more likely that the digit is one instead of 0.9R.

#EDIT#
My thinking is, just because the difference between 1 and 0.9R is incalculable, doesn't meant the difference doesn't exist.

[agingjb]

Another approach.

If 1 and .9R were to represent different real numbers, then their difference would have to be a real number distinct from 0. And this real number would, not being 0, be able to be divided into another real number.

So: what is 1/(1-.9R)

[Mister Earl]

I'd calculate that the same as Pi, with a limited number of digits. It all depends on how accurate you want your answer to be. So say ten digits...
1 / ( 1 - 0.9999999999 )
1 / 0.0000000001 which is calculateable.
So the big question here, which should settle this, is how are numbers that repeat infinitely handled within formulae? What is the standard?

[tofu]

Mister Earl, if you have a pizza with a total area of *exactly* 1 square meter (it's a large pizza) and you cut it into three slices and each slice is *exactly* the same as every other slice...

please tell me the area of one slice.

I think this exercise will help you to understand that 0.99... = 1

[agingjb]

I'd start by looking at the definition of a field, and then at the properties of the quite specific field of the real numbers. But then I wonder (and I'm reluctant to suspect this), do those who claim "No it is not" accept the formal definition and properties of the real field?

If two reals are not equal, then they have a non-zero real difference. Division is defined by all reals except 0, and results in a real quotient.

[Mister Earl]

Mister Earl, if you have a pizza with a total area of *exactly* 1 square meter (it's a large pizza) and you cut it into three slices and each slice is *exactly* the same as every other slice...

please tell me the area of one slice.

I think this exercise will help you to understand that 0.99... = 1

The only way to accurately describe the area of a slice is in fractional form. Decimal is not accurate enough.
3/3 = 1
If you do convert to decimal by fraction, the only way for it to work out is to limit the number of decimal places, like you would when using Pi.

1/3= 0.3333333333
+ 2/3= 0.6666666667
----------------------
= 3/3= 1

[tofu]

Decimal is not accurate enough.

From this statement, it is clear that you do not understand the ... notation.

1/3 = 0.33...

I'm sorry that you can't see that. This is the reason you don't understand the topic of this thread.

[Moose]

My thinking is, just because the difference between 1 and 0.9R is incalculable, doesn't meant the difference doesn't exist.

It's not incalculable at all. The difference is 0. Here's the thing: you're right in that we cannot grind an infinite series as an interative physical process. That's impossible. The best we can do without using limits is to compute an approximation. (Basically the best a computer can do.)

Limit theory, on the other hand, lets us see where increasingly accurate approximations are converging.

The offshoot of this is that, once you've "calculated" the infinith term (if that's even a word), by finding the limit, the result won't be an approximation anymore. It'll be the actual result. But only at the infinith term. Anything short of that is only an approximation, as you're seeing.

That's why 0.999~ isn't an approximation of 1. It's an expression of an infinite progression whose limit is 1. But because it's an infinite progression, and all terms (including the infinith) are present in 0.999~, the limit becomes the reality.

We just can't get there by grinding it.

That's why we can say 1/9th is exactly equal to 0.111~, 2/9 = 0.222~, 3/9 = 0.333~ (= 1/3) and so on, until we get to 9/9 = 0.999~ = 1. The only reason this works is because of limit theory, and that there's no rounding whatsoever involved when you're talking about an infinite progression. There can't possibly be rounding, because there's no final digit to round.

You know that 1/3 is exactly equal to 0.333~, and that 2/3 must be equal to 0.666~ so that 0.333~ + 0.666~ must be equal to 1? (If not, then 1/3 + 2/3 couldn't be equal to 3/3 = 1. Right?)

[Edit: It would appear you don't actually know this. Ouch. Yeah, this is definitely why you're not getting it.]

And if that's true, we also know that 0.333~ + 0.666~ must also be equal to 0.999~ by simple arithmetic (and remembering we can't round an infinite progression.)

Do you see it?

[Moose]

+ 2/3= 0.6666666667


You yourself admitted that an infinite progression doesn't have a final digit. How can you possibly round something that doesn't have a final digit?

[hhEb09'1]

That math isn't proof at all. 1.0000~ = 1, but "-0.99999....." != 0.9R.Sure it is. How do you figure?Careful, you're missing a minus sign there, Moose, but I'm not sure what Mister Earl is objecting to, either. Of course, positive one does not equal negative one, no one has said anything like that.
The only way to accurately describe the area of a slice is in fractional form. Decimal is not accurate enough.
3/3 = 1
If you do convert to decimal by fraction, the only way for it to work out is to limit the number of decimal places, like you would when using Pi.

1/3= 0.3333333333All you are saying there, Mister Earl, is that 0.333~ is meaningless. Of course, if it were meaningless, it would not be equal to anything, but that is not the premise of this thread, which is that it does equal some number under the usual understanding of it. The question is whether that number is 1 or not. Not whether it is meaningful or not--the meaning has already been explained.
You yourself admitted that an infinite progression doesn't have a final digit. How can you possibly round something that doesn't have a final digit?Wait. I round stuff like that all the time, I don't think that there is a problem with rounding, if that is the goal.

[Mister Earl]

From this statement, it is clear that you do not understand the ... notation.

1/3 = 0.33...

I'm sorry that you can't see that. This is the reason you don't understand the topic of this thread.

Actually, I do understand that notation. It's shorthand for a repeating value.
And I'd appreciate it if we could hae this discussion without getting on a personal level or attempting to insult my intelligence by suggesting that since I don't agree with you that I do not belong here.

And Moose, yes. I do see how you're arriving at your destination, but I don't agree with the trip there. For example, 9/9 = 0.9R? That isn't equal at all. I see how you try to explain it by adding the fractions together. But I see it as 0.9R != 1 = 9/9. The reason repeating numbers repeat forever is because there's always a fraction left over that doesn't split neatly into base 10. It's that "little bit left over" that keeps it from becoming 1.

[Moose]

Wait. I round stuff like that all the time, I don't think that there is a problem with rounding, if that is the goal.

Nothing wrong with rounding, so long as you can accept the approximation. But your equality isn't an equality anymore. Not in the strict sense.

So noted about the negative. I didn't spot that.

[Mister Earl]

You yourself admitted that an infinite progression doesn't have a final digit. How can you possibly round something that doesn't have a final digit?

The same way people use 3.14 for Pi or 3.1415963. It isn't perfectly accurate because it is rounded. I consider 0.9R = 1 as rounding. They aren't equal unless rounding happens, which is corner-cutting.

[hhEb09'1]

And Moose, yes. I do see how you're arriving at your destination, but I don't agree with the trip there. For example, 9/9 = 0.9R? That isn't equal at all. I see how you try to explain it by adding the fractions together. But I see it as 0.9R != 1 = 9/9. The reason repeating numbers repeat forever is because there's always a fraction left over that doesn't split neatly into base 10. It's that "little bit left over" that keeps it from becoming 1.So, you'd also say that 1/3 does not equal 0.333~

That's definitely not standard convention.

[Mister Earl]

hhEb09'1

Let's look at it another way, since we're getting nowhere fast :D

If:
1/3 = 0.3R
2/3 = 0.6R
3/3 = 0.9R

Where'd the remainder go?

#EDIT#
That puts us right back to 0.9R = 1. So let me rephrase the question:

Does 0.9R have a remainder? And can it be expressed in fractional form?

[hhEb09'1]

hhEb09'1

Let's look at it another way, since we're getting nowhere fast :D

If:
1/3 = 0.3R
2/3 = 0.6R
3/3 = 0.9R

Where'd the remainder go?Does R stand for repeating? I mean, rather than for remainder.

In which case, there is no remainder. In each case, there is perfect equality.

#EDIT#
That puts us right back to 0.9R = 1. So let me rephrase the question:

Does 0.9R have a remainder? And can it be expressed in fractional form?No, it does not have a remainder. It is equal to 1, by our standard conventions

[Mister Earl]

No, it does not have a remainder. It is equal to 1, by our standard conventions

If it doesn't have a remainder, why do the 9s repeat forever? Why couldn't you stop at 0.999? Or 0.99999? And if this is a standard, what is the mathmatical law called? Is there material somewhere I can reference?

[hhEb09'1]

If it doesn't have a remainder, why do the 9s repeat forever?You didn't ask about a division of two numbers, which is where you get remainders, right? I'm not even sure what you mean by remainder of a number. What is the remainder of 2?
Why couldn't you stop at 0.999? Or 0.99999? And if this is a standard, what is the mathmatical law called? Is there material somewhere I can reference?The sum of an infinite number of terms is explained some here (http://mathworld.wolfram.com/Series.html). The notation may be unfamiliar, but the gist of equation (2) is that 1 + 1/2 + 1/4 + 1/8 + ... continued out forever is equal to 2.

I know you are aware of where the 0.333~ comes from, it is the result of dividing 1 by 3 using long division. Doing it "forever" means that you can make the difference between it and 1/3 as small as you like. Mathematicians say that doing it forever means that the difference is zero (or, closer to how they more formally phrase it, we can make the difference smaller than any number you choose, which means it must be zero), so they're equal.

Part of the reason for stating it so awkwardly is that even strong mathematicians have had objections to the process, similar to yours. The formal explanations have become tighter and tighter, and more abstruse.

[Mister Earl]

You didn't ask about a division of two numbers, which is where you get remainders, right? I'm not even sure what you mean by remainder of a number. What is the remainder of 2?

Badly worded on my part. I mean a remainder as why the number doesn't end at tenths, or hundredths, or thousandths. There's always a remainder that has to be expressed in smaller and smaller values.

#Edit#

Checked out that link. It's been a while since I did any hard-core math, and even that is over my head. I'll have to refamiliarize myself with the operators on that page and work it out in my head.

[hhEb09'1]

Badly worded on my part. I mean a remainder as why the number doesn't end at tenths, or hundredths, or thousandths. There's always a remainder that has to be expressed in smaller and smaller values.That's where "infinity" come is, we effectively eliminate every single one of those smaller and smaller values.

Name one--I can tell you the step at which it was eliminated. Name any one, I can tell you the step at which it was eliminated. Give me a formula for one, I can give you a corresponding formula for determining at which step would be eliminated. Because we can go to infinity, at least in the notation, we can say that all those "remainders" have been eliminated. Eliminating all of them means we have equality.

[Mister Earl]

Ok, I think I'm getting closer to seeing your point of view, hhEb09'1. Now my next question is, how does infinity eliminate the smaller values?

[Moose]

If it doesn't have a remainder, why do the 9s repeat forever?

They don't repeat forever. They repeat to infinity. That's not quite the same thing in the mathematical sense.

Each successive iteration of your approximation has a remainder. The limit does not. The limit is ultimately what we're looking at, because it's only at the limit of that progression where equality really comes into play.

[Edit: Emphasis added to answer your question in the immediately preceeding post.]


Why couldn't you stop at 0.999? Or 0.99999?

Because it's not an equality anymore. It's an approximation. Which is fine for most engineering, most physics, nearly all statistics, and has to be good enough for computer science, but it's not good enough for mathematical proofs. Equal means equal. Not approximately.


And if this is a standard, what is the mathmatical law called?

"Limits". "Limit theory" will get you some hits on google as well.


Is there material somewhere I can reference?

Any competent math textbook at the 12th grade level will cover them. There are online resources that are only a search away that should be good enough.

[Edit: I should add that any freshmen Calculus text will cover them as well, because limits are fundamental to any sort of calculus application you can name.]

[hhEb09'1]

Ok, I think I'm getting closer to seeing your point of view, hhEb09'1. Now my next question is, how does infinity eliminate the smaller values?I thought I explained that. :)

Tell me which smaller value that you are interested in, and I will tell you when it was eliminated, at which step of the process. I can even give you a formula that will compute it for you.

[Mister Earl]

Heck, I'd appriate if you'd go over the entire process. That way I can see what you are thinking from beginning to end.


Any competent math textbook at the 12th grade level will cover them. There are online resources that are only a search away that should be good enough.

Back in the day, my 12th grade school books still talked about Sputnik I being the only artificial satellite. Yeah, ouch.

[hhEb09'1]

Heck, I'd appriate if you'd go over the entire process. That way I can see what you are thinking from beginning to end.Let N be the number of nines in 0.999...999, which is finite, and ends in 9, and is not equal to 1. It differs from 1 by .000...001, which is equal to (.1)N. I suppose that would be what you are calling a remainder.

If you choose a very small value, AVSV, then we can take the base 10 logarithm of it. IF AVSV is less than zero, then the logarithm L will be negative. Round it to the next negative integer, M. If you use that many 9s, M of them, then the remainder will be smaller than your AVSV.

[Mister Earl]

I don't see where infinity is factoring in here and where it is eliminating smaller values?

Meh. I think what I may need to do here is cash in my Montgomery GI Bill money and take night classes to get up to speed in my math so I can follow you all better.

[hhEb09'1]

I don't see where infinity is factoring in here and where it is eliminating smaller values? What happens is, we let N get as large as we like ("go to infinity"). Because we use every possible N (all infinity of them), we never have to worry about any smaller value.

Meh. I think what I may need to do here is cash in my Montgomery GI Bill money and take night classes to get up to speed in my math so I can follow you all better.Good luck with that! and remember what Whitehead said: "The study of mathematics is apt to commence in disappointment." You'll have to get over that somehow :)

[Mister Earl]

Good luck with that! and remember what Whitehead said: "The study of mathematics is apt to commence in disappointment." You'll have to get over that somehow.

There I disagree with you. If the answers came easy, it'd bore me, and the fun part about math is working towards the end, not getting there.

[tofu]

suggesting that since I don't agree with you that I do not belong here.

I didn't say that you don't belong here. I said that you don't understand. And that's true. You do not understand.

Furthermore, your "not understanding" seems to be on purpose, because the example I gave you was very simple, but your response to it was intentionally obtuse. Some people call that hand waving.

1/3 = 0.33...

But you said:


1/3= 0.3333333333

Mister Earl, that is NOT true and you know it. If I had asked you this question a month ago, before you had seen this thread, you would have been honest and said that yes, obviously, 1/3 = 0.33... You've known that all your life, as has anyone who paid attention in a 3rd grade math class.

But because you have taken the (incorrect) position in this thread, and because you can look two moves ahead and see where I'm going with this, you now refuse to say that 1/3 = 0.33... for fear of losing an argument.

I've seen this pattern many times. It happens in religions discussions. It happens in political discussions. It happens when talking to conspiracy theorists. What you're doing is just like a creationist arguing that stars aren't more than 6,000 light years away - not because the creationist actually sees a problem with astronomical observations - but because the creationist doesn't want to lose the bigger argument of evolution.

So, there's the company that you're keeping.

1/3 is exactly equal to 0.33...

It is not equal to any other quantity. I think you need to take a step back and be honest with yourself about why it is that you believe and understand this simple concept, but refuse to admit it.

[Donnie B.]

Mister Earl, may I ask what level of formal math training is the highest you've had? Did you ever study integral or differential calculus at, say, college freshman level?

I don't ask to put you down or intimidate you, only to get a better idea of what sort of notation might be familiar to you. I think it might be possible to express the problem in a way that would allow someone with calculus skills to solve it exactly. (I haven't tried to do this yet so I may be wrong!)

[Mister Earl]

tofu

No, I said far more often that 1/3 = 0.3R. As in .3 repeating. That's the format I was taught. I used 0.33333 for that example alone.

And Donnie, no calculus here. Trig if my memory serves. Thats' why I need to hit the local community college and brush up. Gonna make some calls when I get home.

[hhEb09'1]

Thats' why I need to hit the local community college and brush up. Gonna make some calls when I get home.Good luck Mister Earl, and you're right, it can be fun. :)

[tofu]

tofu

No, I said far more often that 1/3 = 0.3R.

Again, you're being obtuse. Let's take your comment line by line:

The only way to accurately describe the area of a slice is in fractional form.

not true.

Decimal is not accurate enough.

not true. There is a particular notation, "..." that is perfectly accurate.

If you do convert to decimal by fraction, the only way for it to work out is to limit the number of decimal places,

not true. 1/3 is exactly equal to 0.33...

Exactly. Not almost. Not really close. It's not rounded. It is exactly equal.

like you would when using Pi.

not true. Pi does not repeat. You cannot use "..." with Pi. You can round Pi or you can use radians. This is nothing at all like 1/3.

And you must know this if you made it through the 5th grade.

[Mister Earl]

Tofu, what is your problem? You've taken this discussion from civil to personal.
You keep dragging the discussion back to the elipses, which I wasn't even talking about when you quoted me, not to mention taking statements I said out of context. Since my discussion with you is no longer productive, I will cease to respond to your statements.

[tofu]

There's nothing personal in any of my posts. It's not like I called you an ugly poopoo head or made reference to your questionable parentage. And I haven't taken you out of context either. I'm just being direct, calling a spade a spade.

The concept that 1 = 0.99... might possibly throw someone off. That understandable and forgivable. But you're saying that 1/3 does not equal 0.33...

What am I supposed to say about that? You know the truth. You learned it in school just like everyone else did and it never confused you and you never had a problem with it. You just wont admit it now because you can see where it's going and you desperately don't want to lost an argument - on the internet. It's a pride issue. It's exactly like a hoax believer posting a picture of apollo and saying "look everyone, no rover tracks!" but then when someone posts the next or previous picture from the film roll and it clearly shows the tracks, the hoax believer's pride prevents him from admitting what he can see with his own eyes.

[Mister Earl]

Not once have I said 1/3 does not equal 0.3R. You're using one badly worded example as the basis of your entire argument. I did not mean that you cannot express a fraction as a decimal as accurately as a fraction. I misworded that and I was wrong. I was in the middle of explaining my point of view as why .3R * 3 = 1, not 0.9R. Furthermore, you need only look at all my other posts in this section to see that I do understand what the elipses means, but you apparently didn't read those. You twist what I say out of context, then have the nerve to accuse me of hand waving. Want an example? Fine.


like you would when using Pi. (Apparently quoting me here)

not true. Pi does not repeat. You cannot use "..." with Pi. You can round Pi or you can use radians. This is nothing at all like 1/3.

The best part is when I went back to my post and reread what I had written:

I quoted Moose here:
You yourself admitted that an infinite progression doesn't have a final digit. How can you possibly round something that doesn't have a final digit?

And I replied here:
The same way people use 3.14 for Pi or 3.1415963. It isn't perfectly accurate because it is rounded. I consider 0.9R = 1 as rounding. They aren't equal unless rounding happens, which is corner-cutting.

"like you would when using Pi." doesn't even show up in any of my posts! You typed up your quote as something I supposedly said to twist around what I was trying to say to support your argument.

And nothing personal in any of your posts?


And you must know this if you made it through the 5th grade.

Seems to me like you're inferring I'm some slack-jawed hick who isnt' listening to his betters. How would I NOT take offense to that? Not to mention that in the same post you claim not to be getting personal, you try to lump me into the same catagory as some conspiracy theorist.

Try re-reading my posts, and look at the point I was trying to make, instead of ammunition to shoot down an argument I was never part of. And while you're at it, read over hhEb09'1's posts, too. He replies in a constructive fashion, with the purpose of educating and explaining. You do not. Looks like both of us could learn from him.

[tofu]

like you would when using Pi. (Apparently quoting me here)

Yes, I was quoting you. I apologize for using the non-standard bold instead of the regular forum quote. Here is the post I was quoting:

http://www.bautforum.com/showpost.php?p=860766&postcount=1536

In it you say this:


The only way to accurately describe the area of a slice is in fractional form. Decimal is not accurate enough.
3/3 = 1
If you do convert to decimal by fraction, the only way for it to work out is to limit the number of decimal places, like you would when using Pi.

The only way to accurately describe the area of a slice is in fractional form.

incorrect.

Decimal is not accurate enough.

also incorrect.

If you do convert to decimal by fraction, the only way for it to work out is to limit the number of decimal places

totally incorrect. If you limit the number of decimal places, you have not converted the fraction into a decimal. You have converted a different fraction into decimal.

You learned this in school. You Mister Earl, were given a math test and were asked to convert 1/3 into decimal, and you, Mister Earl, wrote 0.33... as the correct answer. We all did. This is not a complicated concept.

like you would when using Pi

Pi is a non-repeating, non-terminating decimal. You cannot use the ellipses with Pi. It's totally different.

[tofu]

Seems to me like you're inferring I'm some slack-jawed hick

You're reading an awful lot into that. I said that you learned this in the 5th grade. I'm pretty sure that's an accurate statement. We all learned this simple math in or about 5th grade. It really is that simple.


He replies in a constructive fashion, with the purpose of educating and explaining. You do not.

Well, I started with you in that vein, but you intentionally gave the wrong answer to my question so that you wouldn't have to face the fact that you're wrong about the 0.99... issue.

Want to try again? You have a pizza with area of 1 unit. You cut the pizza into three exactly equal pieces. What is the area of one piece?

Put some thought into it. You know the truth.

No egos here. Just think about the proposition.

[montebianco]

I'd start by looking at the definition of a field, and then at the properties of the quite specific field of the real numbers. But then I wonder (and I'm reluctant to suspect this), do those who claim "No it is not" accept the formal definition and properties of the real field?

Since the equality of 0.999~ and 1 follows from the formal definition and properties of the real field, I suspect they do not :)

But, it is even more fun than that. From Rudin:


Any two ordered fields with the least-upper-bound property are isomorphic.

So the strategy of declaring that all mathematicians are narrow-minded clods who define the Archimedean number out of existence because they're scared of its implications (a tack tried much earlier in this thread), doesn't work. You want to build your own number system where 0.999~ and one are not equal (their non-equality being claimed as obvious earlier in this thread), you break some of the properties that are equally obvious. It's too bad obviousness does not entail logical consistency.

[worzel]

Not once have I said 1/3 does not equal 0.3R. You're using one badly worded example as the basis of your entire argument. I did not mean that you cannot express a fraction as a decimal as accurately as a fraction. I misworded that and I was wrong. I was in the middle of explaining my point of view as why .3R * 3 = 1, not 0.9R.
Well I'd like to see how you get .3R * 3 = 1, I get .9R :)

[montebianco]

What is the binary representation of this mysterious new number 0.999~?

[Moose]

1.

[montebianco]

1.

I was hoping some other folks would answer :D

But if they do, they can also work on the binary representation of 0.2999~. I'd be interested to hear what it is, in their view. . .

[01101001]

1.

Or... 0.111~2

Like all quantities, it has many representations.


But if they do, they can also work on the binary representation of 0.2999~.

.010011001100110011(0011)2

I like it. It reminds me of Mom. Her friends called her "Dot3".

[worzel]

Not once have I said 1/3 does not equal 0.3R. You're using one badly worded example as the basis of your entire argument. I did not mean that you cannot express a fraction as a decimal as accurately as a fraction. I misworded that and I was wrong. I was in the middle of explaining my point of view as why .3R * 3 = 1, not 0.9R.

Well I'd like to see how you get .3R * 3 = 1, I get .9R :)

Also, exanding on Disinfo Agent's post here (http://www.bautforum.com/showthread.php?t=14593&page=51#1522), when I add 0.9R to itself I get:

0.999999.... (a)
+0.999999.... (+a)
-------------
=1.999999.... (=b)

If I do it by hand I can see with 100% certainty that the nines in 1.9R will go on forever just like they do in the two 0.9Rs I'm adding. I don't need to do it forever, I can put ... to indicate that I am certain the nines will repeat forever just like I know they do in the two 0.9Rs.

But then if I take away one of the 0.9Rs I originally added I get:

1.999999.... (b) (i.e. a+a)
-0.999999.... (-a)
-------------
=1.000000.... (=c)

If I do this by hand I can see with 100% certainty that the zeros in 1.0R will go on forever just like the nines do in the 1.9R and 0.9R I'm subtracting. I don't need to do it forever, I can put ... to indicate that I am certain that the zeros will go on forever - or I could just stop at "1.0", or even "1" as they make no difference (but equally, I'd have to be certain the zeros go on forever to do that).

If I label those numbers a-c then I have:

a+a=b (1)
b-a=c (2)

substituting (1) for b in (2) gives:

(a+a)-a=c

i.e. (because you can do addition with reals in any order you like)

a+(a-a)=c

but a-a=0, therefore:

a+0=c

i.e.

a=c

But a=0.9R and c=1, so that means that:

0.9R=1 !!!


Could you say which step/s you disagree with in that lot, Mister Earl? I.e. which is the first that doesn't follow from the preceding ones and why. I apologise for being tediously laborious, I'm not trying to be patronizing, I just thought it might help to isolate exactly where you disagree with us.

[rahuldandekar]

Maybe it's the concept of doing it inifnitely that's putting people off. Maybe they're thinking "it's going to end sometime", like thinking that 0.000....01 is a real number.

The only way I think there is to get out of this difficulty is limit theory. Please come back after you've done freshman calculus. Limit theory is intuitive, and does not do things infinitely.

Sorry, but, my patience is ending...

[Thomas(believer)]

Maybe it's the concept of doing it inifnitely that's putting people off. Maybe they're thinking "it's going to end sometime", like thinking that 0.000....01 is a real number.

The only way I think there is to get out of this difficulty is limit theory. Please come back after you've done freshman calculus. Limit theory is intuitive, and does not do things infinitely.

Sorry, but, my patience is ending...
But if it is never going to end, that means it never is going to reach the point where it is heading for. And one can say that 0.999~ is a number that is never going to get there, because if it would, it wouldn't be an infinite sequence.
So if you think of 0.999~ as a number where one day you are going to put 9 's at the end, then you are right that you will never reach 1.
If you define 0.999~ as a number where all (whati is all?) the 9's are already there, then by definition 0.999~ equals 1.

[EricNau]

If ".333..." equals 1/3 then ".999..." would have to equal 1. After All, 1/3 * 3 = 1, right?

I voted "Yes."


But really, this is why I use fractions when possible. ;)

[Nicolas]

this mysterious new number 0.999~?

I didn't know you were THAT old ;)

[Nicolas]

And one can say that 0.999~ is a number that is never going to get there, because if it would, it wouldn't be an infinite sequence.

Wrong. If it's an infinite sequence, it means it has already done infinite steps. If it wouldn't have done infinite steps yet, the sequence wouldn't be infinite yet.


If you define 0.999~ as a number where all (whati is all?) the 9's are already there, then by definition 0.999~ equals 1.

That's the point :). Oh and your "all" is an infinite amount. Which you could define as the number of 9's in 0.999~ for it to be exactly equal to 1. Stick to that definition and all is clear. Think about physical reality and all collapses. But then again, these are different worlds. You can't draw a perfect circle, but that doesn't mean we should get rid of the mathematical perfect circle.

[Disinfo Agent]

Since Mister Earl has told us that it's been a while since he was in school, I'm going to assume he belongs to one of the generations that still learnt to do basic arithmetic without the aid of a pocket calculator. So, I have a problem for Earl to think about.

Imagine that an item is on sale for $99.999...9999999... (infinite nines), and you pay for it with a 100 dollar bill. How much change will you get back? ;)

[Moose]

If ".333..." equals 1/3 then ".999..." would have to equal 1. After All, 1/3 * 3 = 1, right?

Exactly right. :)

[rahuldandekar]

Disinfo, I think the problem that 0 =? 0.00000...001 is the same as 0.99999~ =? 1 . It's impossible to accept (for people who cant accept the latter) that some number with a 1 at a decimal place is equal to 0.

[Nicolas]

but 0.00...01 doesn't make sense mathematically. The only way to force it into some sense, is to say it's the same as 0.00... (and adding decimal zeros makes no sense, so that's 0). Because mathematics has no meaning or use for decimals after an infinite repitition. There is no "after" infinite.

[rahuldandekar]

I accept that, but will they?

I still think only limit theory gives the most logical and correct fundamental answer.

[mickal555]

heh

80 pages here for me...

WHY is it such an issue,

Biggest thread in BABBling :lol:

[Mister Earl]

worzel

Now that's just creepy...
0.9R
+0.9R
------
1.9R
-1.0
------
0.9R

Since the 9's never end, you never hit that final 8. So for all intents and purposes, Worzel has shown me a mathmatical proof I can accept that 0.9R = 1.

[rahuldandekar]

worzel

Now that's just creepy...
0.9R
+0.9R
------
1.9R
-1.0
------
0.9R

Very counterintuitive, but I can't refute it!

I like the x = 0.999~, so 10x = 9.999~ and thus x = 9/9 = 1 one better.

[Disinfo Agent]

Disinfo, I think the problem that 0 =? 0.00000...001 is the same as 0.99999~ =? 1 . It's impossible to accept (for people who cant accept the latter) that some number with a 1 at a decimal place is equal to 0.I want Earl (and everyone else) to forget all about math and algebra, and just try to calculate the change as you would if you were standing at the cashier of a store. :)

[hhEb09'1]

I want Earl (and everyone else) to forget all about math and algebra, Trying, trying... unggghhhh

[Moose]

Yeah, I'm with Grapes on this one. Short of a full-frontal lobotomy...

I don't see where you're going with this, Disinfo, but you seem to be introducing rounding into the problem. I think that's going to lead to confusion. Woe. Gnashing and wailing of teeth. Cats and dogs living together. K-Fed going platinum. Maybe even another ten pages to this thread.

Don't do it, Pandora. Carefully set the box down and leave it alone.

( ;) Seriously, where are you going with this? )

[Disinfo Agent]

Trying, trying... unggghhhhNeedless to say, "forget all about math and algebra" was just one of my famous hyperboles (http://www.bautforum.com/showthread.php?p=324700&highlight=hyperbole#post324700). :D

[Nicolas]

The question is not so much where he's going with this, but whether he will reach it or merely come very very close to it.

[Disinfo Agent]

I don't see why not. This is not substantially different from the kind of reasoning that Worzel used in his example, IMO.

[Donnie B.]

worzel

Now that's just creepy...
0.9R
+0.9R
------
1.9R
-1.0
------
0.9R

Since the 9's never end, you never hit that final 8. So for all intents and purposes, Worzel has shown me a mathmatical proof I can accept that 0.9R = 1.
Well, good. :)

I hope that now you may be able to see that this proof (which you accept) is essentially the same as some of the others, like the 10x proof or the 1/3 * 3 proof (which you didn't). To me, those others are much easier to grasp than worzel's, but every brain works a little differently.

[worzel]

Well, good. :)

I hope that now you may be able to see that this proof (which you accept) is essentially the same as some of the others, like the 10x proof or the 1/3 * 3 proof (which you didn't). To me, those others are much easier to grasp than worzel's, but every brain works a little differently.
Mine was just DisInfo's with all the steps spelt out, which itself was just the 10x-x one but with 2x-x and the multiplication done as addition. The concise ones are good if you already believe it. But if you doubt it the gory details are sometimes what it takes to convince yourself there's nothing dodgy hiding in there :)

[Donnie B.]

Okay. But the 1/3 = 0.333... argument uses the very same sort of arithmetic as your proof. Specifically, to understand it, all you need to accept is that dividing one by three yields an infinite sequence of 3s in the result (which anyone who understands long division can prove to herself), and that multiplying 0.333... by three yields 0.999~.

Both those things seem easier to grasp (to me) than that 0.999... + 0.999... = 1.999....

I'd suggest that the conceptual problem had to do with the "0.333... * 3" step, except that I seem to recall that it was the "1/3 = 0.333..." step where we ran into a problem. But if I'm wrong and it was the former, then maybe it would have been easier if expressed as a sum:

0.333...
+0.333...
+0.333...

=0.999...

Like I said, every brain works a bit differently.

[Disinfo Agent]

Okay. But the 1/3 = 0.333... argument uses the very same sort of arithmetic as your proof. Specifically, to understand it, all you need to accept is that dividing one by three yields an infinite sequence of 3s in the result (which anyone who understands long division can prove to herself), and that multiplying 0.333... by three yields 0.999~.

Both those things seem easier to grasp (to me) than that 0.999... + 0.999... = 1.999....I agree with Worzel. His proof uses ideas from arithmetic only. Once you bring fractions and decimal expansions into the "equation", you've already dipped your toe into 5th/6th grade math. And the "x=0.999..., 10x=9.999..." proof is pure algebra.

I do see a hierarchy whereby some people fell more comfortable with the methods they learnt first.

(Of course, if we're very picky, anything with infinities is really calculus, if not higher math. But don't tell that to the students.)

[worzel]

My favourite proof is the epsilon delta one that 1/x=0 in the limit as x goes towards infinity :)

[Disinfo Agent]

...Which directly invokes the Archimedean Property of the real numbers.

[Guillaume]

What about this one :

S = 0.9999... = sum(n = 1 to +infinite ; 9*10^-n)
10*S = 10*sum(n = 1 to +infinite ; 9*10^-n)
= sum(n = 1 to +infinite ; 10*9*10^-n) // multiply each member of the sum by 10
= sum(n = 1 to +infinite ; 9*10^(-n+1))
= sum(n = 0 to +infinite ; 9*10^-n) // changing the indice
= 9 + sum(n = 1 to +infinite ; 9*10^-n) // puting out of the sum its first term
= 9 + S

So 10*S = 9 + S
9*S = 9
S = 1

I'm not sure if this is clear with this notation

[Disinfo Agent]

That's a good one. You can do it in a more straightforward way:

S = 0.999... -> 10S = 9.99... = 9 + 0.99... = 9 + S

And from here on as above.

[Guillaume]

I know but the "S = 0.999... -> 10S = 9.99..." thing is easier to admit if you put it as an infinite sum I think, the demonstration I put seems more rigorous.

[Thomas(believer)]

I just finished the book "A brief History of Infinity" by Brian Clegg and I had to think about this thread. ( Or did I buy the book because of this thread? :think: )

Maybe there are some experts who can explain it better to me, but at this moment I see it as folllows:

We can say that 0.999~equals 1, not because it has been proven that the real numbers are a continuum (it has not been proven!), but because it has been proven that it doen't matter. The rules of the way we calculate things are independant of the continuum hypothesis. They work well anyway.
So I guess I want to to say is that 0.999~ = 1 because it doen't matter, but the proove of this is not trivial.
Or I still don't get it, which can be very true as well.

[hhEb09'1]

We can say that 0.999~equals 1, not because it has been proven that the real numbers are a continuum (it has not been proven!), but because it has been proven that it doen't matter. The rules of the way we calculate things are independant of the continuum hypothesis. They work well anyway.
So I guess I want to to say is that 0.999~ = 1 because it doen't matter, but the proove of this is not trivial.
Or I still don't get it, which can be very true as well.I'm thinking the latter. :) The continuum hypothesis (http://en.wikipedia.org/wiki/Continuum_hypothesis) has nothing to do with the real numbers, except indirectly--it's a conjecture about transfinite numbers. The real numbers are sometimes called the continuum.

PS: This wiki page (http://en.wikipedia.org/wiki/Continuum_%28mathematics%29) says "The continuum hypothesis is sometimes stated by saying that no cardinality lies between that of the continuum and that of the natural numbers." But that's not really why it is called the continuum hypothesis, I think.

PSS: That wiki page says that the word continuum has two distinct meanings in mathematics, which seem to be "a linearly ordered set that is 'densely ordered'", and "any nontrivial locally compact connected Hausdorff space." I don't think that they are that distinct :)

[Disinfo Agent]

The Continuum Hypothesis has to do with sets between the natural numbers and the real numbers, but it does not affect the standard construction of the real numbers, that I know.

[Thomas(believer)]

I'm thinking the latter. :) The continuum hypothesis (http://en.wikipedia.org/wiki/Continuum_hypothesis) has nothing to do with the real numbers, except indirectly--it's a conjecture about transfinite numbers. The real numbers are sometimes called the continuum.

PS: This wiki page (http://en.wikipedia.org/wiki/Continuum_%28mathematics%29) says "The continuum hypothesis is sometimes stated by saying that no cardinality lies between that of the continuum and that of the natural numbers." But that's not really why it is called the continuum hypothesis, I think.

PSS: That wiki page says that the word continuum has two distinct meanings in mathematics, which seem to be "a linearly ordered set that is 'densely ordered'", and "any nontrivial locally compact connected Hausdorff space." I don't think that they are that distinct :)

Let's take the set of integers. We know that we cannot sqeeuze an integer in between let's say 1 and 2. Everyone agrees 1 doen't equal 2.

The set of reals. We know that we cannot sqeeuze a real value in between .999~ and 1. Everyone (sometimes it takes some effort) agrees 0.999~ equals 1.

The same kind of argument, a different result. Why is that?
Would this be true if there was a one-to-one correspondance between the reals and integers? If these sets would have the same cardinality.

[hhEb09'1]

The same kind of argument, a different result. Why is that?One's a continuum, the other is not :)

Would this be true if there was a one-to-one correspondance between the reals and integers? If these sets would have the same cardinality.I was about ready to answer "no" but now I'm not sure what it is that you are asking "Would this be true?" Could you be more specific about what "this" is?

[Thomas(believer)]

One's a continuum, the other is not :)I was about ready to answer "no" but now I'm not sure what it is that you are asking "Would this be true?" Could you be more specific about what "this" is?

To me one-to-one correspondance means that I can replace every real by an integer. The integers are not a continuum, that's why 2 doesn't equal 1. ;)
So I assume that if this was the case the reals couldn't be a continuum. And hence for the reals to be a continuum they must have a higher cardinality than the integers. But maybe I go wrong here already.

[hhEb09'1]

To me one-to-one correspondance means that I can replace every real by an integer. The integers are not a continuum, that's why 2 doesn't equal 1. ;)
So I assume that if this was the case the reals couldn't be a continuum. And hence for the reals to be a continuum they must have a higher cardinality than the integers. But maybe I go wrong here already.Consider this: the rational numbers have the same cardinality as the integers. They're not a continuum, but that property that you are talking about (if there is no element between a and b, then a=b) is true of the rational numbers.

[rahuldandekar]

Consider this: the rational numbers have the same cardinality as the integers. They're not a continuum, but that property that you are talking about (if there is no element between a and b, then a=b) is true of the rational numbers.

It's not a property of continuity, it's a property of denseness.

Thanks for the insight, hhEb.

[Thomas(believer)]

Consider this: the rational numbers have the same cardinality as the integers. They're not a continuum, but that property that you are talking about (if there is no element between a and b, then a=b) is true of the rational numbers.
0.999~ is a rational number, I suppose (9/10+9/100....), 1 is a rational number. Obviously there is no smaller rational number in between the two. So they must be equal.
Now, I was thinking about another question. Maybe someone can answer this:
Is there for every rational number q: q<1 an irrational number r which satisfies:
q< r <1 . I would guess so, because Q is not a continuum.

edit: I guess you can also find a rational number, which satifies this condition. So this is not what I wanted to ask.
I change my question: which is the smallest number <1, a rational or an irrational number

[01101001]

Is there for every rational number q: q<1 an irrational number r which satisfies:
q< r <1 . I would guess so, because Q is not a continuum.

I'm not into reals, but I can think of at least one* (and probably infinitely more can be constructed, as many as there are irrational numbers between 0 and 1):

Let d = 1 - q. d > 0.

Consider i, an irrational such that 0 < i < 1. Consider r = q + d*i.

q is rational. d is rational. d*i is irrational. r is irrational.

d*i is greater than 0. q is less than r.

d*i is less than d. r is less than 1.

r is irrational. q < r < 1.

* I would choose, of course, the irrational, the transcendental, Thue-Morse Constant: .0110100110010110 etc.

Edit: Oh, you changed the question.

Are you sure there is a smallest number less than 1? (Or even a largest number less than 1?)

[Thomas(believer)]

Edit: Oh, you changed the question.

Are you sure there is a smallest number less than 1? (Or even a largest number less than 1?)

That's why I changed the question. I'm pretty sure there are an infinite number of numbers smaller than 1. But not every infinity seems to be the same.

[01101001]

That's why I changed the question. I'm pretty sure there are an infinite number of numbers smaller than 1. But not every infinity seems to be the same.

So what do you mean by "the smallest number <1"? That doesn't seem to be stated correctly.

[rahuldandekar]

Do you mean the smallest number less than 1 or the largest number less than one.

Anyway, these rules may come in handy to form your own conclusions:

1) There is a rational number between every two rational numbers.
2) There is an irrational number between every two irrational numbers.
3) There is a rational number between every two irrational numbers.
4) There is an irrational number between every two rational numbers.

[Thomas(believer)]

Most confusing matter.

What makes the rationals not a continuum?

[rahuldandekar]

The fact that there are irrationals between any two rationals. It "breaks up" the rationals, and they can't form a countinous line.

[Disinfo Agent]

Let's take the set of integers. We know that we cannot sqeeuze an integer in between let's say 1 and 2. Everyone agrees 1 doen't equal 2.

The set of reals. We know that we cannot sqeeuze a real value in between .999~ and 1. Everyone (sometimes it takes some effort) agrees 0.999~ equals 1.

The same kind of argument, a different result. Why is that?Your reasoning is based on the fact that you know that 2 is the integer immediately after 1, and 1 is (at most) the real number immediately after 0.999~ (though the latter two are actually the same).
How do you know the cardinality of the reals is the one immediately after the cardinality of the integers?


Would this be true if there was a one-to-one correspondance between the reals and integers? If these sets would have the same cardinality.There is no one-to-one correspondence between the reals and the integers. That's what the Pythagoreans found out centuries ago, and it's why the reals (with funky results such as 0.999~=1 and all) are so important.



Most confusing matter.

What makes the rationals not a continuum?The fact that there are irrationals between any two rationals. It "breaks up" the rationals, and they can't form a countinous line.Or, stated in another way, that there are sequences of rational numbers which do not converge to rational numbers. For example, the successive approximations to the square root of 2: 1, 1.4, 1.41, 1.414, and so on.

[agingjb]

The fact that there are irrationals between any two rationals. It "breaks up" the rationals, and they can't form a countinous line.

Hmm. There are rationals between any two irrationals. Are the irrationals (and rationals) "dense", well yes, but the measure of the irrationals in an interval is the length of the interval, whereas the measure of the rationals is zero.

Edit: as I was corrected below, both rationals and irrationals are dense, but distinguished by other properties (sparseness, cardinality).

[Disinfo Agent]

The rationals and the irrationals are both dense sets of numbers. (The standard definition of dense is that between any two numbers there is a third, different from each, belonging to the same set.) So is their union, R.

[agingjb]

I suspect that no-one before Cantor gave much thought to the cardinalities of the integers and the reals. The Greeks knew that the square root of 2 is not rational, but not that pi is irrational (and transcendental).

Of course it's easy to show that any denumerable set of numbers cannot exhaust the real line (think of a (numbered) infinite set of intervals whose length sums to as small a value as you like).

[agingjb]

The rationals and the irrationals are both dense sets of numbers. (The standard definition of dense is that between any two numbers there is a third, different from each, belonging to the same set.) So is their union, R.

Indeed, I was thinking of "sparsity", in which the rationals do differ from the irrationals.

[Disinfo Agent]

That is measure theory. ;)

[hhEb09'1]

There is no one-to-one correspondence between the reals and the integers. That's what the Pythagoreans found out centuries ago, and it's why the reals (with funky results such as 0.999~=1 and all) are so important.
I suspect that no-one before Cantor gave much thought to the cardinalities of the integers and the reals. The Greeks knew that the square root of 2 is not rational, but not that pi is irrational (and transcendental).I assume that comment is in reponse to Disinfo Agent's comment about the Pythagoreans. That's true, the Pythagoreans knew that there were numbers that were not rational--but that's a long ways from showing that there is no one-to-one correspondence.

[Thomas(believer)]

Is the sum where the integer N goes from 1 to infinity of 9/10^N and the outcome is a rational number, dense enough to ever reach 1?
Or must you construct a real number 0.999... where ... stands for a greater infinity? Or are these two things the same?
Maybe a stupid question, but you get that after you find out that there is another infinity that is bigger than infinity.
These mathematicians can give you headaches. :razz:

[hhEb09'1]

Is the sum where the integer N goes from 1 to infinity of 9/10^N and the outcome is a rational number, dense enough to ever reach 1?The infinity "used" in that calculation is the infinity of the integers, obviously. There is one term for every integer.
Maybe a stupid question, but you get that after you find out that there is another infinity that is bigger than infinity.No stupid questions

These mathematicians can give you headaches. :razz:Not near as bad as the headaches that mathematicians get :)

[hhEb09'1]

I think we finally have an authoratative answer to this question: Danica McKellar says "yes" (http://www.danicamckellar.com/) :)

(Check out the third question on her Mathematics page)

[Grey]

I think we finally have an authoratative answer to this question: Danica McKellar says "yes" (http://www.danicamckellar.com/) :)

(Check out the third question on her Mathematics page)And she has an Erdős–Bacon number of 6. Pretty impressive!

[kucharek]

I know I'm tickling a sleeping dragon here... :-)

Today is an article in SPIEGEL-Online, Germany's largest print- and online-newsmagazine.
Some schoolgirl got an award for asking thew smartest mathematical question: "Is 0.99999... less than 1?"
http://www.spiegel.de/schulspiegel/wissen/0,1518,549422,00.html

Babelfished "translation":
http://babelfish.altavista.com/babelfish/trurl_pagecontent?lp=de_en&url=http%3A%2F%2Fwww.spiegel.de%2Fschulspiegel%2Fw issen%2F0%2C1518%2C549422%2C00.html

[Neverfly]

Babelfish sure made an interesting translation...:doh:

Among other oddities, I found this gem in the article:

With the Mathelehrerin Lina had discussed likewise as with her nut/mother, a physics teacher - and found the answers unsatisfactory.

Poor Nut.:neutral:

Eh... OT, I know. But I was amused...

[Disinfo Agent]

Automatic translation is more mysterious than any mathematical riddle.

[Neverfly]

Stephen Hawking may have some choice words on this...

[montebianco]

I know I'm tickling a sleeping dragon here... :-)

Go ahead :)

Couldn't help but notice that the score in the poll is now a whopping 3.26 standard deviations better than what you get by flipping a coin...


Today is an article in SPIEGEL-Online, Germany's largest print- and online-newsmagazine.
Some schoolgirl got an award for asking thew smartest mathematical question: "Is 0.99999... less than 1?"
http://www.spiegel.de/schulspiegel/wissen/0,1518,549422,00.html

And we have as an argument against:


Das kann aber doch nicht sein

That was a very popular argument in this thread :(

I have added a new feature to my signature. Be sure to click on the link!

[hhEb09'1]

Couldn't help but notice And since the poll is public, it makes a nice filter :)

[jfribrg]

From what I can make of the translation, the teacher was hard pressed to come up with a simple explanation. If only he had known about this thread. Every conceivable explanation and more are found somewhere in this "discussion".

[mickal555]

Oh yeah lets talk about this again >_>.

Someone start a monty hall problem thread then.

[jfribrg]

Someone start a monty hall problem thread then.

toSeeked (http://www.bautforum.com/off-topic-babbling/18049-simple-intuitive-explanation-monty-halls-puzzle.html) by almost 3 years.

[agingjb]

I'm sure I've said this before, but I find the result of this poll on this site to be profoundly depressing.

[Tobin Dax]

Oh yeah lets talk about this again >_>.

Someone start a monty hall problem thread then.
My gods, he's alive!

Good to see you, Mickal.

[speedfreek]

Surely this question simply shows up the imperfection in the decimal system, where it is impossible to express 1/3 without an infinitely recurring decimal number?

Is the decimal system using a "digital" method to try to quantify an "analogue" universe?

[Disinfo Agent]

It's only digital until you increase the number of digits to infinity.

[speedfreek]

A good point, so as long as we can accept that a decimal number is infinitely close to being 1, then that must mean it is 1. But it's a long winded way of going about it! ;)

[Disinfo Agent]

There's a misconception somewhere in there: all real numbers are decimal. 'Decimal' is just a way to name them. Well, two ways, in some cases. ;)

[agingjb]

What worries me is the high proportion of contributors to this site who completely reject the concept of limit, that, I would have said, is fundamental in the mathematics that enables any serious development of astronomy.

[Sten]

Surely this question simply shows up the imperfection in the decimal system, where it is impossible to express 1/3 without an infinitely recurring decimal number?

Don't think there's a way around that. We could use fractions to represent the numbers instead, but then we miss the irrational numbers.


A good point, so as long as we can accept that a decimal number is infinitely close to being 1, then that must mean it is 1. But it's a long winded way of going about it! ;)

It is, but it's in the nature of the beast; every way of representing numbers I know of either misses some of them, or gets some of the multiple times.

Anyone know a practical system in which every number is included, and has a unique representation?


What worries me is the high proportion of contributors to this site who completely reject the concept of limit, that, I would have said, is fundamental in the mathematics that enables any serious development of astronomy.

The poll results aren't the best advertisement I can think of for a science board.

[Moose]

A lot of those poll votes were early adopters who voted before the thread really got going. Others who argued (and voted) before being convinced. It's still not perfect, but not as bad as it looks.

[Bogie]

There are an infinite number of infinities. One of them is .999 ad infinitum. Another one is 1.000 ad infinitum. They are not the same.

[Neverfly]

I just voted without reading the thread at all..:neutral:
What does that make me?

[Bogie]

I just voted without reading the thread at all..:neutral:
What does that make me?Self assured.

[speedfreek]

There are an infinite number of infinities. One of them is .999 ad infinitum. Another one is 1.000 ad infinitum. They are not the same.

I just know I am going to regret asking this, but:

What is the point of adding an infinite number of zeros after the decimal point? How does the next zero in line make the number any more accurate?

[Bogie]

I just know I am going to regret asking this, but:

What is the point of adding an infinite number of zeros after the decimal point? How does the next zero in line make the number any more accurate?I too may regret responding but each zero simply lines up with each 9 in the infinite series as a means of demonstration that they are not the same all the way out to infinity of each series.

[Moose]

I too may regret responding but each zero simply lines up with each 9 in the infinite series as a means of demonstration that they are not the same all the way out to infinity of each series.

Bogie, have you read the thread yet? You need to, if only the first ten pages or so. Dozens of mathematical proofs and demonstrations for this property have been presented and repeated far too many times already. There's no reason to go through it all again.

[Neverfly]

Bogie, have you read the thread yet? You need to, if only the first ten pages or so. Dozens of mathematical proofs and demonstrations for this property have been presented and repeated far too many times already. There's no reason to go through it all again.

Bogie, you regretted it.

[Bogie]

Bogie, have you read the thread yet? You need to, if only the first ten pages or so. Dozens of mathematical proofs and demonstrations for this property have been presented and repeated far too many times already. There's no reason to go through it all again.Thank you Moose. Now I'm embarassed to say I did real quite a bit of it and saw the proofs and still think they are different.

But like you say, there is no reason to go through it all again. I'll just say that approaching a limit is not the same as reacing it, and if that is proved wrong then just chalk it up as another thing on my list :D.

[speedfreek]

Dare I ask what the "correct" answer to the OP is, and why people here are so depressed by the results of the poll?

EDIT: Just saw the answers above - don't worry about it!

[Bogie]

Bogie, you regretted it.:eh:Did you have to point that out?:mad:

[Moose]

It's proved wrong, I'm afraid. The whole point of limit theory (and ultimately calculus) is to let you compute slopes and areas and complicated whatnot that simply cannot be computed iteratively, which is where you appear to be stuck.

While it's true you can't reach 0 by _stringing_ 9s after a decimal, limit theory lets you skip all that and go straight to an infinite number of 9s. And when there are an infinite number of 9s after the decimal, they are equal. It's not intuitive, but then, it's not really supposed to be intuitive.

Integration depends entirely on limit theory working as advertised. If this property of 1 fails, so does higher math. And engineering. And we know engineers can build stuff.

[Neverfly]

:eh:Did you have to point that out?:mad:

Hey, I'm just spamming at this point... I haven't read the thread myself!

[Bogie]

It's proved wrong, I'm afraid. The whole point of limit theory (and ultimately calculus) is to let you compute slopes and areas and complicated whatnot that simply cannot be computed iteratively, which is where you appear to be stuck.

While it's true you can't reach 0 by _stringing_ 9s after a decimal, limit theory lets you skip all that and go straight to an infinite number of 9s. And when there are an infinite number of 9s after the decimal, they are equal. It's not intuitive, but then, it's not really supposed to be intuitive.

Integration depends entirely on limit theory working as advertised. If this property of 1 fails, so does higher math. And engineering. And we know engineers can build stuff.OK, thanks Moose. I'll just have to stay wrong on this one.

The OP said "exactly" and if iterations are not involved, and if buildings and bridges are at stake, I'll just let that last tiny iteration drown in the infinity of the preceding iterations.

[Moose]

I'll just let that last tiny iteration drown in the infinity of the preceding iterations.

That's the entire point, Bogie. There is no such thing as a last iteration in an infinite series. None whatsoever. If there was a last iteration, it wouldn't be an infinite series at all, and they'd be different values.

[Bogie]

That's the entire point, Bogie. There is no such thing as a last iteration in an infinite series. None whatsoever. If there was a last iteration, it wouldn't be an infinite series at all, and they'd be different values.:) That was my point too. An infinite series is never completed and so never reaches the limit. But I will give. I'm just wrong on this. It is embarrassing. I can't help it.

[Neverfly]

1662 posts in one thread discussing this issue... Dunno if there really is anything to be embarrassed about!:p

That's why I'm just trolling- so that I'll be a pest instead of a fool;)

[Bogie]

1662 posts in one thread discussing this issue... Dunno if there really is anything to be embarrassed about!:p

That's why I'm just trolling- so that I'll be a pest instead of a fool;)And thanks again! :sad:

[tdvance]

There are an infinite number of infinities. One of them is .999 ad infinitum. Another one is 1.000 ad infinitum. They are not the same.

There are an infinite number of infinities, but that has nothing to do with 1.000... and .999.... being the same real number.

[tdvance]

There seems to be a lot of Achilles and the Tortoise type arguments here--so I'll put it this way--if 1.000.... and 0.9999.... are different numbers, then Achilles can never catch up with and pass the tortoise.

[Bogie]

There seems to be a lot of Achilles and the Tortoise type arguments here--so I'll put it this way--if 1.000.... and 0.9999.... are different numbers, then Achilles can never catch up with and pass the tortoise.Go Tortoise.

[HenrikOlsen]

I still prefer the very simple
1/9=0.11111111. . .
1/9=0.22222222. . .
.
8/9=0.88888888. . .
9/9=0.99999999. . .

9/9=1
QED

[Moose]

That's the one that ultimately convinced me, Henrik.

[Bogie]

But 9/9 would never be anything but 1. It is a gimmick to go from 8/9=.888... and then to 9/9=.999... . 9/9 just never equals .999... in my feeble brain.

I concede to the consensus and to the rules employed by calculus. If within the discipline of calculus there is a rule that says that by referring to an infinite series of .999 … we can say that it equals 1, I accept that as a rule.

:D Personally and from a view of infinity that says anything infinite is endless, that means to me that an infinite series of .999s is endless. I take the personal position that endlessly approaching the limit is not the same as being at the limit and so I prefer to be wrong and keep my personal view that 1 is not exactly the same as the infinite series .999… .

[SeanF]

But 9/9 would never be anything but 1. It is a gimmick to go from 8/9=.888... and then to 9/9=.999... . 9/9 just never equals .999... in my feeble brain.
Would you argue that .888... plus .111... is not equal to .999...?

[Kaptain K]

Originally Posted by Bogie
But 9/9 would never be anything but 1. It is a gimmick to go from 8/9=.888... and then to 9/9=.999... . 9/9 just never equals .999... in my feeble brain.
Let me get this straight:

You have no problem with one ninth equaling a decimal point followed an infinite string of ones (1/9=.111...). Correct?

Nor with 2/9=.222...

You are cool all the way up through 8/9=.888...

Yet some how you balk at 9/9=.999...=1

Which part do you disagree with, 9/9=.999... or 9/9=1?

[Bogie]

Would you argue that .888... plus .111... is not equal to .999...?No, it is equal to .999... .

Let me get this straight:

You have no problem with one ninth equaling a decimal point followed an infinite string of ones (1/9=.111...). Correct?Correct.

Nor with 2/9=.222...Correct.


You are cool all the way up through 8/9=.888...Correct.


Yet some how you balk at 9/9=.999...=1Exactly.


Which part do you disagree with, 9/9=.999... or 9/9=1?You're kidding, right? How do you get .999... from 9/9?

I concede to the consensus and to the rules employed by calculus. If within the discipline of calculus there is a rule that says that by referring to an infinite series of .999 … we can say that it equals 1, I accept that as a rule.

Are you saying there is such a rule?

[HenrikOlsen]

You say that you're ok with 8/9=.888...
You say that you're ok with 1/9=.111...
You say that you're ok with .888... plus .111... is equal to .999...

Are you saying you're not ok with 1/9 + 8/9 = 9/9 ?

[Kaptain K]

How do you get .999... from 9/9?
The same way I got .888... from 8/9 which you had no problem with!

[Bogie]

You say that you're ok with 8/9=.888...
You say that you're ok with 1/9=.111...
You say that you're ok with .888... plus .111... is equal to .999...

Are you saying you're not ok with 1/9 + 8/9 = 9/9 ?No, I'm OK with that too.

Keep going and you will see where our views differ.

I am will to concede to the consensus and to the rules employed by calculus. If within the discipline of calculus there is a rule that says that by referring to an infinite series of .999 … we can say that it equals 1, I accept that as a rule.

Are you saying there is such a rule?

[Moose]

It's not a rule, Bogie, it simply is. Your intuition is misleading you. Look:

1) .1111~ + .8888~ = .9999~ (You've conceded this.)

2) .1111~ + 8/9 = .9999~ (You've conceded this.)

3) 1/9 + 8/9 = .9999~ (You've conceded this too.)

4) 9/9 = .9999~ (Which you appear to not be conceding.)

5) 9/9 = 1 (Which you haven't explicitly said one way or another)

6) 1 = 9/9 = .9999~ (Which you're denying.)

Ultimately, you appear to be denying that 1/9 and 8/9 = 9/9, because you've already conceded every intermediate step except the one that leads from 3) to 4), and the inescapable conclusion, all using elementary school arithmetic.

Bogie, .999~ must equal 1, and while it is the very cornerstone of limit theory, none of the proofs require limit theory or some special made up rule. At worse, we're talking algebra. At best, 6th grade fractions and decimals.

[Chuck]

Add 0.999... to 1.0 giving 1.0 + 0.999... = 1.999... Divide both sides by 2 giving (1.0 + 0.999...) ÷ 2 = 0.999... Multiply both sides by 2 giving 1.0 + 0.999... = 2 × 0.999... Subtract 0.999... from both sides giving 1.0 = 0.999...

The real reason that 0.999... = 1.0 is because that's now mathematicians use the notation. Anyone who disagrees has changed the subject by introducing his own definition.

[HenrikOlsen]

Right, I'll keep going, in small steps, you can point out the one you don't like.

(1) 8/9=.888...
(2) 1/9=.111...
(3) .888... + .111... = .999...
(4) 1/9 + 8/9 = 9/9
(5) 9/9 = 1
these you already accepted in various posts.

Combine (1) and (3) to get
(6) 8/9 + .111... = .999...

Combine (2) and (6) to get
(7) 8/9 + 1/9 = .999...

Combine (4) and (7) to get
(8) 9/9 = .999...

Combine (5) and (8) to get
(9) 1 = .999...

Which step do you think doesn't hold?

[Bogie]

It's not a rule, Bogie, it simply is. Your intuition is misleading you. Look:

1) .1111~ + .8888~ = .9999~ (You've conceded this.)

2) 0.1111~ + 8/9 = .9999~ (You've conceded this.)

3) 1/9 + 8/9 = .9999~ (You've conceded this too.)

4) 9/9 = .9999~ (Which you appear to not be conceding.)Right, because 9/9 = 1


5) 9/9 = 1 (Which you haven't explicitly said one way or another)I agree with that.


6) 1 = 9/9 = .9999~ (Which you're denying.)Correct.


Ultimately, you appear to be denying that 1/9 and 8/9 = 9/9, because you've already conceded every intermediate step except the one that leads from 3) to 4), and the inescapable conclusion, all using elementary school arithmetic. Because given two possible answers you must decide which is correct. I simply decide that the simplest choice is correct, i.e. 9/9 = 1, not .999... .


Bogie, .999~ must equal 1, and while it is the very cornerstone of limit theory.But it is not a rule?
..., none of the proofs require limit theory. At worse, we're talking algebra. At best, 6th grade fractions and decimals.My point puts me in the "wrong" cateagory. Darn, and I hate being wrong. Oh well, :).

Here is the thing, whether in math or science, if you have two possibilities to decide between, choosing the simplest one makes the most sense to me.

The choice that you present, 1= 9/9 = .999~ is two possible outcomes to choose from. To me 9/9 =1, not .999~. Thanks for letting me be wrong on this.

[Bogie]

Right, I'll keep going, in small steps, you can point out the one you don't like.

(1) 8/9=.888...
(2) 1/9=.111...
(3) .888... + .111... = .999...
(4) 1/9 + 8/9 = 9/9
(5) 9/9 = 1
these you already accepted in various posts.

Combine (1) and (3) to get
(6) 8/9 + .111... = .999...

Combine (2) and (6) to get
(7) 8/9 + 1/9 = .999...

Combine (4) and (7) to get
(8) 9/9 = .999...

Combine (5) and (8) to get
(9) 1 = .999...

Which step do you think doesn't hold?Step (8).

Going from step 7 to step 8 is where the gimmick creeps in. In step seven you added .888~ to .111~ and get .999~ and I agree because there is only one choice.

At step 8 there are two choices, .111~ + .888~ = .999~ and .111~ + .888~ = 9/9. Though .111~ + .888~ does equal .999~, 9/9 does not equal .999~ it equals 1 if you choose the simplest of the two possibilities.

[Moose]

Here is the thing, whether in math or science, if you have two possibilities to decide between, choosing the simplest one makes the most sense to me.

No. I haven't given you two possibilities at all. I've given you one. Only one.

Henrik's example is even clearer than mine.

1/9 + 8/9 = 9/9.
9/9 = 1
1/9 = .1111~
8/9 = .8888~
.1111~ + .8888~ = .9999~

All of these are true, and you've conceded every single one of these.

But because of this, and because math works, then .9999~ = 9/9 = 1. They are absolutely equivalent. Math is an all or nothing proposition. It's either entirely right, or entirely wrong. You cannot pick and choose which equalities you like and which ones you don't. Equal is equal.

What you're doing, ultimately, is committing the "argument by personal incredulity" logical fallacy. This is the exact same logical fallacy that creationists commit when they ignore the contrary evidence and insist that the world is 6000 years old.

[Disinfo Agent]

You're kidding, right? How do you get .999... from 9/9?How about using the long division algorithm (http://www.mathpath.org/Algor/algor.long.div.htm) to confirm it?

[Kaptain K]

The choice that you present, 1= 9/9 = .999~ is two possible outcomes to choose from.
It is not two possible outcomes! It is an equality!

1 equals 9/9 equals .999

Just like 2+2 equals 4 equals 2x2.

[Bogie]

Add 0.999... to 1.0 giving 1.0 + 0.999... = 1.999... Divide both sides by 2 giving (1.0 + 0.999...) ÷ 2 = 0.999... Multiply both sides by 2 giving 1.0 + 0.999... = 2 × 0.999... Subtract 0.999... from both sides giving 1.0 = 0.999...

The real reason that 0.999... = 1.0 is because that's now mathematicians use the notation. Anyone who disagrees has changed the subject by introducing his own definition.Obviously I'm not a methematician :). Let the feeding frenzy end.

[SeanF]

In step seven you added .888~ to .111~ and get .999~ and I agree because there is only one choice.
Step (7) does not use .888~ and .111~, it uses 8/9 and 1/9.

[Chuck]

Obviously I'm not a methematician :). Let the feeding frenzy end.

No! This can't end until everyone agrees that they're equal.

[Bogie]

Step (7) does not use .888~ and .111~, it uses 8/9 and 1/9.Oops, sorry. Then step seven has two choices.

8/9 plus 1/9 equals 9/9 or .999~ and I choose 9/9 which equals 1.

[Moose]

You know, I've never liked Pi. I'm summarily redefining it.

Oh, not to an integer, that's too easy. I just think we don't need so many named constants. From today onward, I declare that Pi = e = Avogadro's number to be equivalent and interchangeable. And before you ask, I haven't yet decided about the gravitational constant and the speed of light, but pencil it into your textbooks and reality will give full credit either way.

[Bogie]

No! This can't end until everyone agrees that they're equal.What if I die before I agree. Problem solved, right :p.

[Moose]

What if I die before I agree. Problem solved, right :p.

Might just happen if someone tries to build something using your selective math. (I'm joking, but not very much.)

[Chuck]

What if I die before I agree. Problem solved, right :p.
I'm sure we can find someone to take over for you.

[Bogie]

You know, I've never liked Pi. I'm summarily redefining it.

Oh, not to an integer, that's too easy. I just think we don't need so many named constants. From today onward, I declare that Pi = e = Avogadro's number to be equivalent and interchangeable. And before you ask, I haven't yet decided about the gravitational constant and the speed of light, but pencil it into your textbooks and reality will give full credit either way.OK, good idea. And why don't we insist that all people over 6' marry only people under 5' so genetics will eventually make us all between 5' and 6'. We could eliminate plus and minus sizes.

[Disinfo Agent]

I'm sure we can find someone to take over for you.I noticed that Bogie has already cast his vote in the poll. I guess his mind is made up.

[Bogie]

Might just happen if someone tries to build something using your selective math. (I'm joking, but not very much.)I know.



I'm sure we can find someone to take over for you.But not as much fun I bet :dance:.

[Bogie]

I noticed that Bogie has already cast his vote in the poll. I guess his mind is made up.And not only that but I'm willing to be wrong on this.

[Bogie]

Off topic: I wish half of you had taken half the effort you've take here to argue with me about my current ATM thread :D.

[Disinfo Agent]

And not only that but I'm willing to be wrong on this.What's the point? That won't improve our ratings now... :p


Off topic: I wish half of you had taken half the effort you've take here to argue with me about my current ATM threadYou're ATM here, too.

[Moose]

And not only that but I'm willing to be wrong on this.

You're wrong, you admit you're wrong, you know why you're wrong, and yet you're camping out on incorrect elementary-school arithmetic. While adults routinely camp out on blatantly untenable positions, there's got to be a personal stake in there somewhere. Did Arithmetic cheat on your sister or something?

Are you trolling us?

[Bogie]

You're wrong, you admit you're wrong, you know why you're wrong, and yet you're camping out on incorrect elementary-school arithmetic. While adults routinely camp out on blatantly untenable positions, there's got to be a personal stake in there somewhere. Did Arithmetic cheat on your sister or something?

Are you trolling us?No. I'm really sure that 9/9 = 1 and not .9999~. Sorry. Say what you want about me but personal attacks are against the rules :).

[Moose]

So is trolling. In any case, nothing at all in what I said was a personal attack. Your insistence on holding a position you know is wrong makes no sense at all. I'm trying to figure out what's motivating you to do this.

[Bogie]

So is trolling. In any case, nothing at all in what I said was a personal attack. Your insistence on holding a position you know is wrong makes no sense at all. I'm trying to figure out what's motivating you to do this.I didn't mean to imply that you had said anything that was a personal attack. I used a smiley. Do you have a tendancy to jump to conclusions and if so might your view of 9/9 be too hasty (just kidding Moose, a sense of humor is totally wasted sometimes, IMHO)?

[hhEb09'1]

Bogie, what's 0.444... divided by 2?

[worzel]

Step (8).

Going from step 7 to step 8 is where the gimmick creeps in. In step seven you added .888~ to .111~ and get .999~ and I agree because there is only one choice.

At step 8 there are two choices, .111~ + .888~ = .999~ and .111~ + .888~ = 9/9. Though .111~ + .888~ does equal .999~, 9/9 does not equal .999~ it equals 1 if you choose the simplest of the two possibilities.
Step (8) is a necessary consequence of step (4) and step (7).



(4) 1/9 + 8/9 = 9/9

(7) 8/9 + 1/9 = .999...

Combine (4) and (7) to get
(8) 9/9 = .999...

To disagree with it is to either claim that

8/9 + 1/9 = 1/9 + 8/9 is false,

or that at least one of steps (4) and (7) are incorrect.

So which is it?

[Bogie]

Bogie, what's 0.444... divided by 2?I appreciate the efforts to help me grow my understanding and I'm sure that my answer of .222... will lead to another step or two that would convince someone of normal competence that 9/9 equals .999~. But I will just have to be deemed wrong on this and add it to the long list of misconceptions about reality that I must be harboring :).

[SeanF]

Oops, sorry. Then step seven has two choices.
It's not a choice, it's a statement.

8/9+1/9=.999...

Do you disagree with that statement? Note that there's no 9/9 anywhere in it, so if you have a problem with 9/9, it's not in this step.

[Bogie]

Step (8) is a necessary consequence of step (4) and step (7).


To disagree with it is to either claim that

8/9 + 1/9 = 1/9 + 8/9 is false,

or that at least one of steps (4) and (7) are incorrect.

So which is it?I have replied to this already. Review the thread after the post you quoted and let me know if you see my reply.

[Bogie]

It's not a choice, it's a statement.

8/9+1/9=.999...

Do you disagree with that statement? Note that there's no 9/9 anywhere in it, so if you have a problem with 9/9, it's not in this step.I disagree then. 8/9 + 1/9 can be 9/9 so there is the choice as to whether it is .999~ as your statement says or if it is equal to 9/9.

Given two choices I choose the simplest one, i.e. 9/9 which to me = 1.

[worzel]

I have replied to this already. Review the thread after the post you quoted and let me know if you see my reply.
I did already. Saying that it is a choice is a non-answer. You said step 8. If only step 8 is incorrect then you are, in effect, saying that

8/9 + 1/9 does not equal 1/9 + 8/9

Is that really your answer?

[Moose]

Do you have a tendancy to jump to conclusions

As much as anybody else does (I can't remember how early I voted, but I had to see the arithmetic proof to understand why I was mistaken about this). I'm certainly not married to my intuition.

My intuition doesn't agree with the Monty Hall problem, for example, but the math doesn't give a flying rat's fig how well I understand it. The math works, I can demonstrate it accurately, I just don't 'get' it yet. I can either face reality and keep trying to figure it out, or be obstinate about it. I don't consider the latter option to be worthy of my time.

Where I'm having a problem with this discussion is that I dislike obstinacy only slightly less than I dislike repeating myself, especially when it takes on unreasonable proportions like this thread. I dislike when people try to "agree to disagree" with math rather than understand how we came to stand on the shoulders of thousands of years of math giants.

The fact that I can't seem to resist having my time wasted on a necrophilic thread when I could be doing something useful (like translating dull documentation nobody's going to read in either language) isn't doing much for my temper today either.

[Bogie]

I did already. Saying that it is a choice is a non-answer. You said step 8. If only step 8 is incorrect then you are, in effect, saying that

8/9 + 1/9 does not equal 1/9 + 8/9

Is that really your answer?Here is the post I thought you missed.
http://www.bautforum.com/off-topic-babbling/14593-do-you-think-0-9999999-1-infinite-9s-57.html#post1229174

[SeanF]

I disagree then.
Okay.

Now, you had earlier said you agreed with statements (1), (2), and (3):

(1) 8/9=.888...
(2) 1/9=.111...
(3) .888... + .111... = .999...

If statement (1) is true, then can't we just substitute 8/9 for .888... in (3) without changing the validity of equation (3)? I mean, isn't that exactly what the "=" in statement (1) means?

[Bogie]

As much as anybody else does (I can't remember how early I voted, but I had to see the arithmetic proof to understand why I was mistaken about this). I'm certainly not married to my intuition.

My intuition doesn't agree with the Monty Hall problem, for example, but the math doesn't give a flying rat's fig how well I understand it. The math works, I can demonstrate it accurately, I just don't 'get' it yet. I can either face reality and keep trying to figure it out, or be obstinate about it. I don't consider the latter option to be worthy of my time.

Where I'm having a problem with this discussion is that I dislike obstinacy only slightly less than I dislike repeating myself, especially when it takes on unreasonable proportions like this thread. I dislike when people try to "agree to disagree" with math rather than understand how we came to stand on the shoulders of thousands of years of math giants.

The fact that I can't seem to resist having my time wasted on a necrophilic thread when I could be doing something useful (like translating dull documentation nobody's going to read in either language) isn't doing much for my temper today either.There probably is a sub-conscious reason that I refuse to say the 9/9 equals both .999~ and 1. I could worry about it or I could chose to have a mental block. I chose the mental block :).

I have problems with discussions like this too because the arguments that I made all seem to come to the final conclusion that to me 9/9 can be presented as = 1, or =.999~. In my feeble mind, regardless of the vast opposition, that is two different answers if only because one consists of the number 1, and the other doesn't.

I find 1 the simpler answer and even though that only makes sense to me, I choose what seems to be the simplest answer to me.

[Bogie]

Okay.

Now, you had earlier said you agreed with statements (1), (2), and (3):

(1) 8/9=.888...
(2) 1/9=.111...
(3) .888... + .111... = .999...

If statement (1) is true, then can't we just substitute 8/9 for .888... in (3) without changing the validity of equation (3)? Yes we can. But when we get to 8/9 + 1/9, and say it =.999~, my feeble mind says 8/9 + 1/9, simple fractions, equals 9/9 which reduces to 1, not .999~. Sorry but that is my problem.

[Moose]

Yes we can. But when we get to 8/9 + 1/9, and say it =.999~, my feeble mind says 8/9 + 1/9, simple fractions, equals 9/9 which reduces to 1, not .999~. Sorry but that is my problem.

Still trying to figure out your block here, but why do you think they can't be both?

[Bogie]

Still trying to figure out your block here, but why do you think they can't be both?Well doctor, of all my feeble arguments, I think it has to do with infinity not being attainable and so we can never reach the limit. I know that you don't see it that way.

[Nick Theodorakis]

I'm going to try a different approach (well, different than the last 6 or so pages; I'm not going to wade through 60 pages to see if someone else did it);

Bogie, do you agree that 0.333... equals 1/3?

And 3 time 1/3 equals 1?

So if you multiply 3 times 0.333... (or els add 0.333... three times) you will see that you get 0.999...

Nick

[Bogie]

I'm going to try a different approach (well, different than the last 6 or so pages; I'm not going to wade through 60 pages to see if someone else did it);

Bogie, do you agree that 0.333... equals 1/3?

And 3 time 1/3 equals 1?

So if you multiply 3 times 0.333... (or els add 0.333... three times) you will see that you get 0.999...

NickI see all of that.

Can you see that 1/3 + 1/3 + 1/3
= 9/9 and 9/9 reduces to 1?

OK. Now can you let me live with the problem that 3 time 0.333~ = 0.999~ is not as simple as reducing 9/9 to 1. I'm choosing the simplest of two methods of resolving the equations which is reducing 9/9 to 1.

[SeanF]

I have problems with discussions like this too because the arguments that I made all seem to come to the final conclusion that to me 9/9 can be presented as = 1, or =.999~. In my feeble mind, regardless of the vast opposition, that is two different answers if only because one consists of the number 1, and the other doesn't.
Wait, wait, wait. You believe that 1 and .999... are two different numbers but 9/9 is equal to both of them?

[Bogie]

Wait, wait, wait. You believe that 1 and .999... are two different numbers but 9/9 is equal to both of them?SeanF, why is it so important that you convince me.

Answer a question for me. Can infinity ever be attained. For example if space was infinite could you ever get to the end of it?

[Moose]

Well doctor, of all my feeble arguments, I think it has to do with infinity not being attainable and so we can never reach the limit. I know that you don't see it that way.

Forget the infinity part of it for a minute. It's a red herring and isn't relevant to the problem.

Here are the propositions you've agreed to in prior posts so far:

A: 0.9999~ = 0.1111~ + 0.8888~
B: 0.1111~ + 0.8888~ = 1/9 + 8/9
C: 1/9 + 8/9 = 9/9
D: 9/9 = 1

You've agreed to every single one of these. Right?

Formal logic says (and this is arguably the single most fundamental property of formal logic) A -> B -> C -> D is the same as saying A -> D. This property has been proven so many times, it's a bit worn around the edges.

Apply formal logic to our proposition and we get:

0.9999~ = 0.1111~ + 0.8888~ = 1/9 + 8/9 = 9/9 = 1

Commutativity, a fundamental property of addition, guarantees D', C', B', and A', and D' -> C' -> B' -> A' are all valid, as must be D' -> A'. In plain English, 0.9999~ must be equal to 1, and the reverse must be equally true.

[Tango Cat]

Wow, it took me a long time to get through this one.

Bogie, I agree with you that 1 is a simpler representation for 9/9 than 0.999~, just as 3/2 is a simpler representation for that number than 1,091,382/727,588. But just because one is simpler than the other, does not mean they must represent different numbers.

I will take a different spin than most of the people here who are arguing against you. The symbols 0.999~ and 1 mean the same thing, because that is the way they are defined in modern mathematics. I think many people are taking the perspective that numbers are real objects, whose properties can somehow be discovered. They aren't. Numbers are abstract quantities invented by humans, and they behave the way humans specify, subject only to the constraint of logical consistent. It would be nice also, tho, if numbers were to be useful for modeling real world phenomena.

So that's the standard mathematical answer. 0.999~ is equal to 1 because it is defined that way. Whether you can travel an infinite distance in the universe or not is irrelevant. These are creatures of the mind, and I can travel an infinite distance in my mind without any trouble at all.

Now, if the choice is just arbitrary, why didn't mathematicians define 0.999~ as being different than 1, as you would like to? For a long time, some of them wanted to. But careful analysis of the situation led them to realize that if they did define 0.999~ as different than one, then this new number couldn't possibly obey the rules of arithmetic. That is what most of the proofs offered here actually show - they don't show that your choice is wrong, they just show that a logical consequence of your choice is that your numbers don't obey standard arithmetic rules. And numbers that obey those rules have been vastly more successful in explaining real world phenomena than numbers that don't obey those rules.

So I say, you can choose 0.999~ to be different than one, or you can choose the standard rules of arithmetic. You cannot choose both, though - if you do, you believe in things that contradict each other. Modern mathematics has made the second choice. If you make the first choice, you are free to do so. However, you must throw away some basic arithmetical rules if you do so.

[Disinfo Agent]

Hello, Tango Cat, welcome to the forum. I must take issue with something you've written:


The symbols 0.999~ and 1 mean the same thing, because that is the way they are defined in modern mathematics.That's not accurate.

In modern mathematics, 0.999~ is defined as the limit of the sequence (0, 0.9, 0.99, 0.999,...) It can be proven that this limit equals 1.

This is an important point because it shows that mathematicians did not decide in advance that 0.999~ and 1 had to be the same. Their equality is a consequence of more general properties of the real numbers. That's why we can't just ignore it, and exchange it for a more intuitive convention (for some).

[Moose]

SeanF, why is it so important that you convince me.

Because we're all educators in our own way. For my part, it's unacceptable to me to "agree to disagree" on something so fundamental and immutable as arithmetic. I won't tacitly condone such a thing by my inaction.

Because I ultimately closed the door on far too many of my dreams because there was a time where I allowed myself to believe as you do, and nobody cared enough to take the time I needed to help me understand.

Because math is the most basic building block to science, and this block of yours is completely incompatible with what should be your desire to eventually progress your ATM work to mainstream acceptance. You have to fix this.

[Bogie]

These arguments about math, proofs, and logic are necessary obviously to establish the credibility of higher math. In the real world where math can only attempt to describe reality, not determine it, infinity cannot be acheived.

My position is that I speculate about reality and so I need to accept the reality of infinity; it cannot be reached.

The argument that you can mentally reach infinity is exactly what is necessary for a mathematician because math must tie out. If you can't use an infinite progression then math won't work. So I agree that under the rules of math with the proof that 1 and .999~ are equal, but I don't agree that in the real world that is true.

[Tango Cat]

Hello, Tango Cat, welcome to the forum.

Hello Disinfo Agent, thanks for your welcome.


I must take issue with something you've written:

That's not accurate.

In modern mathematics, 0.999~ is defined as the limit of the sequence (0, 0.9, 0.99, 0.999,...) It can be proven that this limit equals 1.

Maybe we are stuck on semantics, but I think that is entirely consistent with what I said (which for some reason doesn't get included when I try to quote you). The definition I refer to associates the symbol 0.999~ with the limit of the sequence of numbers you describe; you refer to this as a definition yourself. I agree with everything you say in the last paragraph, but to me, it sounds consistent with what I said.


This is an important point because it shows that mathematicians did not decide in advance that 0.999~ and 1 had to be the same. Their equality is a consequence of more general properties of the real numbers. That's why we can't just ignore it, and exchange it for a more intuitive convention (for some).

This also makes me think any disagreement we have is semantic, because I agree with this completely, and think it is completely consistent with what I said. In particular, the part that I put in bold. If you accept the properties of the real numbers (and the definition of 0.999~ as the limit), then you have no choice but to conclude that it is equal to one. If you define 0.999~ as something other than the limit of the series you described, then you can do that, but there is no reason to think this newly invented number is a real number, that obeys the properties real number obey; and as many people have shown, you can prove that it doesn't. So I say there are two choices - define 0.999~ as the limit, and get as a consequence all the properties of the real numbers, or define it in some other way, and get new numbers that don't have the properties of real numbers.

So I guess I don't see where we differ. We both seem to agree that definition of 0.999~ as the limit of 0.9, 0.99, 0.999 and so on, requires that it equals one. So you can have inequality, provided you define 0.999~ in some other way, and live with the consequences - a number that doesn't obey the laws of real numbers.

[Tango Cat]

The argument that you can mentally reach infinity is exactly what is necessary for a mathematician because math must tie out. If you can't use an infinite progression then math won't work. So I agree that under the rules of math with the proof that 1 and .999~ are equal, but I don't agree that in the real world that is true.

I take the opposite view. If you have a system of math in which 1 and 0.999~ are not equal, your math system doesn't describe the real world very well.

How do you conclude that 0.999~ is an infinity that cannot be reached? I just walked across a room to reach this computer. To get here, I had to cover 0.9 of the distance, then 0.99 of the distance, then 0.999 of the distance, then 0.9999 of the distance, and so on. This kind of infinity, I reach all the time, in the real world.

[Nick Theodorakis]

How do you feel about irrational numbers?

Nick

[Bogie]

I take the opposite view. If you have a system of math in which 1 and 0.999~ are not equal, your math system doesn't describe the real world very well.

How do you conclude that 0.999~ is an infinity that cannot be reached? I just walked across a room to reach this computer. To get here, I had to cover 0.9 of the distance, then 0.99 of the distance, then 0.999 of the distance, then 0.9999 of the distance, and so on. This kind of infinity, I reach all the time, in the real world.You didn't cover an infinite distance.

[Bogie]

How do you feel about irrational numbers?

NickI love them.

[Nick Theodorakis]

How do you feel about irrational numbers?

I love them.

Do you believe they exist in the "real world?"

Nick

[SeanF]

SeanF, why is it so important that you convince me.
I'm wondering why it's so important to you that you not be convinced. :)


Answer a question for me. Can infinity ever be attained.
In a certain context, this one for example, yes.

If you disagree with the idea that 0.999... equals 1 because of the infinite 9s, then you ought to also disagree with the idea that 0.333... equals 1/3.

[Disinfo Agent]

These arguments about math, proofs, and logic are necessary obviously to establish the credibility of higher math. In the real world where math can only attempt to describe reality, not determine it, infinity cannot be acheived.Infinity doesn't make as much difference as you think.

There are many ways to write real numbers: as fractions, as decimal expansions... or as continued fractions (http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/cfINTRO.html). One of the beauties of continued fractions is that every rational number can be expressed as a terminating continued fraction.

Find the continued fraction expansion for 0.999... What do you get?


If you accept the properties of the real numbers (and the definition of 0.999~ as the limit), then you have no choice but to conclude that it is equal to one. If you define 0.999~ as something other than the limit of the series you described, then you can do that, but there is no reason to think this newly invented number is a real number, that obeys the properties real number obey; and as many people have shown, you can prove that it doesn't. So I say there are two choices - define 0.999~ as the limit, and get as a consequence all the properties of the real numbers, or define it in some other way, and get new numbers that don't have the properties of real numbers.We are in agreement. :)

[Bogie]

Do you believe they exist in the "real world?"

NickNick, do you really care what I think about them or are you interesting in classifying me in regard to my mathematical acumen?

[Bogie]

Infinity doesn't make as much difference as you think.I agree we are dealing with some tiny increments after awhile aren't we. You do not indicate that you give much weight to my view about infinity, i.e. it is not attainable. And yet that is my logic to distinguish between math and reality. So what can I say to convince you that the difference is meaningful. Probably nothing, right.

[Bogie]

I'm wondering why it's so important to you that you not be convinced. :)


In a certain context, this one for example, yes.What about in my example?


If you disagree with the idea that 0.999... equals 1 because of the infinite 9s, then you ought to also disagree with the idea that 0.333... equals 1/3.Maybe, but not in my feeble thinking. 1/3 is .333~. Why would I not agree? 9/9 equals 1. You agree. 9/9 > than .999~ if you consider that in the real world an infinite series is never complete and so the limit is never reached. In math you can define .999~ equals 1 and it works fine for math, so we agree on what you insist on and not on what I insist on.

[Moose]

You agree. 9/9 > than .999~ if you consider that in the real world an infinite series is never complete and so the limit is never reached.

No. What you're describing is 0.9999...9 Not 0.9999~. Remember, I told you earlier that an infinite progression does not end. The non-existent last slice is infinitely small.

Here's where you're causing yourself grief, Bogie. You insist that infinity doesn't directly apply to the real world, and you're absolutely right. Math does not describe the real world. Physics describes the real world.

Math is a symbolic language. It's a tool that must be perfect and absolutely self-consistent at all times so that it may be used by other disciplines to model the real world.

I'm not splitting Hares here; the non-existent 'last slice' of that race is infinitely small and is run in an infinitely small amount of time. This is the point (and remember, points are infinitely small) where the Hare passes the Tortoise. A dead heat. 1.

Math uses symbolic constructs to let us understand that the Hare must pass the Tortoise. This is how, and why, limit theory and infinity both very much apply to the real world.

And one of the properties of mathematics, the perfect and self-consistent symbolic language, is that 1 = 0.9999~. It must. This strange property of one is required for self-consistency. And it's a good thing it is, because of how many things (including elementary school arithmetics) depend on it being true.

[Disinfo Agent]

So what can I say to convince you that the difference is meaningful. Probably nothing, right.I'm already well convinced that it isn't meaningful.

One way to talk about infinity in math by using the notion of a sequence. A sequence of real numbers can be defined as a map from the set N={0, 1, 2, ...} of the integers into some subset of the set R of real numbers. But actually we only need sequences of integers here. To represent all these numbers at once, we can use a formula that gives each of them. Examples:

an = 1/(n+1) represents the infinite sequence (1, 1/2, 1/3, 1/4, ...)

bn = 1/2n represents the infinite sequence (1, 1/2, 1/4, 1/8, ...); this one shows up in Zeno's paradoxes

cn = (n+1)2 represents (1, 4, 9, 16, ...); the sequence of the perfect squares

dn = 1+1/1+1/2+1/6+...+1/n! represents (1, 2, 2.5, 2.66..., 2.70833..., ...); this sequence converges to the number e;

and so on.

To keep track of the (possibly infinite) digits in a decimal expansion, we can always identify it with the sequence of digits. So:


the decimal expansion of 1/8 is given by the sequence of coefficients d0=0, d1=1, d2=2, d3=5, and with the remaining digits being zero, d4=d5=...=dn=0 for each integer n that follows. The value of the expansion is of course d0+d1 x 0.1+d2 x 0.01+...+dn x 0.1n+...=0.125.

the decimal expansion of 1/3 is no different; it's given by a sequence of coefficients such that d0=0, and d1=d2=d3=...=dn=3 for each integer n that follows. The value of the expansion is d0+d1 x 0.1+d2 x 0.01+...+dn x 0.1n+...=0+ 0.3+0.03+0.003+...=0.333...

for the expansion that interests us, 0.999..., it's the same. We can describe it as a sequence such that d0=0, and d1=d2=...=dn=...=9. To see why the value represented by this expansion, 0+9 x 0.1+9 x 0.01+...+9 x 0.1n+..., equals 1, see the first page of this thread (http://www.bautforum.com/off-topic-babbling/16961-two-reasons-why-999-1-a.html). :)

[Bogie]

No. What you're describing is 0.9999...9 Not 0.9999~. Remember, I told you earlier that an infinite progression does not end. The non-existent last slice is infinitely small. At least we agree that there is a distinction between 0.999...9 and 0.999~.

Where we part company is at the concept that something can be infinitely small.

I often challenge the BBT/GTR people who jump to the conclusion that the universe started as an infinitely dense zero volume point origin. BBT and GTR don't include that "infinitely small" origin.


Here's where you're causing yourself grief, Bogie. You insist that infinity doesn't directly apply to the real world, and you're absolutely right. Math does not describe the real world. Physics describes the real world.

Math is a symbolic language. It's a tool that must be perfect and absolutely self-consistent at all times so that it may be used by other disciplines to model the real world. Agreed, but ...


I'm not splitting Hares here; the non-existent 'last slice' of that race is infinitely small and is run in an infinitely small amount of time. This is the point (and remember, points are infinitely small) where the Hare passes the Tortoise. A dead heat. 1.That is fiction. Nothing can be infinitely small.


Math uses symbolic constructs to let us understand that the Hare must pass the Tortoise. This is how, and why, limit theory and infinity both very much apply to the real world.I would say it this way; Limit theory and infinity are both applied to the real world.


And one of the properties of mathematics, the perfect and self-consistent symbolic language, is that 1 = 0.9999~. It must. This strange property of one is required for self-consistency. And it's a good thing it is, because of how many things (including elementary school arithmetics) depend on it being true.That is a property of mathematics. We agree it must allow such a definition to achieve the self-consistency.

In the real world though, nothing can be infinitely small, IMHO.

[Bogie]

I'm already well convinced that it isn't meaningful.I'm convinced that if your are convinced it isn't meaningful then the discussion has no where to go.


One way to talk about infinity in math is the use the notion of a sequence. A sequence of real numbers can be defined as a map from the set N={0, 1, 2, ...} of the integers into some subset of the set R of real numbers. But actually we only need sequences of integers here. To represent all these numbers at once, we can use a formula that gives each of them. Examples:

an = 1/(n+1) represents the infinite sequence (1, 1/2, 1/3, 1/4, ...)

bn = 1/2n represents the infinite sequence (1, 1/2, 1/4, 1/8, ...); this one shows up in Zeno's paradoxes

cn = (n+1)2 represents (1, 4, 9, 16, ...); the sequence of the perfect squares

dn = 1+1/1+1/2+1/6+...+1/n! represents (1, 2, 2.5, 2.66..., 2.70833..., ...); this sequence converges to the number e;

and so on.

To keep track of the (possibly infinite) digits in a decimal expansion, we can always identify it with the sequence of digits. So:


the decimal expansion of 1/8 is given by the sequence of coefficients d0=0, d1=1, d2=2, d3=5, and with the remaining digits being zero, d4=d5=...=dn=0 for each integer n that follows. The value of the expansion is of course d0+d1 x 0.1+d2 x 0.01+...+dn x 0.1n+...=0.125.

the decimal expansion of 1/3 is no different; it's given by a sequence of coefficients such that d0=0, and d1=d2=d3=...=dn=3 for each integer n that follows. The value of the expansion is d0+d1 x 0.1+d2 x 0.01+...+dn x 0.1n+...=0.333...

for the expansion that interests us, 0.999..., it's the same. We can describe it as a sequence such that d0=0, and d1=d2=...=dn=...=9. To see why the value represented by this expansion, 0+9 x 0.1+9 x 0.01+...+9 x 0.1n+..., equals 1, see the first page of this thread. :)
I don't disagree that there is a way to deal with infinity in math. It is a construct for the reasons we have discussed; it must be defined mathematically in order for math to be credible. But math is used to model reality, it is not reality.

In reality, infinity cannot be achieved.

[Disinfo Agent]

You said earlier that mathematics is just a model, it doesn't represent anything real. If that is the case, then the number represented by 0.999~ is just as non-real as the number represented by 1, and mathematicians are free to do as they please with both of them. So what's stopping them from saying 1 and 0.999~ are the same number?
Surely not reality! According to you, reality is irrelevant for mathematics.

[Bogie]

You said earlier that mathematics is just a model, it doesn't represent anything real.I might have said that mathematics is used to model reality. In that sense I'm sure I didn't say it doesn't represent anything real.
If that is the case, then the number represented by 0.999~ is just as non-real as the number represented by 1, and mathematicians are as free to do as they please with both of them. So what's stopping them from saying 1 and 0.999~ are the same number?Nothing. They do say that. It is necessary for math to be credible and consistent, as I agreed.
Surely not reality, according to you!No, you misquote me.

[pghnative]

If I were a moderator, I'd ban kucharek for resurrecting this thread.;)

(Length of ban to be "for 99.99...% of a lifetime")

[grouchybastid]

I love the fact that this kind of discussion is the first thread I came across upon joining the forum. :D




In the real world though, nothing can be infinitely small, IMHO.

Once you start talking about subatomic physics, there are plenty of things that could be considered (in the classical sense) to be infinitely small. Electrons, for example...gotta admit those exist in the real world, seeing as how you're reading these words on a computer screen.

[Moose]

In the real world though, nothing can be infinitely small, IMHO.

Zero is infinitely small.

In any case, it doesn't matter that you can't make an infinitely small slice of a physical object using physical processes. That has no bearing whatsoever on math.

Limit theory says that as your series advances to an infinite number of terms, the difference between the result of your progression and the value it's approaching (the limit) also becomes infinitely small. This property lets you treat the (infinite) series as perfectly equivalent and utterly consistent with the limit itself.

It's what lets you plan the exact amount of chlorine you'll need to treat the water in a large and curvy pool at some future luxury hotel, as well as figuring out exactly how much concrete you'll need to pour at some future date.

It's real because it works in the real world, and it works symbolically because it's as real as it needs to be. And you really can't get away from that, no matter how much you'd prefer to not like it.

[Bogie]

I love the fact that this kind of discussion is the first thread I came across upon joining the forum. :D




Once you start talking about subatomic physics, there are plenty of things that could be considered (in the classical sense) to be infinitely small. Electrons, for example...gotta admit those exist in the real world, seeing as how you're reading these words on a computer screen.I'm pretty sure we won't be allowed to discuss this very long here, but if space is infinite and filled with energy that is infinitely fine, then electrons are huge amounts of energy; far from being infinitely small.

[Disinfo Agent]

I love the fact that this kind of discussion is the first thread I came across upon joining the forum. :DWelcome aboard. :)


But math is used to model reality, it is not reality.

In reality, infinity cannot be achieved.Our ideas are probably not as far apart as you might think. I'm the same guy who got into trouble a while ago for suggesting that physical infinities do not exist (http://www.bautforum.com/general-science/39964-how-often-every-day-do-you-encounter-infinities-physical-reality-kind.html). I still maintain that they have never been observed.

[Bogie]

Zero is infinitely small. Isn't that a mathematical concept :).


In any case, it doesn't matter that you can't make an infinitely small slice of a physical object using physical processes. That has no bearing whatsoever on math.True.


Limit theory says that as your series advances to an infinite number of terms, the difference between the result of your progression and the value it's approaching (the limit) also becomes infinitely small. This property lets you treat the (infinite) series as perfectly equivalent and utterly consistent with the limit itself.That is not being argued, but I would phrase it differently, "This property lets you treat the infinite series as equivalent and consistent with the limit itself."


It's what lets you plan the exact amount of chlorine you'll need to treat the water in a large and curvy pool at some future luxury hotel, as well as figuring out exactly how much concrete you'll need to pour at some future date.I don't think it is that decisive in calculations at that level, but I will defend your right to say so.


It's real because it works in the real world, and it works symbolically because it's as real as it needs to be. And you really can't get away from that, no matter how much you'd prefer to not like it.I like it. You are not exactly portraying our differences. Since we have different perspectives I wouldn't assume that you would go too far to give my view any credibility.

[Moose]

*shrug* It's meaningless, Bogie, 1 = 0.999~. You've been shown why. I'm not interested in getting into an ATM discussion with you.

[Bogie]

Welcome aboard. :)

Our ideas are probably not as far apart as you might think. I'm the same guy who got into trouble a while ago for suggesting that physical infinities do not exist (http://www.bautforum.com/general-science/39964-how-often-every-day-do-you-encounter-infinities-physical-reality-kind.html). I still maintain that they have never been observed.Yikes. Interesting. Maybe after I get agreement that math and reality are two different things I'll jump in over there (not):).