Simple mathematical problem (?)

[Iamme]

DanishDynamite---Just ONE measely example. This is like talking about the theory of a thing called 'pi' and nobody can provide evidence that there is a use for it, like Pi X D = C.

 

[DanishDynamite]

DanishDynamite---Just ONE measely example. This is like talking about the theory of a thing called 'pi' and nobody can provide evidence that there is a use for it, like Pi X D = C.

Iamme, the list is endless.

Every single electronic apparatus (including your PC) that you have ever used, including the Internet, has been possible only because of the mathematics behind it. (Did you know that one of the first practical applications of imaginary numbers was within electronics?)

Newton's Laws.
Einstein's theories of relativity.
Quantum Mechanics

Seriously, Iamme, the list is endless.

 

[xouper]

Iamme: Xouper---I aplogize for getting you so steamed, about me and others not comprehending the concept.

Considering that you continue to ridicule that which you don't understand (mathematics in particluar), your apology seems hollow and insincere.

 

[patoco12]

Just ONE measely example. This is like talking about the theory of a thing called 'pi' and nobody can provide evidence that there is a use for it, like Pi X D = C.

I believe that black holes were discovered because we have a consistent model of mathematics which allows for infinite values.

Math is our "telescope" on the deep universe and the deep universe does weird stuff, so why should every mathematical fact be intuitive and immediately practical? That seems short sighted to me.

 

[Iamme]

Interesting guys. I hate to be gang-tackled, so I think I'll back off, and let some of the other nincompoops who were getting xouper to have steam come out his ears, to post. I don't want to make any enemies around here.

 

[Globert]

I can't believe I read the whole thread. My eyeballs need alka-seltzer. I believe 0.999...=1 as a fact. I believe.


IIRC any thing past 0.0000001 is beyond our capacity to measure directly with current technology.(what's the wavelength on a scan tunneling microscope BTW?)






Globe

 

[patoco12]

I believe 0.999...=1 as a fact. I believe.

You don't need to believe. It is a fact.

IIRC any thing past 0.0000001 is beyond our capacity to measure directly with current technology.(what's the wavelength on a scan tunneling microscope BTW?)

We don't need to measure anything to prove that .9~=1. We have Math, which is much stronger.

 

[69dodge]

Initially, I considered the ".999... = 1" question to be purely a matter of definition: ".999..." denotes, by definition, the limit of a certain sequence; since that limit is 1, the issue is settled. This is the "standard" way to approach the question, I think. But some people here seemed to be looking at the question differently, and I wondered if their alternate approach could also be made precise. Specifically, instead of asking what limit the sequence .9, .99, .999, . . . tends to, and then somewhat arbitrarily defining ".999..." to denote that limit, is there some way we can directly ascribe meaning to ".999... with its infinity of 9s already there"?

Here's what I came up with. Let's associate each digit of the decimal expansion with an appropriately-sized segment of the real number line: the first 9, in the tenths place, gets associated with the segment, of length .9, from 0 to .9; the next 9, in the hundredths place, gets associated with the adjacent segment, of length .09, from .9 to .99, and so on. Then, we take the union of all these segments (there are an infinite number of them) and ask for the length of the result.

The answer is 1. But there's an interesting little twist. Regardless of which endpoints we choose to include in each of the individual segments (left only, right only, neither, both), the union does not contain its right endpoint, at 1. Of course, the open interval (0, 1) has length 1, just as the closed interval [0, 1] does, because a single point has zero length. But I wonder if the feeling some people have, that .999... doesn't quite equal 1, might be related to the fact that the union doesn't include the point at 1, though its length is 1.

 

[Globert]

You don't need to believe. It is a fact.

We don't need to measure anything to prove that .9~=1. We have Math, which is much stronger.

Lighten up . It is a fact.
It would take a while to measure directly. Nor did I suggest one try to. I was inquiring how small we actually could see.

 

[xouper]

Globert: It would take a while to measure directly.

I assume you know that this equality (.999... = 1) is a mathematical abstraction and thus is not amenable to empirical measurement. If you wish to change the subject, I was going to suggest starting a new thread, but I see you already did that here.

 

[geni]

Ok lets keep this simple


0.9999...=1

I'm sory but does anyone else have any sugestions for putting this thread to sleep?

 

[LFTKBS]

Ok lets keep this simple

0.9999...=1

I'm sory but does anyone else have any sugestions for putting this thread to sleep?

1) Let's stop posting here.
2) Enyone who still doesn't think that 0.9~ = 1, just stop, stop, stop. You are wrong, and that's okay. But we're not making it up as we go along here. Everything is under control. Now stop asking.
3) See 1.

 

[Suggestologist]

You don't need to believe. It is a fact.

We don't need to measure anything to prove that .9~=1. We have Math, which is much stronger.

Math is a human construct, it has no absolute reality beyond what people can empirically apply it to. The universe is not continuous -- infinities are nonsensical when you look at the real world. In any real system, there is a point at which you cannot move any closer to a position, without actually being at that position.

Think about the quantum mechanics metaphor which goes: You have a full cup of water. Take half the water out of a cup of water and continue removing half of the remainder. Eventually you will only have one water molecule left. You can split up the single molecule into it's atoms and subatomic particles -- but if you remove half of these particles -- then at this point, it is no longer water. You no longer have any water in the cup.

Similarly, there is a point before which you reach a position (1) at which you cannot move closer without reaching it: 0.999.... seems like the correct representation for this point. I'm not saying you cannot believe that .99999... = 1 and that such a belief can be useful for some mathematics. And one would have to appeal to raw dogmatic obsinance to do anything other than acknowledge the validity of other views about what 0.9999... means which do have utility in mathematics.

In any real system, you will have a finite number of digits for the real answer. Thus, it must be the case that having a certain finite number of 9's after the decimal point may as well equal 1 -- due to quantization -- in a real physical system. What percentage of water can you remove from the cup -- before you cannot remove any more without removing it all? This is the analogue of the distinguishability of 0.9999.... from 1 in the real number system.

 

[Walter Wayne]

Math is a human construct, it has no absolute reality beyond what people can empirically apply it to.

So if you have an apple in a basket, and put another apple in there, how many apples are in the basket?

The universe is not continuous -- infinities are nonsensical when you look at the real world. In any real system, there is a point at which you cannot move any closer to a position, without actually being at that position.

Universe is not continuous? Where is the paper in which experiments showed that space-time was quantized.

Infinities are nonsensical when you look at the real world?
If move my hand a 1 m/s towards a wall 1 m away, how many times do I half the distance between my hand and the wall consecutively in half a second? Once. In three-quarters of a second, I half it twice consecutively. In seven-eighths I half thrice consecutively. In one second how times has that distant been halved, an infinite number. For any time less that a second I halved it a finite number. Which means we have a physical situation where 0.5+0.25+0.125... = 1. Until such time as space-time is no longer considered continuous, infinities make perfect sense.

Walt

 

[xouper]

Suggestologist: ... blah blah blah ... What percentage of water can you remove from the cup -- before you cannot remove any more without removing it all? This is the analogue of the distinguishability of 0.9999.... from 1 in the real number system.

Regarding this latest nonsense from Suggestologist, I have a question for the gallery - We can lead the horse to water, but apparently we can't make him drink. At what point do we stop trying, and instead lead him to the glue factory?

 

[Skeptoid]

None so blind as those who will not see.

Perhaps we could ask Hal to retitle this thread to "The 0.999... = 1 Affair" and let it run to 44 pages.

 

[BillyJoe]

The last thread on this topic ran to only two pages. I'm not sure what this says about the relative IQs of the participants

 

[LuxFerum]

We can lead the horse to water, but apparently we can't make him drink. At what point do we stop trying, and instead lead him to the glue factory?

Yes we can, it takes more time, maybe some use of force, but it is possible.

 

[slimshady2357]

Math is a human construct

Ok, ya. I'm fine with that. In fact it's a whole whack load of different human constructions.

One such construction is the Real number system. It is well defined.

Similarly, there is a point before which you reach a position (1) at which you cannot move closer without reaching it

This is not true for the Real number system. It is well defined. This is a fact about the human construction called the Reals.

I'm not saying you cannot believe that .99999... = 1 and that such a belief can be useful for some mathematics.

No, what you're saying is just plain wrong, it contradicts the human created DEFINITION of the notation for the Reals.

And one would have to appeal to raw dogmatic obsinance to do anything other than acknowledge the validity of other views about what 0.9999... means which do have utility in mathematics.

Your NOTATION is WRONG! AGH!

But I understand what you mean. I agree that we should all acknowledge different number systems (such as the Surreals). HOWEVER, they use a different NOTATION and do not represent the types of numbers you are thinking of in DECIMAL notation. Getting it yet?

This is the analogue of the distinguishability of 0.9999.... from 1 in the real number system.

No, it might be the analogue of a number in the Surreal number system, but the Reals are well defined and the notation is well defined and quite simply 0.999... = 1.

Once you accept that you are using the wrong notation to talk about numbers which are not part of the Reals, maybe we can talk to each other, instead of past each other.

Adam

 

[Suggestologist]

Ok, ya. I'm fine with that. In fact it's a whole whack load of different human constructions.

One such construction is the Real number system. It is well defined.



This is not true for the Real number system. It is well defined. This is a fact about the human construction called the Reals.



No, what you're saying is just plain wrong, it contradicts the human created DEFINITION of the notation for the Reals.



Your NOTATION is WRONG! AGH!

But I understand what you mean. I agree that we should all acknowledge different number systems (such as the Surreals). HOWEVER, they use a different NOTATION and do not represent the types of numbers you are thinking of in DECIMAL notation. Getting it yet?



No, it might be the analogue of a number in the Surreal number system, but the Reals are well defined and the notation is well defined and quite simply 0.999... = 1.

Once you accept that you are using the wrong notation to talk about numbers which are not part of the Reals, maybe we can talk to each other, instead of past each other.

Adam

How can you argue from definition, when definition contradicts actual usability?