What if our numbering system was not based on 10?

[matt grime]

What you've said there is

"if <state something that is false and contradicts the notation> then <the notational assumption is contradicted>"

The last bit of that post refers to "the 9". Which of the (possibly infinitely many) nines are you referring to with the definite article? ("the 9" apparently is finite too...?)

I mean in 0.99... there is a number after the first 9, the second 9,... and so on.


Or perhaps you're just pointing out that if we take a terminating decimal (ie one with a "last" non-zero digit) then multiply by two, the answer must be a terminating decimal too. So...?

 

[Gamish]

Why must such a simple thing be so complicated? When I meant is that it is innacurate to say that a number can occur after an infinite number.

BTW, you express an infinite number by ()? like 1/3=.(3)?

 

[matt grime]

that is the convention that dextercioby explained to you. One may indicate repeated digits in many ways.

It is not inaccurate to say that a number can occur after an infinite number, it is just plain wrong when talking about decimals. These things are not complicated, and if you took the time to learn the things other people would explain to you rather than presuming to tell them they are wrong then you wouldn't make such mistakes nor would you try and explain the real numbers in layman terms to a professional mathematician.

 

[Alkatran]

The most common way to denote it on a computer is:
.999~ / 2 = .4999~

So, here's my challenge to you:

You say that there is a 5 at the end of the string of 9s for (.999~/2). I want you to write out the number. When you reach the 5, tell me.

 

[chroot]

The fundamental disconnect here is that Gamish is stuck to the notion that a system's behavior in approaching infinity should be the same as that system's behavior at infinity. A simple confusion of the limit with the value at the limit. A calculus course should clear that right up; in the meantime perhaps a few examples of functions with
"discontinuous limits" would be educational.

- Warren

 

[Manchot]

Yes, the ~ is a common way of denoting a repeating decimal on a computer. And yes, .9~/2 = .5. You can prove it algebraically this way:

x = .9~ (we are defining x to be .9~)
10x = 9.9~ (by moving over the decimal point)

Then, we can subtract Equation 1 from Equation 2.
10x - x = 9.9~ - .9~
9x = 9
x=1

Obviously, then, .9~/2 = 1/2 = 0.5.

The trap that you've fallen into is a very common one, which I think is behind the reasoning that some people have. You see, when most think of division, they think of it as an algorithm that they learned as a child rather than a function. What is that algorithm, you ask? Long divison. When dividing two numbers, you're not performing long division, you're just performing the division function. It's hard to explain, but do you see what I mean?

 

[Gamish]

OK, I understand it a lot more now. I am just confused with one remark from matt grime.

It is not inaccurate to say that a number can occur after an infinite number, it is just plain wrong when talking about decimals

You said that it can be accurate to have a number after infinity, but it is not accurate to have a number at the end of infinity when talking about decimals. What do you mean by that?

You say that there is a 5 at the end of the string of 9s for (.999~/2). I want you to write out the number. When you reach the 5, tell me.

My answer is kinda a joke, but I would say .49~5

 

[Curious3141]

OK, I understand it a lot more now. I am just confused with one remark from matt grime. You said that it can be accurate to have a number after infinity, but it is not accurate to have a number at the end of infinity when talking about decimals. What do you mean by that?

I don't know exactly what he was referencing, but my guess would be the theory of transfinite numbers (cardinals and ordinals). Read up on them.

 

[Alkatran]

OK, I understand it a lot more now. I am just confused with one remark from matt grime. You said that it can be accurate to have a number after infinity, but it is not accurate to have a number at the end of infinity when talking about decimals. What do you mean by that?



My answer is kinda a joke, but I would say .49~5

Alright, let's break the rules a bit and say that 5 is there.

The value it represents is 5*10^(-infinity) = 5/10^infinity = 5/infinity = 0

So it's entirely superfluous and might as well be another 9. (A bit of Occham's Razor I sees)

 

[Chronos]

Even an irrational number can be expressed as an infinite series of rational numbers.

 

[matt grime]

You said that it can be accurate to have a number after infinity, but it is not accurate to have a number at the end of infinity when talking about decimals. What do you mean by that?

No, I didn't say that. I said that saying it was "inaccurate" was not the best way of saying it, and that it is plain wrong to say that there is a number after infinity, though I can see how there may have been some confusion.

Crappy analogy: "it's inaccurate to say that a camel is a dodge charger" . Inaccurate is not a good word here as it indicates somehow there is some degree of accuracy involved when it is actually that you've just got the wrong idea entirely because you think they can both be modes of transport. Better to say "wrong" or even "idiotic". Litotes aside, of course.


So far no one else but you has talked about having something "at infinity" or "after infinity": mathematicians won't tend to say that since there is not, in this situation, any such place labelled "infinity".

There are for instance infinite ordinals where one can count "beyond infinity" of course the layman will be confused by this idea because he thinks that we actually mean you have to count as in count beans when you do maths. Witness the disbelief in Cantor's diagonal argument etc.

 

[Gamish]

So, I guess it was a miscommunication of words on your behalf. I was not trying to prove that a number can occur beyond infinity, I was trying to ask why not? With all this said, it would be wrong to say .9~5 meaning an infinite nuber of 9's to the right of the decimal, with a 5 at the end.




void main {if (pf==1)functions.goto.href("http:\\www.physicsforums.com");}
/*hehe*/

 

[DaveC426913]

As an outside observer, I implore the initial poster to recognize his/her responsibility in the confusion of this thread due to "miscommunication of words".

It would be very helpful to have a basic understanding of the vocabulary and notation of a subject, but it is not essential - if one is open to learning.

But to invent one's own numerical notation (and even words (lamen??)), and to then feign condescension when one thinks others have not explained it well enough - is quite puerile.

Does the poster want to learn, or does the poster want to argue?

 

[matt grime]

So, I guess it was a miscommunication of words on your behalf. I was not trying to prove that a number can occur beyond infinity, I was trying to ask why not? With all this said, it would be wrong to say .9~5 meaning an infinite nuber of 9's to the right of the decimal, with a 5 at the end. /

If you were going to claim it was a standard decimal representation of a real number, of course it would be wrong. It appears you've not bothered learning the meanings of the symbols involved: Real numbers ARE NOT decimals. Decimals are a useful model of the real numbers, that is all. The objects you are asking about are not in that model, that is all, that is what's wrong.

I can give different presentations (of the real numbers) where it is prefectly reasonable to talk about things positioned after an infinite number of terms, but we aren't dealing with them, are we?

I don't think you're in any position to claim anything about a mildly ambiguous sentence in English being a miscommunication.

 

[Gamish]

If you were going to claim it was a standard decimal representation of a real number, of course it would be wrong. It appears you've not bothered learning the meanings of the symbols involved: Real numbers ARE NOT decimals. Decimals are a useful model of the real numbers, that is all. The objects you are asking about are not in that model, that is all, that is what's wrong.

I can give different presentations (of the real numbers) where it is prefectly reasonable to talk about things positioned after an infinite number of terms, but we aren't dealing with them, are we?

I don't think you're in any position to claim anything about a mildly ambiguous sentence in English being a miscommunication.

Well, just because we all use the english language doesn't mean it we cannot have some sort of miscommunication. If you read the above post, there was a miscommunication of the work "inaccurate". I know what real numbers are, but that is rather irrelevant to the argument I was establishing which was the theory that a number can occur after an infinite decinmal. I guess it would be expressed as .9~5

But, this thread seems to be an attempt to disprove the validy of my question, which is rather simple.

 

[Alkatran]

You can't have a number at the end of an infinite series of number: there is no end. Period. By your logic I could:

.9~5~45454545~~~~~~~~~~~

Now what in the world would that mean?!

 

[BobG]

What if our numbering system was not based on 10? What if it was based in 12, or 99, who knows. Would math theoreticly still work just as our mathematical system of 10 seems to work flawlessly?

Thanks in advance

BTW, what is .9i/2? is it .49i5? i=infinite!

Yes, all the mathematical laws (commutative property, distributive, etc) would still apply, regardless of the base.

The Babylonians were pretty good mathematicians and they used a dual base system with base 60 as their primary base and 10 as a secondary base (in fact, this is where we got our 60 minute hours and 60 second minutes and our 60 minute degrees and 60 second minutes).

Base 5 numbering systems were most likely the first counting systems developed, followed by dual base 10-5 systems, and then other variations (Hindu and Chinese base 10 systems, Mayan base 20 systems, etc). Historically, the base of the numbering system has had little to do with the sophistication of the mathematics.

Base 5 was always a popular base because humans theoretically can perceive a maximum of 4 objects before they have to resort to counting. Only pre-school kids can be tested since knowing how to count biases the test results, so I'm not positive that's really the maximum number of objects an adult could perceive, but that idea is supported historically by the popularity of base 5 systems and the virtual non-existence of base 6 systems.

 

[StatusX]

Let me see if I can explain this both rigorously and simply. As you go down this list:

0.45
0.495
0.4995
0.49995
...

the numbers get closer and closer to 0.5 in a specific way. For any other number 'x', you can find a number 'e' such that all numbers on the list above a certain point are more than e away from x. For example, if x=0.51, you can pick e=0.009 and all numbers on the list are more than e away from x. If x=0.4999, you can pick e=0.00004, and all numbers on the list after the third are more than e away from x. But for x=0.5, there is no number e that will work. Go ahead, try to find it. Mathematicians say this means 0.5 is the "limit" of the series. And under decimal notation, the number 0.49...5 (although not usually defined, as others have mentioned) can be thought of as defined to be equal to the limit of the above series. So 0.499...5 = 0.5 in the same way 1/2 = 0.5. They are just different ways of writing the same number.

 

[matt grime]

Prove that if you allow this "new representation" into the decimals that the algebraic operations still work.

Can we just forget this 0.4999...5 garbage? At least forget it in the context of the real numbers.

Incidentally how can someone be familiar with the defintion of the reals (complete totally ordered field, unique up to canonical isomorphism) and think that 0.9i is a reasonanle notation for a recurring decimal, never mind even start such a thread as this.

 

[StatusX]

are you talking about my "new representation" matt? If so, I only mean it to show the real reason the 5 at the end is meaningless: it doesn't affect the limit. To you, someone familiar with infinity, it is obvious that there can be nothing at the end. But I thought if I took it back to the basics and showed that infinity is only defined with limits, it would be more clear. You could have 0.(34)(56)(78) if you wanted, and define it as the limit of:

0.345678
0.343456567878
0.343434565656787878

obviously nothing would matter after the first ellipses, but it shows why it doesn't matter.

 

[matt grime]

You've just introduced new and completely extraneous information into representatives of elements of an analytically constructed algebraic object. You don't feel just a little compelled to explain why it "works" in terms of the algebra and analysis of the reals? I don't think it does work, for what it's worth, and cannot see why you'd introduce the idea. I also don't agree with saying "infinity is defined with limits" since "infinity" is an ambiguous term that we ought to avoid using.

 

[StatusX]

I should have said infinitely repeating decimals. And as for algebraic rules: drop everything after the first repeating group and then proceed as normal.

So yes, there is extraneous information, and I certainly don't intend for anyone to use something like this for any practical or theoretical purpose. The only reason I mentioned 0.499...5 is because to a lot of people new to the subject, this seems intuitively to be half of 0.999..., and in a sense, they aren't completely wrong. If you think of 0.999... as the limit of the sequence {0.9, 0.99, 0.999,...}, then the sequence in my first post is obtained by halving each term. In a way, this sequence can be thought of as 0.499...5, and the limit of the sequence is 0.5. It is just another way of showing that the system is consistent; that whether you choose to use 0.999...=1 and half that to get 0.5 or do it the way I did, you get the same answer.

 

[matt grime]

"In a way, this sequence can be thought <WRONGLY> of as 0.499...5," my addition.

Why teach people bad habits and wrong maths?

 

[StatusX]

Ok this is my last try to get you to see what i was doing. Are you really suggesting there isn't an intuitive way in which the sequence {0.45, 0.495, 0.4995, ...} could be represented as 0.4999...5? Forget anything else, just think "representing the sequence." Of course there is. But when you see how the decimal notation is defined with limits, you see this is no different than 0.4999... (this is no more useful, by the way), which is no different than 0.5. Once they see that, they'll see why they should never use 0.4999...5 again, instead of just having to take a mathematicians word for it. But since no one seems to have benifited from it, forget I said anything.

 

[matt grime]

I see perfectly what you're doing, and it is wrong. Why teach something that is just plain wrong? It isn't intuitive to me since it creates new unexplained symbols in positions that clearly cause confusion. So don't teach them something that is wrong. If you wish to think of it in terms of limits of the finite examples then do it properly:

0.\underbrace{9 \ldots 9}_n
divided by two gives a number that differs from 0.5 by k*10^{-n} for some constant k, hence in the limit the difference shrinks to zero.

And then if your attitude is that they see why they shouldn't use it, why teach them to think in terms of it? Teach them the proper definitions and deductive reasoning and half the cranks who can't grasp these simple things wouldn't have a leg to stand on since no reasonably educated person would fail to see why they're wrong.

 

[StatusX]

You're right, I should have been more careful about it, and showed why their intuition of 0.499..5 is wrong without ever using that symbol myself. Instead, I should have just referred to the sequence and talked about it's limit. In any case, it seems all the confused people are gone, so this is all academic.

 

[ahrkron]

This seems a good point to close this incarnation of the .99... = 1 issue.
As usual, most probably it won't take long before it's reborn... oh, well.