326 replies to this topic

[m4x]

Yeah, you're right it's all gettin' pointless.

[genesis]

Alright, everyone just consider this:

Does (1 + 1/n)^n ever equal e (2.718...)? No, as n approaches infinity (1 + 1/n)^n tends towards e, by definition. There is no argument, it is mathematically defined that way. Its like saying 0! = 1, it is an agreed mathematical result.

Now, how do you construct 0.999...? You start of with 0.9 and then keep adding 9's onto the end. So, 0.999... = 9/10^1 + 9/10^2 + ... + 9/10^n. As n approaches infinity, 0.999... tends towards 1 (as above), but NEVER touches it BY DEFINITION. Voila! Now someone is going to say 'what if n IS infinity' and we have to do this all over again. But listen up, what if I told you (like for 1/n, when n = 0) that n = infinity is undefined?

And now for the moment of truth, do you see the symmetry between 1/0 = infinity and 1/infinity = 0? Here's the punchline: they are both undefined! Back to definitions, infinity and undefined are synonymous!

[Overlord]

0,999.... = sum for n=1 to +inf of (9/10)^n = by definition lim k->inf sum for n=1 to k of the same thing (if the limit exists)
the value of a seria is defined as the limit of the partial sums, if it's exists

same as integral from 0 to +inf is equal by definition to limit (k->inf) of integral from 0 to k, if it exists.
hake's theorem : the KH-integral from a to b of f(x) dx exists iff lim c->a KH-integral from c to b of f(x) dx, and if it exists, then the integ from a to b is equal to the lim of the integ from c to b

[genesis]

You're right, I see the flaw in my argument...

"If two cyclists, 2 km apart, ride towards each other at 60 km/h and a fly, starting on one, flies back and forth between the two at 120 km/h, how far does the fly travel before the cyclists meet?"

The distance travelled is an infinite sum, but intuitively, the cyclists meet in 2 minutes and the fly is travelling 120 km/h, so the fly DOES travel 2 km. What makes this problem so interesting is that an infinite sum must occur in a finite time, suggesting that it terminates at some stage. Any ideas?

[Marco]

"If two cyclists, 2 km apart, ride towards each other at 60 km/h and a fly, starting on one, flies back and forth between the two at 120 km/h, how far does the fly travel before the cyclists meet?"

The distance travelled is an infinite sum, but intuitively, the cyclists meet in 2 minutes and the fly is travelling 120 km/h, so the fly DOES travel 2 km. What makes this problem so interesting is that an infinite sum must occur in a finite time, suggesting that it terminates at some stage. Any ideas?

The fly has to change it's direction infinite times and fly back, true, but the time needed for this also will tend to 0 the shorter the distance of the cycles become.

As for the sum, it's not infinite. The sum of the fly's way is sum of all (2/3)^i for i=1 to +oo. Because: way = x. We'll change the problem to a fixed wall and a cycle coming with 60km/h instead of 2 cylces (it's the same). The cycle comes with 60km/h and the fly has 120km/h, so they'll met when the fly traveled 2/3 of the way. The new way the fly has to pass then is 2/3 of the original way only. And this infinite times until the cycle crashed the wall.

So just check it out: the sum of all (2/3)^i is exactly 2.

[R00KIE]

We'll change the problem to a fixed wall and a cycle coming with 60km/h instead of 2 cylces (it's the same)

In that case the cyclist would need to run at 120 Km/h, so you need to redo your math, sorry.

[Marco]

In that case the cyclist would need to run at 120 Km/h, so you need to redo your math, sorry.

Nope, if the cyclists travel with 60 km/h towards each other, each has a speed of 30 km/h (-> I doubled it already)

Anyway, I made another mistake: it's not the same as a cycle with twice the speed and a fixed wall, because when the fly reached the cycle and flies back to the wall, the distance would be the same, whereas the distance to the second cycle would shorten (the double speed of the first cycle has no effect the).

OK, so I'll redo maths, hoping it'll be correct this time
... tommorow may be

[Marco]

The fly doesn't travel 2km, but 4km (120km/h * 2/60 * h = 4km). For the sum:

1. Distance "x" be the distance from one cycle to the other, when the fly is at one of the cycles.

2. The way the fly has to travel to pass x by be "f", a cyclists speed "a" and the fly's speed "b".

3. For a distance x, the fly's way is x*b/(b+a) to meet a cycle

4. Meanwhile both cycles traveled a way of x*a/(b+a), thus the new distance x' the fly has to pass by is x*(b+a-2a)/(b+a) = (b-a)/(b+a) now.

5. Therefore, when the fly passed by the distance between the two cycles n times already, it's way f will be (b-a)/(b+a)^n * x*b/(b+a) this time. The total way the fly moves thus is the sum of all x*b/(b+a) * ((b-a)/(b+a))^n over all n for 0 to +oo

6. For the example this means: x=2km, a=30km/h (the speed of one cycle when both ride towards each other with 60 km/h), b=120km/h.

--> The fly's way is the sum over all 2km*4/5*(3/5)^n, and that's 4km. To check this with your calc (with some n not +oo, but big enough):

0->S
For 0->N To 99
S+2*4/5*(3/5)^N->S
Next
S_

[genesis]

60 km/h "EACH"... And for 'infinite sum', I mean 'infinite number of terms'. Sorry for the ambiguity!

[Marco]

No problem, it's a general form. Just set a=60km/h, and x*b/(b+a) * ((b-a)/(b+a))^n becomes 2km*2/3*(1/3)^n. So replace S+2*4/5*(3/5)^N->S by S+2*2/3*(1/3)^N->S, prog should return 2 then.

But cyclists will meet in 1 min in this case, that's why I thought you'd mean 60km/h relative to each other.

[huhn_m]

please!! Have some pitty and don't bring this up again ---

And so 1/inf is 0 ?

[2072]

Thank you very much for those clarications

[huhn_m]

the term 'inf' is not defined as an element of the set of real numbers

Saw this a bit late. Sorry

but 0.9999... is? Is 0.999... real? I thought it has infinite 9s
It might be "real" in the definition of maths but in truth it is not.
In truth this number would not exist and simply be 1.
But in math it exists and your example of cats and dogs is lame.

If you want to argue so then lets do something:

GraphicsCard = 1
Laptop = 2

GraphicsCard+GraphicsCard = Laptop (?!?)

So I give you 2 Graphic cards and you give me a laptop?
Numbers do not stand for ITEMS but for a COUNT of items at most.
Learn some math yourself

and I DO know maths. Maybe not as well as you but you don't seem to get
sacrasm

And to the second guest. We had this argument already.

YOu cannot multiply aqnd endless number by 10 since you loose precission.

[Marco]

It's OK that we have to distinguish between an idea or information and the representation of it. What makes an idea being what it is is not that we represent it in a certain way, but the relations it has to other ideas (you could call it the idea's semantic).

A "dog" for example isn't a "dog" for me because I call it "a dog", but because I know it has four legs, a tail and it barks. Arbitary I also could call it "a bottle". (here I remember a saying "the identifier 'dog' doesn't wag with it's tail").

Nevertheless I can't agree you about irrational numbers:

Real numbers don't start for counts because they can't be counted, they're just a set of THINGS that we represent using strings like "1" or "cat" or "dog" or "laptop".

Irrationals can't be counted, true (there's a way for rationals however). But BECAUSE they can't be counted, they (most at least) cannot exist i think. BECAUSE: you have neither a way to represent them, nor have they any meaning for you (they have no relations identifying them exactly; note: counting numbers is one possible way to represent them).

Only SOME irrational numbers, such as "pi", "sqrt 2", "ln 74578924", "e^(sinh 65.2)" and so on exist (despite we can't count them), because we can represent them ^^, AND we know about their relations / what they mean (as we have instructions how to built them). For example "ln 74578924" is irrational, nevertheless I can find an expression representing it, I know how to build it, what it means et.c. and it's a valid information for me therefore. But that's not possible for ALL number classified as "irrational"

Imagine a completely randomized number such as "45346.83625871388650392..." represented by an infinite amount of digits that would never end and you also have no given relations for this number such as for "ln 2" (e.g. you don't know how to built this number).

You don't know what this number is, how to use it and so on. The only way how to represent it was to use an infinite amount of digits. This number's information is infinite complex therefore, thus doesn't exist. Because (I said this some postings above) you're unable to use/process an infinite complex information (= all things neccessarry to describe it exactly; no matter weather you use a numbering system for representation or do it another way). ...

Let's get back to that number "45346.83625871388650392...". You would round it let's say after the 200 billionsth digit at least, only then you would be able to use/process it. BUT: then it would not be an irrational number anymore, you lost precision here. It's a ROUNDED RATIONAL number =/= your original irrational number.

----
Note: I used decimal system for this example, but the point is, that it's not impossible with decimal system only to represent this number, but it's impossible with toher numbering systems too and it's impossible even IN GENERAL. You will NEVER find a way to represent it if you have no given algorithm at least how to built it (such as for pi, ln2 and so on = irrationals that exist)

[huhn_m]

Real numbers don't start for counts because they can't be counted, they're just a set of THINGS that we represent using strings like "1" or "cat" or "dog" or "laptop".

I think you misunderstood me there ... numbers are USED to count things.
They represent a count of things.

e.g. you can have 5 cows or dogs or as you like.

[Marco]

@firas: So I think now that you're right.

I said that you can't represent an irrational number like "45346.83625871388650392..." and that it doesn't exist therefore. You said in fact a number that can't be represented is not an irrational, because we always need an algorithm to create a number, and if we have one, we also can use this algorithm for representing that number (and vice versa; I totally agree here). So following your arguments, this hypothetical number "45346.83625871388650392..." also does not exist, if we really assume it had no such algorithm.

Though I think both comes out to the same, I've never regarded it your way. My fault was here: there are no irrationals that we can't represent and that don't exist therefore. But things that we can't represent and that don't exist therefore are not defined to be irrationals because they are *nothing* for us then. Only information we can create by algorithms can be called information and regarded to be irrationals (note: we can regard every information to be numbers). That's the best solution for the problem: simple and logic

----
Though, there's one thing: we can create completely randomized strings, randomized in so far that they are not reproducable by an algorithm (thus "real" random, not random that appears us random only). In quantum physics you have events that will appar with a certain probability only, but you can't say exactly if they will or will not (such as "the electron is within this space" or "the nucleus will decay within this period of time" and so on). Scientiets say: it's not only that we wouldn't know about the cause of this events, but they really have no cause. So for example there is a cause for the decay of a nucleus. But there's no cause wether it will decay now or in five minutes or in five years.

We can observe such random events and convert them into a string. We had a perfect randomized string then (and this is the fact even used today for quantum cryptography). However, this doesn't matter for my random numbers argument, because we can create random numbers, but we can't create one that has an infinite amount of digits (-> we can't create an infinite complex information, as well as we can't use/process one).

----
I have one question left now. What does "random" (I mean real random such as in quantum physics) mean? I think it means "chaos" or "lack of information". In how far can we regard such "random information" to be information then?

[genesis]

But things that we can't represent and that don't exist therefore are not defined to be irrationals because they are *nothing* for us then. Only information we can create by algorithms can be called information and regarded to be irrationals (note: we can regard every information to be numbers).

<{POST_SNAPBACK}>

Hehe... that's just confirming sayings like 'If you can't see it, its not there..." and "A tree falling in an empty forest makes no sound...". Infrared radiation disproves the first saying, while the second... well, its more philosophical! If you define irrational numbers as numbers we can represent, there may be others we can't, while if you define irrational numbers as others, there may be irrationals we can't represent, not unrelated to quantum theory...

[TacoFred]

"A tree falling in an empty forest makes no sound..."
if there are huge cushions to insulate the sound, maybe.....but that'd be a LOT of cushions....
LOL

[CakkuZ[FIN]]

I don't understand that peolple still stay that 0.99999... is different that 1 after all the mathematics demonstrations given

it is not a question, it is a fact, as 2+2=4, 0.99999999(...)=1
with the infinite there is no 1 at the end of 1-0.9999999... because it would say the number is finished
so 1-0.99999..=0

wrong 1-0.99...=0.00...(infinite zeros here)..01

[Overlord]

haha very funny

[Marco]

^^ still wrong: 0.99~ + 0.99~ = 1.9~8 != 1.98~.
(assuming that an infinite amount of digits would be possible at all, which you know I think is not anyways).

[Andy.Davies]

Hence why i dont get into maths debates

[R00KIE]

nothing in the real world relies on maths

You should be very very carefull when making this kind of statements, or do you mean that everything we learn in physics isn't true? and what about any kind of engennering? we do rely on math to acurately predict the behaviour of things you know.

[Marco]

nothing in the real world relies on maths

World IS math

[Orwell]

I can't agree with that.

Like liquid said, Math and Physics are intensively used to describe the world and the reality; they can only provide a description, a mathematical representation of a real phenomemon. This description is accurate more especially as the results we can get from it are close to the reality, but it will never be the reality.

The role of the physician is to find some new (simple) mathematical models to represent as better as possible a (complex) reality. The role of the engineer is to choose the more appropriate model among those provided by the physician, and to apply it in concrete situations, so he get able to predict with more or less accuracy what should happen in the reality, depending on the way he designs his products etc

or do you mean that everything we learn in physics isn't true?

There is almost nothing "true" in physics. You can only say if a model is more or less accurate (and again, this depends strongly on the considered situation).
For example, you may say "The speed of a falling object is constantly increasing". This is (generally) true, it's a real phenomemon.
However, you cannot say that the formula "h(t) = h0 + v0*t + g*t?/2" is true. It is only a "good" mathematical approximation of this phenomenon.