# .999999999999... = 1 ?

### #161

Posted 16 February 2005 - 09:28 AM

### #162

Posted 16 February 2005 - 10:14 AM

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!

### #163

Posted 16 February 2005 - 08:45 PM

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

### #164

Posted 18 February 2005 - 10:44 AM

"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?

### #165

Posted 18 February 2005 - 03:37 PM

"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.

### #166

Posted 18 February 2005 - 11:23 PM

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.

### #167

Posted 19 February 2005 - 12:56 AM

Nope, if the cyclists travel with 60 km/h towards each other, each has a speed of 30 km/h (-> I doubled it already)In that case the cyclist would need to run at 120 Km/h, so you need to redo your math, sorry.

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

### #168

Posted 19 February 2005 - 11:37 AM

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_

### #169

Posted 21 February 2005 - 01:35 PM

### #170

Posted 21 February 2005 - 06:56 PM

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

### #171 Guest_firas_*

Posted 26 February 2005 - 03:28 AM

I can't believe you guys argued about whether 0.9999999... = 1 for five pages!

The important thing to understand is that the real numbers are an uncountably infinite set of THINGS. We can never list them all. But the point is they are THINGS that we can give some kind of NAME.

For example the real number "pi", or the real number "sqrt(2)", or the real number "234234", or the real number "4/10", or the real number "45 + 3", or the real number "1" or the real number "0.999999...".

These are just NAMES for elements inside a set! The important thing for us is to understand when two NAMES refer to the same number, for example:

"5" is "3 + 2" is "lim x-> inf (5x / x)" is "4.999...". These are all different NAMES for the same thing. Don't go crazy like this, it's all perfectly logical!

Good luck!

### #172 Guest_firas_*

Posted 26 February 2005 - 03:32 AM

### #173

Posted 26 February 2005 - 05:42 PM

And so 1/inf

**is**0 ?

### #174 Guest_firas_*

Posted 26 February 2005 - 06:27 PM

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

And so 1/infis0 ?

Have some pity and please, learn more math! Sorry, but arguing about whether something true is true is just scary! I'm sure you all love math, so I'm sure you'll learn more and you'll understand why 1 = 0.9999... and why that's completely logical.

As for 1/inf = 0, you have to understand that REPRESENTATION is arbitrary. I could call 1 "cat" and 2 "dog" and we could say "cat" + "cat" = "dog" and it would be true. "dog" and "cat" represent ELEMENTS in the SET called "real numbers," that's all. In our agreed-upon system of representation called "algebra", the term 'inf' is not defined as an element of the set of real numbers. Thus, 1/inf DOES NOT REPRESENT a real number. Since 0 does represent a real number, that equation is false. But it has nothing, NOTHING to do with the FACT that in our system of representation REQUIRES that 0.999... = 1. We AGREE this is true, because IF IT ISN'T, algebra doesn't work!

The reason we have this problem, ie we have multiple representations for the same value, is that the real numbers can't be listed, they can't be listed in order from smallest to largest, it's proven that that is impossible. Thus, there is no single algorithm to enumerate them like there is for the listable integers. As a result, we get these apparent messy things like 1 = 0.999.... But that is not illogical, it's just something you have to assume in order for your LANGUAGE that you use to represent elements in the set of real numbers to work. it has NOTHING to do with the actual numbers, just our way of writing them.

### #175 Guest_firas_*

Posted 26 February 2005 - 06:33 PM

"sqrt(2)"

you're using a FINITE symbol, the string "sqrt(2)", to represent an ELEMENT in the set of REAL numbers which, in our system of base 10, would require an INFINITE number of symbols. The same way, when we write:

"0.9999..." the "..." part (or in proper math, there would be a bar over the 9's) is just one of the SYMBOLS which makes up the STRING "0.9999...". This STRING represents an ELEMENT in the SET of real numbers. Another string that can represent that element is "1".

### #176

Posted 27 February 2005 - 01:19 PM

### #177 Guest_Guest_*

Posted 02 March 2005 - 02:27 PM

(... is infinity)

Plug in 0.999999999999... for A:

A = 0.999999999999...

A x 10 = 9.99999999999...

9.99999999999... - A = 9

Now plug in 1 for A:

A = 1

A x 10 = 10

10 - A = 9

See 0.99999999999999999... could act as 1 but could also be an another equivalent to 1. Look at the equations. You both get 9 for each problem. You take the answer for A x 10 and subtract that by A and you get 9. So 0.9999999999999999999999... could equal 1.

### #178

Posted 02 March 2005 - 02:58 PM

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.

### #179 Guest_firas_*

Posted 02 March 2005 - 03:29 PM

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 simplybe1.

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.

Sorry, huhn_m, but you're just completely wrong here. Get an abstract algebra textbook, or at least a linear algebra or real analysis textbook.

A number DOES NOT stand for a "count" of things if it is a real number! You're once again confusing representation of numbers with the *actual* numbers. The actual numbers DON'T HAVE a representation. Until you assign an element in the SET of real numbers some string (for example, "1", or "0.9999...") it has no name! That's the WHOLE POINT OF REAL NUMBERS! You can't count them! You can count integers, you can count the natural numbers, but you *cannot* count reals (they're even called "uncountably infinite").

"In truth it is not"? What is the "truth" to the real numbers? There is none, they don't exist, all we have is a LANGUAGE called algebra to consistently represent them. Representation =/= actual, real numbers.

Your graphics card and laptop example is perfectly fine. You missed my point, the string "cat" or the string "laptop" or the string "x" can stand for a REAL NUMBER in algebra, haven't you noticed? (eg x = 6, y = 4 etc.) 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". My point here is representation means nothing, you can use anything to represent it. "0.999..." isn't a "true" anything, it's just something we made up to represent a thing.

### #180 Guest_firas_*

Posted 02 March 2005 - 03:36 PM

Is "0.3333..." a real number? Yes! It's even rational! You don't disagree with that do you? And guess what, 6 * "0.333..." = 1/3 * 6 = 6/3 = 2. Do you disagree there?

You don't "lose precision" when you multiply a number like "0.3333...", so how could you "lose precision" when you multiply a number like "0.9999..."? The two situations are exactly the same. You're getting caught up in *representation*, you seem to think "0.9999..." is some kind of special "true number" just because it LOOKS different, but the actual *truth* is that unless "0.9999..." = "1", writing something like "0.999..." would be meaningless, it would represent absolutely nothing.

0.5 = 1/2. Those two sides of the equation look different. Are they actually different?

### #181

Posted 02 March 2005 - 05:45 PM

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:

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).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".

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)

### #182

Posted 02 March 2005 - 06:13 PM

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.

### #183 Guest_firas_*

Posted 02 March 2005 - 06:41 PM

We have no algorithm to enumerate them (eg. the diagonalization argument for rationals, peano's postulates for naturals etc.) because such an algorithm does not exist, the set of irrational numbers is a "turing unrecognizable" problem. This is all true.

However, this says nothing about any single number. Your "random" irrational number argument has one major flaw: there is no such thing as a "random" string of digits, you'd have to give me an algorithm that generates random numbers, and that's *impossible*, so you'd need a better argument. The important thing to realize is that the set of real numbers does exist, in so far as first order logic and the church-turing thesis algorithms exist (look up Hofstadter's proof of Goedel's Incompleteness Theorem for an explanation). You can *never* prove that any individual real number can't be represented, because you can simply make a symbol for it. Example: pi has an *infinite* number of digits which *we don't know*. You can never give me a real number that you can *prove* doesn't have an algorithm to represent it (by Goedel's Thm), thus we can't claim there exist irrationals which can't be represented.

huhn_m: I understood that you meant you can use numbers to count things, but the real numbers are far more powerful than counting - they're continuous and dense!

As for 0.999... = 1, show me how 1/3 is not the same as 0.3333... and your argument will hold. Show me that the number represented by 0.9999... is NOT THE SAME as the number represented by 0.9999... = 9/9 = 1. Prove to me that 0.99999... is different from 1 and from 9/9 and you'll have a point. (This is rhetorical, you obviously can't).

### #184 Guest_firas_*

Posted 02 March 2005 - 06:46 PM

### #185

Posted 03 March 2005 - 11:10 AM

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?

### #186 Guest_firas_*

Posted 03 March 2005 - 03:31 PM

I think that it's too early to tell at this point, given the 7 major quantum mechanics interpretations, whether the collapse of the wave function really is random, or if the wave function is actually linear (try looking up "everett interpretation" or "many-worlds" or "multiverse", and then if you haven't seen it already check out the "copenhagen" interpretation, which are probably the two most popular quantum mechanics interpretations of the wave function's behaviour). I think that "random" will probably always have to be defined in some context; the closest thing we have to "randomness" is our understanding of "entropy", but once again, if we consider a universe where the wave function is complete, this randomness doesn't actually exist, it only appears to us that way because we can't see all the branches.

### #187

Posted 04 March 2005 - 03:38 PM

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...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).

### #188

Posted 18 March 2005 - 11:26 PM

if there are huge cushions to insulate the sound, maybe.....but that'd be a LOT of cushions....

LOL

### #189

Posted 26 April 2005 - 10:35 AM

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

### #190

Posted 26 April 2005 - 05:49 PM

### #191 Guest_bob_*

Posted 22 June 2005 - 09:04 PM

### #192 Guest_Bob_*

Posted 22 June 2005 - 09:11 PM

sorry i made a mistake it is all wrong sorry

### #193 Guest_Bob_*

Posted 22 June 2005 - 09:25 PM

I understand the algebra and the maths and how they came to this conclusion but think about it this way if 1 is 0.9999* then 1+1= 1.999999999*

But then it does make sence like they have mentioned before because if you try to convert 0.99999* to a fraction it is 1

so you lot stop spending time and pages on this because it is just like religon either you belive in it or dont

**Andy.Davies:**Deleted Smileys, too many used and without a point, and you are wrong in your math:

0.99* + 0.99* = 1.98*

**Edited by Andy.Davies, 22 June 2005 - 10:46 PM.**

### #194

Posted 23 June 2005 - 11:52 AM

(assuming that an infinite amount of digits would be possible at all, which you know I think is not anyways).

### #195

Posted 23 June 2005 - 06:46 PM

### #196 Guest_ArbiterOne_*

Posted 01 July 2005 - 10:55 PM

Let me just put it this way:

After a certain degree of precision, there's no point in going further. It's not that you don't feel like it, it's that it can't represent anything in the universe. It's called the Planck length- and nothing can be smaller than that, otherwise quantum mechanics can't tell whether it exists or not- simplified explanation.

Or, alternatively, the way my math teacher said it:

A mathematician and a physicist were in a bar, and a very attractive girl was at one end. She said that they could come over- but each time they moved, they had to cover half of the remaining distance to her.

The mathematician stayed where he was, saying, "There's no point. You'll never get to her."

The physicist stood up and said, "But after a certain time... I will be close enough."

### #197 Guest_Guest_liquid_*_*

Posted 15 July 2005 - 03:11 AM

maths is just made up by humans, so u can let .9999999...=1 if you want, or not let it equal also, remember maths is used to represent the real world, BUT it IS NOT the real world,,,,,,nothing in the real world relies on maths,,,,,,the maths is used to represent the qualities of the real world as good as we can thats all.

### #198

Posted 28 September 2005 - 01:51 PM

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.nothing in the real world relies on maths

### #199

Posted 30 September 2005 - 09:31 AM

World IS mathnothing in the real world relies on maths

### #200

Posted 30 September 2005 - 10:40 AM

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

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

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.

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