a short note about a comment made earlier. when i said you can't do arithmetic with infinity, countable or not, is that you can't have infinite bananas. you can have any real number of bananas or fractions of bananas or any other real number of bananas. you can't have infinite of them. arithmetic is what you do with real numbers. R. you can extend some functions that work in the R set, to work in the extended real set, but that is not arithmetic in both the strictest and the practical sense.

Quote from Woz :

"There are many proofs that 0.999… = 1, of varying degrees of mathematical rigour. A short sketch of one rigorous proof can be simply stated as follows. Consider that two real numbers are identical if and only if their difference is equal to zero. Most people would agree that the difference between 0.999… and 1, if it exists at all, must be very small. (WOZ: but as it exists it must be able to represent.. hence bodge) By considering the convergence of the sequence above, we can show that the magnitude of this difference must be smaller than any positive quantity (WOZ: Why?), and it can be shown (see Archimedean property for details) that the only real number with this property is 0. Since the difference is 0 it follows that the numbers 1 and 0.999… are identical. The same argument also explains why 0.333… = 1⁄3, 0.111… = 1⁄9, etc."

"By considering the convergence of the sequence above, we can show that the magnitude of this difference must be smaller than any positive quantity (WOZ: Why?)"

Because lets say you get to a billion, trillion, dodechedronillion 9s. The difference is now tiny. No divide that tiny tiny difference by a billion, trillion, dodechedronillion. The difference is even tinier. Divide the new difference by a billion, trillion, dodechedronillion. And keep divided the new difference by a billion, trillion, dodechedronillion an infinite number of times.
You then end up (if ending up were possible when using infinities) with NO DIFFERENCE WHATSOEVER, and so an infinite number of 9s is the same as a whole number, and thus 0.999... (note this is not the standard gramatical ellipsis, but the recurring notation that is referenced in the Wiki page as being 100% legit) = 1

1/0.999...=1.000...

so therefore 0.999...=1

Quote from george_tsiros :

great results i see there. did you notice that it says 'extended' real line? as in, "stuff you can do, theoreticaly, about quantities that can not be measured" ? you can add stuff up. countable stuff. you can't count to infinity. you can't do arithmetic with infinity. and by 'arithmetic' we mean "add quantities up and come up with real results".

You can add inifinity to the set of real numbers, and then define addition and multiplication operations on it. With this extended set you'll still be able to do calculations and come up with consistent results. Consistent means "not leading to internal contradictions". It may not be consistent with your intuitive concept of numbers, but it still works.

Note that, according to Cantor's diagonal argument, you can't even count from 0 to 1 in the real numbers.

Also note that the real numbers are not real in the sense that they represent our physical reality. Nature is essentially quantified, whereas set R has infinite detail. But you can still use R to do useful calculations. Likewise, you can calculate with imaginary numbers, and get results that match real-world phenomena.

Quote :

it says "These new elements (+/-inf) are not real numbers"

... meaning that they are not in set R.

Quote from Becky Rose :

Are you sure, that would imply negative infiinity equals 0.

No, because 10^0 equals 1, not zero.

Quote :

No, because 10^0 equals 1, not zero.

Screw me you're right. I just learned something new. I think it's daft though and makes only vague sense. 10^0's only combination is 10, so it's power of 0 is 1. OK I get that, but it's silly. I'm going to rewrite maths to make more Becky-sense.

Quote from Woz :

I started out wanting an answer and have yet to get one.

No, you got many answers, including proofs accepted by generations of mathematicians, but you rejected each and every one of them.

Quote :

Just saying the difference is so close to zero we will call it zero is not proof.

No, the difference is smaller than any positive number, hence it is zero.

Quote from wsinda :

... meaning that they are not in set R.

where is arithmetic defined?

also, we do calculate with complex numbers, but in many areas (for us physicists at least, i'll mention electromagnetism and quantum mechanics) we keep the real part of the number because that is the one that is representative of physical reality.

i insist that you can do arithmetic (in the strict sense) only with real numbers. infinity is a concept, not a real number. that is why the set including +/-inf is not R.

Quote from Woz :

"Misinterpreting the meaning of the use of the "…" (ellipsis) in 0.999… accounts for some of the misunderstanding about its equality to 1. The use here is different from the usage in language or in 0.99…9, in which the ellipsis specifies that some finite portion is left unstated or otherwise omitted. When used to specify a recurring decimal, "…" means that some infinite portion is left unstated, which can only be interpreted as a number by using the mathematical concept of limits."

Let me repeat that VITAL part... "When used to specify a recurring decimal, "…" means that some infinite portion is left unstated, which can only be interpreted as a number by using the mathematical concept of limits."

You do realise that YOU are the one this paragraph talks about? Did you even understand what it says? For some reason you think there is a tiny difference between 1 and 0.9~ (to use your notation), which there is not. There would be one if 0.9~ was finite, which it isn't. There is no botching whatsoever going on, since this is the only way to write a number with a recurrence.


Now, I'll try to explain this by how I think about it, sorry if I don't make sense:

What do you think is inbetween 0.9~9 (some '9' digit of the recurrence) and 1? On a finite number, you could say, "ha, there is 0.9...91, 0.9...92 [...] 0.9...99." Note that the finite number stops at some point (only continuing with '0' if you want so), which gives you "room" to have a final digit. But on .9~ we are infinitely repeating the 9, there is simply no point where you ever cut off the line of 9s to make room for something (a difference) to exist between 0.9~ and 1. The difference between 0.9~ and 1 is 0.

*sigh* anyway...

Quote :

The difference between 0.9~ and 1 is 0.

Technically the difference is 0.0(0)1.

Conveniently, that's the exact 'mass' I have decided to award all numbers in Becky-math.

Can't follow you there, at school they don't teach Becky-math. I don't see how it is valid to put a digit after the recurrence, since this would stop the recurrence by definition. So if we allow this for whatever reason, then the difference would only be 0.0(0)1 if we were talking about 0.9(9)9, which we aren't.

Anyway, I'm out of here. That's like discussing if Pi and 3.141592...(insert Pi digits here) are the same thing. The former is simply the only way to write it fully

Ah Becky-math is brilliant because it replaces any hard concept with 0. You should try it sometime.

Quote from Woz :

Just saying the difference is so close to zero we will call it zero is not proof

Eh?

No wonder why you think the way you do. You got the concept wrong.

It's not just "saying the difference is so close to zero we will call it zero". The difference is infinitesimally small that it is zero! Note the word infinitesimally: ie. there is no difference.

Can you define a value for infinity? If you can define a value for infinity, then you can define the value of the difference between 0.999... and 1.

But the fact remains that it is impossible to define a value for infinity. It is infinite. Therefore, the difference between 0.999... and 1 is 0.

once i asked my father if anybody had ever represented infinity and 0 with the same symbol he said not

1 month later i read this comic gilgamesh where the main characther met this alien culture that represented the dot symbol as:

. = god, inifnity and zero

Quote from lerts :

once i asked my father if anybody had ever represented infinity and 0 with the same symbol he said not

1 month later i read this comic gilgamesh where the main characther met this alien culture that represented the dot symbol as:

. = god, inifnity and zero

Alas, that wisdom does not help us mere mortals.

:doh:

Quote from lerts :

once i asked my father if anybody had ever represented infinity and 0 with the same symbol he said not

1 month later i read this comic gilgamesh where the main characther met this alien culture that represented the dot symbol as:

. = god, inifnity and zero

In m-brane theory the achieve infinity you can only reach infinity, a mandatory requirement for omnipotence, omniprecense, and omnisience (ala God), by reducing one dimension of a membrane to 0 so that it achieves infinite length. Thus suggesting that God is 1 dimensional, a bit like spaghetti really.

Whilst one might argue this is provenance for Pastafarianism, it's also evidence that your comic books are correct and you should believe everything they say.

Here is a video to provence that i'm not just talking rodneys: http://www.youtube.com/watch?v=A4LrbAXb4FQ
I must confess now that it is a waste of 6 minutes 47 of your life, but this and the follow up video do highlight than an infinitately long membrane must have no width.

Of course this assumes that the universe is constructed of membranes, which is merely a theory.

Quote from Becky Rose :

Technically the difference is 0.0(0)1.

And by that exact definition, the recurring zeros never end, so the 1 never happens, thus 0.0(0)1 = 0.0

oh becky thankyou so much im going for scissors and tape

the devils put some playboy video into the links, what the hell was doing in string theory, isnt it odd, probably they wanted to make me miss this amazing piece of info thanks again

by the way being 0 and infinite equal i thought it before reading the comic so should i start believing my ideas because thats the origin of the thought of infinite and 0 being the same thing

edit:

haha the porn video link was due to a string tanga, oh come on thats so much bs im not even watching it i bet the lady is ugly

i just did this:

http://es.youtube.com/watch?v=A4LrbAXb4FQ

and my mind buggled how can i cut a moebius strip along and instead of two halves it gets longer

hell why i didnt learn this in school

Quote from lerts :

oh becky thankyou so much im going for scissors and tape

That's not me you numb nuts! I do have hair on my arms but c'mon!...

Quote :

by the way being 0 and infinite equal i thought it before reading the comic so should i start believing my ideas because thats the origin of the thought of infinite and 0 being the same thing

They're diametrically opposed, not equal, like + and - signs. However, God - should he exist and m-theory be correct - would be both infinite and nothing at the same time, as infinity cannot exist without 0 under m-theory. Thus making God a paradox.

yes but the only thing you can have an infinite amount of is of nothing

and nothing has energy and is infinite therefore energy in the universe is infinite

But if nothing was infinite, how come there is a finite amount of 'stuff'? Therefore nothing must also be finite and so energy is finite.

Edit: Becky, it is, I believe, suggested that matter is merely energy by E=mcc

Quote from lerts :

yes but the only thing you can have an infinite amount of is of nothing

Correct, 1 point.

Quote :

and nothing has energy and is infinite therefore energy in the universe is infinite

Incorrect, -1 point.

Unless you are a God botherer and believe God is a form of energy, in which case, you yourself would be energy. Making the whole universe little more than a Duracell. Maybe the Matrix had it right, copper top ?

I wonder if the amount of energy in the universe is 0.(9) ... ;P

Quote from Tristy :

Edit: Becky, it is, I believe, suggested that matter is merely energy by E=mcc

But that does not imply that nothing is energy, and neither is nothing quantified as existing. If nothing exists then it would be infinite, but as nothing does not exist it therefore is not energy. Or if nothing does exist and is energy then God is nothing.

Quote from Jakg :

Errrrr.

1/3 (as in the fraction 1 over 3) = 0.3333333... (recurring - with me so far?)

1/3 x 3 = 1 (obviously - three thirds of something will always equal 1 - slice a cake in 3 ways, and keep all the slices. How much cake is left? All of it)

0.3333333... x 3 = 0.99999.... (recurring)

However if 0.333333... = 1/3, then surely 0.9999.... = 1?

At the same time, 0.33333 x 30 = 9.99999.....

9.999999 - ( 3 x 0.33333) = 9



EDIT - Bad idea explaining stuff when tired.

meh i just remembered what its about

That explaination kicked my memory into gear

Thanks anyway