9/9 ?????????????????????

[PsyVision]

hiya all,


Well i was in another boring maths lesson with my dumb ass maths teacher who's about 90. Anyway, whats the answer to this do us people thinks ?

0/9 = 0
1/9 = 0.1111 (Re-Occuring)
2/9 = 0.2222 (Re-Occuring)
3/9 = 0.3333 (Re-Occuring)
4/9 = 0.4444 (Re-Occuring)
5/9 = 0.5555 (Re-Occuring)
6/9 = 0.6666 (Re-Occuring)
7/9 = 0.7777 (Re-Occuring)
8/9 = 0.8888 (Re-Occuring)

9/9 = 0.9999 (Re-Occuring) Or 1 ?????

[crptcblade]

As a general rule, anything divided by itself = 1, but I see your view.

[kedaman]

too bad i can't vote for both because 1 = 0.999999..

[Fox]

*agrees*

[parksie]

Oh bollocks...here we go again

[kedaman]

*hehe*

[CiberTHuG]

I just keep thinking of it as a limit. As it approaches 9/9 it approaches 1. I'm not sure if it really plots that way, but once you get to 9/9, you've gotten to 1.

[Sam Finch]

I went for 0.9... but it's both.

We've had this discussion on the maths board, we never settled it because we got bored, but I'll offer you a proof that 0.9999... = 1.


First we show that for any 2 real numbers a,b where a<b there must exist a number c expressable as a non recuring decimal such that a<c<b.



since a<b we can conclude that b-a > 0.

and therefore there must exist 1/(b-a) > 0

Now define a number N = 2^p * 5^q
where p and q are positive whole numbers and N > 1/(b-a)


N > 1/(b-a)
=> 1/N < b-a

N = 2^p * 5^q
=> 1/N * m is expressable as a non recurring decimal for any whole number m.

now set m as the greatest whole number such that m/N <= a

by definition of m
(m+1)/N > a

as m/N <= a
b - (m/N) >= b-a
b - ((m+1)/N) >= (b-a) - 1/N > 0 (as 1/N < b-a)


b - ((m+1)/N) > 0
=> b > ((m+1)/N

and hence a < ((m+1)/N < b

and so there exists a number ((m+1)/N expressable as a non recuring decimal such that a<((m+1)/N<b.


So if 0.9... < 1 there must exist a non recurring decimal c such that

0.9.... < c < 1

clearly this is not the case so 0.9... = 1.

[CiberTHuG]

Originally posted by Sam Finch
First we show that for any 2 real numbers a,b where a<b there must exist a number c expressable as a non recuring decimal such that a<c<b.

Why does it have to be non-recuring?

[Sam Finch]

it doesn't have to be, but there is a non recurring one, but I think it's just a little bit clearer to say a non recurring one when we say there's no decimal between 0.99.... and 1, I don't know why I think that's clearer but I didnit anyway.

[CiberTHuG]

Ah... well... I don't accept your rational. I think there is a descrete number between 1 and 0.999.... I also think that number is equal to the width of a point.

But I do this simply so I feel better about having defined an actual point. I hate to think we go through university working with points that don't physically exist. I say we should let them exist, but so as they don't disturb anything, they are the smallest measurable thing. They are 0.000...001.

Which I guess lends itself to being able to say you have the span of ten points when you have 0.000...010.

*shrug*

[kedaman]

well eh, i don't think a point have a width

[Sam Finch]

If there is a terminating decimal between 0.9... and 1 what is it?

or can you see a hole in the proof that there necasserally is one?

[Pix]

Does it matter what the answer is. If we're not careful there'll end up being an arguement about whether or not 1 is a prime!

[CiberTHuG]

There is not necessarily a terminating decimal betwee them, but I don't see why there has to be. There is a descrete difference between the two numbers, and for that reason, I do not consider them equal. And I consider 9/9 to be equal to 1.

I would say that 8.999... over 9 would be 0.999.... As you make that descrete change from 8.999... to 9, you change from 0.999... to 1. Sort of like a limit.

I just think that the postulate a<b<c where b is a terminating decimal is assuming too much.

Besides, measuring a point is much more romantic. And I think that arguement alone would be able to sway Einstein and or Hawking (maybe not Sagan). Sorry, all the big number crunchers I know of are related to astrophysics in some way since thats the only reason I care about numbers.



One is prime and zero is positive? What?

[CiberTHuG]

'Course, this arguement is interesting...


1/3 = 0.333...

2/3 = 0.666...

1/3 + 2/3 = 3/3

0.333... + 0.666... = 0.999...

3/3 = 0.999...

There is also...

1/9 = 0.111...

8/9 = 0.888...

1/9 + 8/9 = 9/9

0.111... + 0.888... = 0.999...

9/9 = 0.999...

Nah... but it is cute. And this stuff works only with 3 and 9, since they are our special screwed up numbers.

I'm sure the Catholics appreciate that.

[Sam Finch]

If you say I'm assuming too much what is is I'm assuming that you disagree with?

I have given a mathematical proof and unless you can find a flaw in it you can't dispute it. I'm very happy for you to dispute this, but I can't seem to identify what you're disputing.

[Active]

0.99999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999 9999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999 9999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999 9999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999 9999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999 9999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999 9999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999 9999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999 9999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999 9999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999 9999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999 9999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999 9999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999 9999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999 9999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999 9999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999 9999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999 9999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999 9999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999 9999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999 9999999999999999999999999999999999999..... is just as equal to 1

But what is not equal to 1 is..

9999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999 9999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999 9999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999 9999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999 9999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999 9999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999 9999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999 9999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999 9999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999 9999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999 9999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999 9999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999 9999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999 9999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999 9999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999 999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999 8 9999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999 9999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999 9999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999 9999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999 99999999999999999999999999999999999999

[CiberTHuG]

Oh no, the proof is accurate given your postulates. I disagree with the postulate.

But I've been thinking about this... and here is why 0.999... is not equal to 1 (in my opinion).

I feel that you are dropping a bit. That you have a concept and the numbers on the page fail to represent it accurately.

Lets say there were no such beast as fractions. You could not have the part of a whole number, no decimals. Lets say you had 10 units of something. You want to divide that group into 3 (or 9) equal parts. You get 3 groups of 3 each, with 1 left over. That 1 would be split in a world with fractions, but in this world, you can't do that, so it gets ignored. When you put the groups together that you partitioned out, you only get 9 back. You dropped that one.

I feel that in the world with fractions, that one you dropped is that single point. No matter how you slice it, 10 and 1.0 only have an even number of divisions. When you start to break it into thirds or ninths, you have to break the remainder again and again. At some point (0.000...001) you can't break any more. At some infinately small point you find there is nothing smaller.

That is a point. And that singular point gets dropped when you represent 1/3 as 0.333... and 2/3 as 0.666....

This is a case inwhich the numbers tell you one thing becuase they can't truely represent the real world.

In computers, this is an underflow error. You couldn't represent a number small enough, so you represented one that you could. The one you represent is the largest number you can without going over the target.

0.999... is the underflow approximation of 1.

[kedaman]

here's what i think:

choose a more suitable base to express the values: 9

0/10=0
1/10=0.1
2/10=0.2
3/10=0.3
4/10=0.4
5/10=0.5
6/10=0.6
7/10=0.7
8/10=0.8
10/10=1.0

[Bonker Gudd]

How do I vote for 42?

[Behemoth]

Nature abhors a singularity.

[Gary.Lowe]

So who's up for a pint

[Behemoth]

mines a sweet sherry.

d'oh!

[Gary.Lowe]

Behemoth

You must be a drinking buddy od Mr IanPBaker

[simonm]

Sam Finch,

All your proof states is that there is no finite difference between the numbers 1 and 0.9999...

If we can accept the concept of an infinitely recurring decimal, why can we not accept the concept of an infinitely small quantity? Such a number would be greater than zero but less than any finite number you could ever name.

It would look something like this: 0.000...1

[CiberTHuG]

My point exactly, Simon.

[kedaman]

infinetely small number, there's no such thing as infinitely small number, especially when there's no number infinty either.

[simonm]

There must be no such thing as an infinitely recurring decimal...Therefore 0.9999... is NOT equal to 1!

[Guv]

Sam: I disagree with your proof. Note the following.

Recurring .9999 is a shorthand notational representation of the sum of a series, namely the following.

DecimalSeries(N) = 9/10 + 9/100 + 9/1000 + 9/10000 + . . . + 9/10^N

Consider the following series, which corresponds to recurring .FFFFF in hex notation.

HexSeries(N) = 15/16 + 15/256 + 15/4095 + 15/65536 + . . . + 15/16^N

I have coined the notation DecimalSeries(N) & HexSeries(N) to represent the sum of the first N terms of the respective series.

For all N, DecimalSeries(N) < Hexseries(N) < 1

QED: there is a real number greater than recurring .9999 and less than one.

[CiberTHuG]

Originally posted by kedaman
infinetely small number, there's no such thing as infinitely small number, especially when there's no number infinty either.

There is no number infinity? Why? I say there exists a number so large there is no number bigger and a number so small there is no number smaller.

Mind you, you can't get to these number save by saying everything and point. In other words, you can never count up to or down to either of these infinities, but they exist.

And when I think of this, I think of someone telling me that the universe is expanding. After some clarification I understand that they are not saying that matter in our universe is traveling away from each other, taking more and more room in Space, but that the void Space itself is expanding. Not only are the stars moving apart, but the distance between the stars is increasing of its own accord, Space and everything in it (The Universe) is expanding.

So I ask... what is it filling?

I have no idea. But if you can buy into the idea that the Universe is expanding, the concept of a singular infinity is right next door.

[kedaman]

eh, are you sure it isn't like this?

DecimalSeries(N) = Hexseries(N) = 1

[kedaman]

nope, there's always a number bigger than the earlier one, and also a smaller than the one before, why should the next be any different? infinity is not a number, and is used in other referense frames

[Sam Finch]

CyberThug

There is no number infinity? Why? I say there exists a number so large there is no number bigger and a number so small there is no number smaller.

if there's a biggest number then what is double that number?

if there's a smallest number then what's half that number?

I did try to develop a theory like the one you proposed, with a number sigma to represent the largest number. and 1/sigma to represent the smallest number. However it lead to a contradiction. (I can't remember what the contradiction was though.) I'm sure it's an Idea that's beeen proposed before and it doesn't work.

It also start making some serious blows at calculus.

Guv

I did specificly mention that there was a terminating decimal between the two. 0.9... is not a terminating decimal. So your argument is invalid.

[CiberTHuG]

Sam, if you can double it, then it isn't infinity. But to ask yourself the same question, if I doubled Life Universe and Everything, what do I get? If I was able to double it, did I really have everything?

If you had a point in space, can you halve it? If so, did you really have a singular point?

Infinity and Infinitesimal exist. You can't represent them with any number. That leads to the underflow problem that makes people think that 0.999... is equal to 1.

[Fox]

You can't compare these two things... you in fact can not half a point because - as you said - it's already the smallest thing.

But numbers are defined by human, someone just told "let's say 1 + 1 = 2..." and that's it. Mathematical said 0.999... is equal to 1 and that's what counts. You may or may not associate this with the laws of reality, but remember, numbers werde invented by human and are not real, or did you every eat 0.1 chocolate? No, because it's a piece of chocolate and not the chocolate itselves! See the point?

And by the way, you're not talking about infinity, you're talking about infinite approximization, that's different... of course you can't double infinite values, but 0.999... isn't infinite, it's just a number between 0 and 2! (which would be at best 4)

[kedaman]

Originally posted by CiberTHuG
Sam, if you can double it, then it isn't infinity. But to ask yourself the same question, if I doubled Life Universe and Everything, what do I get? If I was able to double it, did I really have everything?

If you had a point in space, can you halve it? If so, did you really have a singular point?

Infinity and Infinitesimal exist. You can't represent them with any number. That leads to the underflow problem that makes people think that 0.999... is equal to 1.

also everything is being defined by human, oh yeah and if you want to apply something on reality, go find it first. Good explanation fox

[Guv]

Sam: How can you state that my proof is invalid without showing where it breaks down?

Note that there is no radix point involved in my proof.

DecimalSeries(N) = 9/10 + 9/100 + 9/1000 + . . . + 9/10^N

HexSeries(N) = 15/16 + 15/256 + 15/4096 + . . . + 15/16^N

The above explicitly specify what recurring .99999 and recurring .FFFFF really represent, without using radix notation which is only a convenient shorthand notation.

For all values of N, it is obvious that DecimalSeries < HexSeries < 1

Hence there is a real number greater than DecimalSeries(N) and less than one, namely HexSeries(N).

Where does the above break down?

For what value of N does the above become invalid?

Perhaps you want to claim that HexSeries(N) = 1 for some value of N. Such a claim still leaves DecimalSeries(N) less that one.

[PsyVision]

I thought that would get you lot going.

[kedaman]

hmm when i look closer, how can you express 0.9999... with one of those series and have a value at N at the same time?