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the world's smallest number?  RSS feed

 
blacksmith
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James Christian:

Yet again, you have only succeeded in proving my point.
In the case there are no employees the share which is added to each employee's pay is but the money does not, repeat NOT slip away silently.


Check the code again. Notice that the surplus has been decremented away - the correct behavior in the normal case when there are employees. The money has disappeared.

There are no remainders in floating point arithmetic.
 
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I know. Dividing a floating point by 0 in java gives POSITIVE_INFINITY when you divide a positive number and NEGATIVE_INFINITY when you divide a negative number. So not only are we making up names for non existant numbers but we are also giving them both negative and positive values. What will they think up next to keep themselves entertained???

The fact remains that implementing the fact that when you divide by 0 the original amount remains in tact works perfectly for your scenario and every other scenario that I can possibly think of.
 
James Christian
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posted April 07, 2005 10:42 AM Profile for Ryan McGuire Send New Private Message Edit/Delete Post Reply With Quote Just to reopen the original can of worms...

Concerning .999(9) == 1...
- Do we all agree that the proofs given so far are sufficient?
- Have there been counter-proofs?
- Are people still unconvinced but unable to explain why?
- Somthing else?

I tried to make that list un-biased to either the They're Equal or They're Unequal camps, but since I count myself as being on the Equal side of the fence (to mix my metaphors with a 9.999(9) foot pole), I may have failed.



In answer to Ryan's question "have there been counter-proofe?"... Just one or two.

If we really must as educated scientists entertain our fantasies and accept that a number with an infinite number of 9's exists, then yes (0.999(9) == 1) would equate to true in my opinion because a number that is infinitely close in value 1 has by definition, no finite difference in value; and hence the closest number to 0 would be infinitely small and so it would be equal to 0 also.

BUT if you were to ask me would 0.999(9) pass the (0.999(9) instanceof 1) test then the answer would be certainly NOOOO!!!

The reason being 1 is a number that exists and 0.999(9) is a fictitious concept that helps people get their phd's when there is nothing left to study that helps mankind. :roll:
 
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Originally posted by James Christian:
So not only are we making up names for non existant numbers but we are also giving them both negative and positive values.



Hi James,

Can you explain 0(zero)?

What is Zero ?? What is the existance of Zero ?

I would really like to know.
Yes, in your own words, you can take any practical example to explain Zero.

Thanks in advance.
 
R K Singh
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Originally posted by James Christian:
I'll try and make it easy to understand. If I have five oranges and I divide them among 5 people each person gets 1. If I divide them among 2 people each person gets 2 and 1 is left over. Optionally of course they could break the remainder in half but that's beside the point. If I divide the 5 oranges between 0 people the 5 oranges remain 5 oranges. They are not divided into an infinite number of infinitely small pieces and shared out to nobody. Hence, scientifically 5/0 = 0 remainder 5 not infinity remainder 0.



By the time search for definition of Zero is over, let me have the pleasure of disecting this para.

if x/y = z and remainder is r, airthmetic rule of divison says that r < y, if r > y it means that it can be divided further.

Now if I take your calculation then 5/0 = 0 remainder 5, as remainder 5 > 0 so I must divide remainder 5 with 0 again so again I divide 5/0 = 0 remainder 5, as remainder 5 > 0 so I must divide remainder 5 with 0 again so again I divide 5/0 = 0 remainder 5, as remainder 5 > 0 so I must divide remainder 5 with 0 again so again I divide 5/0 = 0 remainder 5, as remainder 5 > 0 so I must divide remainder 5 with 0 again so again I divide 5/0 = 0 remainder 5, as remainder 5 > 0 so I must divide remainder 5 with 0 again so again I divide 5/0 = 0 remainder 5, as remainder 5 > 0 so I must divide remainder 5 with 0 again so again I divide ....

opsss... now I cant divide any more, one can divide it further for his own pleasure.



Now let us divide 5 oranges among 2 people. In practicality division will never happen if there are no people(Note: I did not use numeric term 0). So as other have already told that either divide by zero is infinity(another term for undefined). Its known fact that infinity is undefined like Zero.


It was wrong to hard code errors for division by 0 into hardware as the normal function would have worked fine 5/0 = 0 remainder 5 and likewise 5%0 = 5.

As remainder 5 > 0 it can not be answer of 5/0.

I think rules of division need to be refreshed again.
Atleast I studied that remainder cant be greater than divider.

It was even more of a mistake to create an exception devoted to this operation,
This operation cant be performed in real life so as per airthmetic an exception was needed, and fortunately we have one "DivideByZeroException".

which is normal in real life and happens every day,
I would like to see an example of 0(Zero) in real life and would like to know where it happens.

Division by 0 is one of the most fundamental concepts which upholds our very existence. Our bodies are continually divided by 0 until our death and subsequent decomposure.
Yaa, very much possible at the time of accidental death, body is suddenly divided by infinity or 0000000 ** n times.

It i the reality that dividing by 0 = 0 remainder (still in tact) that holds the fabric of our precious universe together.
May I put this quote on my site.
[ April 07, 2005: Message edited by: R K Singh ]
 
James Christian
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Dear Mr. Singh,

please read the discussion in the rest of the discussion in the forum.

We have already unanimously agreed that division by 0 presents an exceptional circumstance to the division algorithm. What we are now discussing is the correct implementation of said exceptional circumstance and its relevance and practicalities to OOP in java virtual machine.

In particular with relevance to the existance of a smallest number larger than 0. How do YOU feel in this regard? Is there a smallest number larger than 0? If so what is it? And what practical use does it bring us in life, the universe and everything.
 
lowercase baba
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James,

you seem to be forgetting something... the people who implemented all this were doing one thing (well, a lot of things)... the were trying to get the machine to follow the rules of mathmatics. Addition had to follow all the known rules of Addition, etc.

Now, for hundreds of years, mathematically, it has been agreed upon that you are not allowed to divide by 0, following the strict rules of mathematics. Therefore, when they implemented it on the machine, following these rules, they decided to not allow it.

I think it was Jim who first said it... they COULD have implemented it the way you suggest, but they COULDN'T then have called it division... it's something else that's DARN CLOSE, but fundamentally different.

The problem with returning an object with a quotient and a remainder is that you now have one basic arithmatic function that returns an object while the rest return a primitive. now it starts getting hard to do 7 + (8/3). not impossible, but difficult.

would you then have all arithmatic functions return an ArithmaticResult object? i guess for the other three basic functions, you'd return the sum/difference/product and a 0, while with division you'd return a quotient and a remainder?

one last thing... i don't think we're debating the CORRECT implementation... just ALTERNATIVES. Perhaps the folks who did all this originally defined "correct" to be "follows all mathmatical rules of the day", in which case, the current IS correct.
[ April 08, 2005: Message edited by: fred rosenberger ]
 
R K Singh
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Originally posted by James Christian:
Is there a smallest number larger than 0? If so what is it?



It is: Lim x->0

And still I am looking for definition of Zero ??

And how do you know that 5/0 will result in 0 and remainder 5.

Please correct me if I am wrong, result could be anything from following infinte values

5/0 = 0 remiander 5
5/0 = 1 remiander 5
5/0 = 2 remiander 5
5/0 = 3 remiander 5
5/0 = 4 remiander 5
.
.
.
5/0 = N remiander 5 (N is any number)

Could you please suggest me, which one is the correct answer and why ??

Again thanks in advance.
 
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Originally posted by James Christian:

BUT if you were to ask me would 0.999(9) pass the (0.999(9) instanceof 1) test then the answer would be certainly NOOOO!!!

The reason being 1 is a number that exists and 0.999(9) is a fictitious concept that helps people get their phd's when there is nothing left to study that helps mankind. :roll:



I'm just trying to clarify what you're saying; I'm not disagreeing... yet.

Would 0.111(1) also count as a fictitious concept? Do all repeating decimal numbers? ...or is .999(9) a special case?

Ryan
 
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[JC]: Is there a smallest number larger than 0? If so what is it?

[RKS]: It is: Lim x->0


Perhaps you mean the limit of x as x->0? The limit is zero.

I disagree with James about the validity of writing 0.99999... - but we agree that if such a number does exist (and I say it does), it would be equal to 1.
 
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I suppose PI would also be considered a fictitious concept. We should ban any and all numbers that don't end!

Stupid math people making life so darn difficult for the rest of us. Who do they think they are anyways.
 
Ryan McGuire
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Originally posted by Steven Bell:
I suppose PI would also be considered a fictitious concept. We should ban any and all numbers that don't end!

Stupid math people making life so darn difficult for the rest of us. Who do they think they are anyways.



Or we could just pass a law to set their values to something a little more palatable.

Ryan
 
R K Singh
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Originally posted by Jim Yingst:
The limit is zero.



If I remember correctly then x will tend towards zero but will never be equal to zero. So it will be moving towards zero and will always be greater than zero.
 
Jim Yingst
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We say that x tends to zero, but the limit is zero. The term "limit" is invoked in order that we may transition from the "tending to" to "is". A limit can be a well-defined number (zero, in this case) even though x may never quite reach that number.
 
James Christian
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I suppose PI would also be considered a fictitious concept. We should ban any and all numbers that don't end!


Now this one merits an answer.

The fundamental difference between PI and 0.999(9) (or any infinitley recurring number i.e. 0.111(1) ) is that PI DOES have a FINITE value. With 0.999(9) we are defining the number as finishing with infinite 9's that never end and so by definition the number is not finite and hence does not exist.

While with PI the case is a completely different matter. The value PI is most certainly finite as it defines a ratio relationship between two other FINITE values. The only problem is is that using base 10 maths all of our attempts to determine the value's ultimate limite have thus far been unfruitful because all of our technology has its current limits of precision but that is not to say that one day such technology with enough precision will never exist.

In conclusion, the value PI is my opinion a FINITE value (even tough we are not absolutely sure of its exact value) while 0.999(9) is by definition INFINITE (without defined limits).

 
James Christian
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Dear Mr. Singh,

Definition of Zero:

I'm sure you are aware of the physical laws of conservation of matter and conservation of energy. Just in case you are not simpy put they state that the universe contains a finite amount of energy and a finite amount of matter which never changes. The only thing that changes is their state or position.

So subtraction and addition in the universe in sense of magically making energy disappear or magically creating mass that hadn't previously existed in one form or another are rights which are exclusively reserved to God.

But addition and subtraction in the sense of adding and removing energy or matter from one subsystem to another are operation which we mere mortals also can carry out.

Hence in applying such a case to orange scenario let us say that in the supermarket there exists a law of conservation of oranges (for simplicity let's not talk about oranges being eaten or decomposing)

The supersystem of the supermarkets's oranges holds several subsystems such as "oranges purchased, oranges stolen, oranges thrown away, oranges given to staff, etc."

While moving oranges from one subsystem to another does not affect the number of oranges in the supersystem it does affect the number of oranges in the subsystem itelf.

e.g. Until oranges are sold out there is always a finite number of oranges in the subsystem "oranges in stock". BUT when the oranges are sold out the subsystem contains 0 oranges. This is not to say that the operations of subtraction and division have magically obseleted the matter and energy which makes up the oranges but that they have been removed from that subsystem to any number of other subsystems.

Hence I define Zero as being the mere abcense of something in a defined subsystem.

So when we divide a number of oranges between 0 people and the result is 0 what we are saying is that the number of oranges in a portion to be shared out is 0. i.e. we are not sharing out any oranges and thus the remainder is equal to original quantity.

The inverse of this would be to share out 100 oranges between 1 person. 100/1 = 100 remainder 0. The size of the portion is 100 and the number of oranges that remain in the oranges in stock subsystem is 0.

P.S. WE HAVE ALREADY UNANIMOUSLY AGREED THAT DIVISION BY 0 PRESENTS AN EXCEPTIONAL CIRCUMSTANCE FOR THE DIVISION ALGORITH IMPLEMENTED BY THE HARDWARE.

Dear all,

I wouldn't suggest enforcing all numerical operations to be an object with two values (quotient and remainder) as implicitly they already are as these two expressions illutrate:

 
Steven Bell
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1/3 is right out. So make sure you never try and divide something between three people. It's no longer allowed.

P.S. PI is not a rational number
[ April 09, 2005: Message edited by: Steven Bell ]
 
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Wikipedia: Irrational Numbers

It can readily be shown that the irrational numbers are precisely those numbers whose expansion in any given base (decimal, binary, etc) never ends and never enters a periodic pattern

 
Jim Yingst
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[Steven]: P.S. PI is not a rational number

I don't think James said it was. He said it was a ratio, but that only implies rational if the numbers in the ratio are integers. (Or if we can multiply numerator and denominator by some other number, such that we get a ratio of two integers.)

Reading James' penultimate post, I'm having a hard time making any sense of it; I think the problem may be that James is using very different definitions of "finite" and "infinite" than most of us are accustomed to.

So James:

Is the value of 1/3 infinite?

Is the value (not representation) of 0.333(3)... infinite?

If you say it's infinite - could we at least agree it's somewhere greater than 0 and less than 1? Does that not mean it has finite limits?

Regarding pi: are you saying you think it's possible we will one day discover that pi has an exact nonrepeating decimal representation? (Presumably, a really long one, but not infinitely long?)

Are you going so far as to say that such a representation must exist (because pi is "finite") even if we don't know exactly what it is?
 
Nick George
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James, you and your hypothetical, abstract models! Who's ever heard of an orange that can neither be eaten nor decompose? You can make all the abstract models you like, but in the real world you eat your oranges!
 
James Christian
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posted April 09, 2005 02:04 PM Profile for Nick George Email Nick George Send New Private Message Edit/Delete Post Reply With Quote James, you and your hypothetical, abstract models! Who's ever heard of an orange that can neither be eaten nor decompose? You can make all the abstract models you like, but in the real world you eat your orang



Yes Nick, you certainly do! But the matter and energy which made it up still exist.
 
James Christian
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Is the value of 1/3 infinite?

Is the value (not representation) of 0.333(3)... infinite?

If you say it's infinite - could we at least agree it's somewhere greater than 0 and less than 1? Does that not mean it has finite limits?

Regarding pi: are you saying you think it's possible we will one day discover that pi has an exact nonrepeating decimal representation? (Presumably, a really long one, but not infinitely long?)

Are you going so far as to say that such a representation must exist (because pi is "finite") even if we don't know exactly what it is?



Hi Jim, how's it hangin?

As regards PI, as far as I'm aware it has never been demonstrated that there is a repeating pattern. Do I believe that there is an end to numbers after the decimal point? Now that's a difficult question! I've never looked into so deeply but the fact that its value represents a definite geometric relationship seems to suggest to me that its value is precise and therefore finite (in the sense that it is not an abstract value)

1/3 = 0.333(3)

Now, that's even trickier to explain! At first glance the number appears to fall into the 0.111(1) and 0.999(9) category of abtract non-existant numbers, or at least in base 10.

But allow me to perform the same operation in base 12 where the range of single digit integer is:
0,1,2,3,4,5,6,7,8,9,a,b

b / 3 = 4 (12/3)
so 1 / 3 = 0.4

A clearly defined and representable number. However, I fear the same does not hold true for 0.111(1) or 0.999(9). With what figure shall we miltiply 0.111(1) to obtain 1 or with what amount shall we divide 1 to obtain 0.999(9). For me these figures represent two amounts that are not finite in the sense that there precise quantity cannot be defined. And the reason they cannot be defined is quite simple. They do not exist.
 
James Christian
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If we further this argument we obtain interesting results.

Using base 10 it has been argued 0.999(9) (the closest number to 1) is a real and vaild number equal to 1 on the basis that 0.333(3) * 3 = 0.999(9) and 1/3 * 3 = 1.

However using base 12 as has been seen 1/3 = 0.4 and 0.4 * 3 = 1, no problems. BUT the closest number to 1 in base 12, which if does exist should be equal to our mythical 0.999(9) in base 10. However, in base 12 our mythical number is 0.bbb(b) and only at a far stretch of the imagination could 0.4 * 3 equal 0.bbb(b) in base 12 whereas 0.444(4) * 3 could theoretically fulfill the relationship. Translating 0.444(4) into base 10 we arrive at a value which is greater than 0.333(3). Hence our mythical 0.999(9) when divided by 3 give two different values. Does that seem mathematically acceptable to anybody? Surely if a number is finite (i.e. has a value) it would not matter what base we perform the operation in, the result would always be the same. Why, on this very basis and very firm foundation all of computer science rests itself. If we cannot be sure that mathematical operations hold true irrespective of their base we could not entrust so much in the hands of the binary system. Yet we do, and with very fruitful results.

The plain fact that the closest number to 1 when divided by 3 gives two very different results in two different bases is undeniable evidence that the quantity does not exist unless concrete parameters are installed. In the physical universe, everything is finite and everything has a value.

Returning to the arrow analogy in an abstract way it would be possible to say that its distance to the target halves eternally. Yet the arrow hits the target. How can these two observations ever be compatible?

Well the fact is that distance itself is an abstract concept which is however always implemented in finite terms, no matter how small or how large they may be. We may measure in kilometres or nanometres but the unit of measure is always finite. When we ask the question what is the smallest number larger than 0 our minds are free to run wild and invent all kinds of absract concepts as our imaginations are not bound by the laws of the physical universe.

However, when presented with question: What is the shortest distance an arrow can be from its target without touching it? we now enter into reality and are forced to consider the question from a realistic and physical point of view. Although I would find it hard to prove such a concept the fact is that physical motion while it appears to the eye to be infinitely smooth, which is too slow to detect real motion as Walt Disney as so often proved., is in fact a series of minute translations from one position to another much in the same way we animate objects on a computer screen. The breadth of these translations is goverened by the size of atomic particles and electro-magnetic fields and so the shortest distance that an arrow can be to its target without touching it is in fact finite and determined by physical laws.

For example, I remember at school in algebra when we had to draw one of those curly graphs that approach the axis but never touch it. After many failed attempts at not touching the axis I resorted to super sharpening my pencil until it put surgeon's scalpals to shame and having another bash at it. But try as I might the damn curve always touched the axis... and long before it reached infinity I might add. All this clearly because the distance between my curve and the axis was limited and defined by the breadth of my pencil tip.

So to draw my curve I would need an infinitely sharp pencil it would seem. And if the point of my pencil was infinitely sharp then quite simply put,... it wouldn't exist and it doesn't so no surprises there.
 
James Christian
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P.S I hope I don't come across as being opinionated.
 
Nick George
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I know you're not a big fan of proofs, but for what it's worth to you,
Proof that Pi is irrational
 
Steven Bell
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1/9 = 0.1111...

As a guess the only repeating decimal out of:
0.111...
0.222...
0.333...
ect...

that can't be reproduce in a fraction would probably be .999... and that is because it = 1.

It's just a matter of finding a number x that will go into y*10 the correct number of times with y as the remainder.
[ April 09, 2005: Message edited by: Steven Bell ]
 
Warren Dew
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Nick George:

I know you're not a big fan of proofs, but for what it's worth to you,
Proof that Pi is irrational


I think you're going to have to explain more clearly that this implies that the decimal representation of pi is both infinitely long and nonrepeating. Remember you're talking to a man here who thinks 3x4=11.
 
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Originally posted by James Christian:
If we further this argument we obtain interesting results.

However using base 12 as has been seen 1/3 = 0.4 and 0.4 * 3 = 1, no problems. BUT the closest number to 1 in base 12, which if does exist should be equal to our mythical 0.999(9) in base 10. However, in base 12 our mythical number is 0.bbb(b) and only at a far stretch of the imagination could 0.4 * 3 equal 0.bbb(b) in base 12 whereas 0.444(4) * 3 could theoretically fulfill the relationship. Translating 0.444(4) into base 10 we arrive at a value which is greater than 0.333(3). Hence our mythical 0.999(9) when divided by 3 give two different values. Does that seem mathematically acceptable to anybody?



I think where you are going wrong is
0.bbbbbb(b) in base 12 is not equal to 0.99999(9) in base 10

Actually, 0.bbbbb(b) in base 12 = 11.99999(9) in base 10
 
Jim Yingst
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Working backwards, mostly...

[Jayesh]: I think where you are going wrong is
0.bbbbbb(b) in base 12 is not equal to 0.99999(9) in base 10

Actually, 0.bbbbb(b) in base 12 = 11.99999(9) in base 10


Where did that come from? .bbb(b) in base 12 and .999(9) in base 10 are both equal to 1. 11.999(9) in base 10 would be equal to 12. No idea what that has to do with anything though...

[Steven]: As a guess the only repeating decimal [...] that can't be reproduce in a fraction would probably be .999... and that is because it = 1.

Well, it can be reproduced as a fraction. It's just that that fraction can also be simplified to 1, which is usually the preferred representation.

In decimal:

.111(1) = 1/9
.222(2) = 2/9
.333(3) = 3/9 = 1/3
.444(4) = 4/9
.555(5) = 5/9
.666(6) = 6/9 = 2/3
.777(7) = 7/9
.888(8) = 8/9
.999(9) = 9/9 = 1

It's hard to look at this and understand why .333(3)... would be a "clearly defined and representable number" but .111(1)... and .999(9)... would be "abstract and nonexistent". It will be interesting to see what, ummm, rationalization will come next.

For comparison, in duodecimal:

.111(1) = 1/b
.222(2) = 2/b
.333(3) = 3/b
.444(4) = 4/b
.555(5) = 5/b
.666(6) = 6/b
.777(7) = 7/b
.888(8) = 8/b
.999(9) = 9/b
.aaa(a) = a/b
.bbb(b) = b/b = 1

James: any repeating decimal can be translated into a ratio of two integers, and we could also find a representation in some other base(s) which would terminate in a finite number of digits - just like you did for 1/3. For example, .111(1) in decimal can be represented as .14 in duodecimal. Or as .1 in base 9.

[James]: However, in base 12 our mythical number is 0.bbb(b) and only at a far stretch of the imagination could 0.4 * 3 equal 0.bbb(b) in base 12

I guess my imagination stretches farther than yours. It's true that 0.bbb(b) is not the first way I would think to state the answer, especially when the question is formed in this way. But that doesn't mean it's not true.

whereas 0.444(4) * 3 could theoretically fulfill the relationship.

Only if you think (as Warren alludes) that 4 * 3 = 11, or b in duodecimal. I'd think 0.444(4) * 3 would equal .ccc(c) (since 4 * 3 is 12, not 11( except of course all those c's are too big for duodecimal, and each c would beome 10 in doudicimal, carrying over to yield a result of 1.111(1). Yeah, I know using the c temporarily was a bit unorthodox, but at least I got the right result.

Does that seem mathematically acceptable to anybody?

It works just fine of you do the math right.
 
Nick George
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[Warren Dew]I think you're going to have to explain more clearly that this implies that the decimal representation of pi is both infinitely long and nonrepeating. Remember you're talking to a man here who thinks 3x4=11.

It's absolute gibberish to me, and therefore useless as a proof, per se, but it sounded as though James beleived pi to have an end. Though I cannot explain exactly how the proof works, its existence demonstrates that pi cannot be expressed as a ratio of two integers.

But, come to think of it, all the proofs listed here so far have fallen by the wayside.
 
James Christian
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Does that seem mathematically acceptable to anybody?

It works just fine of you do the math right.



Yeah, sorry guys. I really screwed that one up big time.

But the question still remains with what do we divide 1 to obtain 0.999(9) and for them matter to obtain 0.000(0)1 and furthermore seeing as the basic unit in each base has a different magnitude after the decimal point (i.e 0.1 > 0x0.1) is not 0.000(0)...1 greater than 0x0.000(0)...1 . If the smallest number greater than 0 does exist and it expresses the difference between 1 and 0.999(9) (which we have seen = 1), how can the smallest number greater than 0 have two different values in two different bases?

Surely by now we must be unanimous that the smallest number greater than 0 does not exist as there will always be a number smaller than it.
 
James Christian
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posted April 09, 2005 10:23 PM
--------------------------------------------------------------------------------
I know you're not a big fan of proofs, but for what it's worth to you,
Proof that Pi is irrational
--------------------



Thanks Nick, that just made things SOOOOOOOOO clear.

I'm sure you're right that PI cannot be expressed as a ratio of two finite integers but the basic point is that circles and spheres exist. In fact they seem to be God's favourite shape. Look at the Sun, the stars, the planets, magnetic fields, atoms etc.. I know they are not perfect spheres but we do seem to see a recurring pattern in the design of the universe.

I don't know if one exists in the world but I presume that a perfect circle has an area which is finite. Maybe with our limited maths and technology we cannot precisely define what that value is but nontheless the value is constant. Nothing more and nothing less than whatever it is. And if the relationship between the area and the radius is defined by PI then this seems to suggest to me that the value of PI is exact. You cannot use a value which is lesser or greater and still obtain the correct result. And seen as God seems to have recurrently used this value in his geometric calculations when he founded the universe seems to give further proof that not only is the value existant and precisely defined but also that it is a very practical value.

In fact using RADIANS as opposed to DEGREES has proven very useful to engineers all over the world. Perhaps it is true that PI isn't definable in any base that we can conceive, except of course for base PI where PI = 1.

However, returning to the original topic I fear that the same arguments cannot be presented for the existence of the smallest number greater than 0.

What practical use does it have?
Does it help engineers?
Is it a finite value?
Are there any patterns in the universe which depend on its value?
Can it be defined in terms of geometric relationhips as can PI?
Can it be demonstrated that there is no smaller value?
Can you think of one practical usage of the number other than highly entertaining philosophical debates?
 
James Christian
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whereas 0.444(4) * 3 could theoretically fulfill the relationship.

Only if you think (as Warren alludes) that 4 * 3 = 11, or b in duodecimal. I'd think 0.444(4) * 3 would equal .ccc(c) (since 4 * 3 is 12, not 11( except of course all those c's are too big for duodecimal, and each c would beome 10 in doudicimal, carrying over to yield a result of 1.111(1). Yeah, I know using the c temporarily was a bit unorthodox, but at least I got the right result.



I should have written 0.3bbb(b) * 3 = 0.bbb(b). Well spotted Jim.
 
James Christian
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But, come to think of it, all the proofs listed here so far have fallen by the wayside.



Does wayside mean "on deaf ears" ?
 
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I don�t know what you guys are talking about, so I hope what I say can help.

Just as 3.1605 cannot be defined as an integer object, Pi cannot be defined as a float or integer object. It can however, be defined as a function, which is a special type of number object.

The question to ask is can 0.9999.. infinity be defined as an floating point number and can this number be the of value 1.

Ask me this if you was to give a child a card with a number 0 plus infinite number of 9 cards and told him to lay out the 9 cards until all the nine cards magically disappear and the 0 turns into a 1, he will most likely to say that will never happen even if infinity is reachable.

Something that is infinite cannot equal something that is finite like an integer or float number.
[ April 10, 2005: Message edited by: Gerald Davis ]
 
Steven Bell
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Originally posted by James Christian:


But the question still remains with what do we divide 1 to obtain 0.999(9)


That is easy, you divide by 1 because 1 == 0.999(9)


and for them matter to obtain 0.000(0)1 and furthermore seeing as the basic unit in each base has a different magnitude after the decimal point (i.e 0.1 > 0x0.1) is not 0.000(0)...1 greater than 0x0.000(0)...1 .


The problem is you are trying to create the smallest number greater than zero which we know does not exist. When you write 0.000(0)...1 by putting the 1 there you say that the number terminates at some point and so I can make a smaller number by dividing by 2 or adding another zero.


If the smallest number greater than 0 does exist and it expresses the difference between 1 and 0.999(9) (which we have seen = 1), how can the smallest number greater than 0 have two different values in two different bases?
Surely by now we must be unanimous that the smallest number greater than 0 does not exist as there will always be a number smaller than it.


Yes, the smallest number greater than zero does not exist, but I don't see how that invalidates the numbers
.111(1)
.222(1)
.333(3)
.444(4)
ect...
in some way, or shows that PI actually has an end point.

Saying that the value of PI is an irrational number does not mean that the area of a circle is infinite you can still bound PI by 3.15 and 3.13. If you can bound a number that number is not infinite, but that is not the same as saying it has an infinite number of digits after the decimal place, which it does.
 
James Christian
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Whether PI has an end point or not I'm not exactly sure as much as I would like to I can't make head nor tail of the link which provided the "proof". My original intention was always to show that the difference between PI (whether it ends or not) and the smallest number greater than 0 or the greatest number lesser than 1 is that while PI is a definable and practical constant the other two values indefinable and therefore do not exist. It seems that we all hold the same opinion yet are coming from different directions in our analogy's.

I agree that 0.333(3) etc represent the nearest we can express in decimal, fractions such as 1/3 and before merely aimed to demonstrate that while these figures in base 10 seem indefinable in other bases they give easily managable values. However the same does not hold true the mythical smallest number greater than 0 and greatest number lesser than 1.

P.S. I still think I'm right about the JVM behaviour of division by 0 though.
 
Ryan McGuire
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Originally posted by Gerald Davis:

Ask me this if you was to give a child a card with a number 0 plus infinite number of 9 cards and told him to lay out the 9 cards until all the nine cards magically disappear and the 0 turns into a 1, he will most likely to say that will never happen even if infinity is reachable.

Something that is infinite cannot equal something that is finite like an integer or float number.



But .999(9) isn't infinite. It just has an infinite number of digits. The value doesn't "go on forever", but the representation of that value does. In this respect it's similar to PI.

Let me state the obvious:
There are many ways to express any given number.

For instance: If you divide up a pie into 9 pieces, you could say each piece is "1/9" of the original. Or it might be "0.111(1)" of the originial. Or it might be "0.1 (base 9)" of the original. The fact that in base 10 the piece of pie has a repeating representation while in base 9 it has a terminating representation doesn't change the piece itself; it's still just one ninth of the pie.


Now concerning .999(9) (base 10) being equal to 1:
A. Would you agree that 1/9 and 0.111(1) are two representations of the same value?
B. Would you agree that 2/9 and 0.222(2) are two representations of the same value?
C. Would you agree that 2 * 0.111(1) = 0.222(2)?
D. Would you agree that 9 * 0.111(1) = 0.999(9)?
E. Would you agree that 9 * (1/9) = 0.999(9)?
F. Would you agree that 9/9 = 0.999(9)?
G. Would you agree that 1 = 0.999(9)?

For anyone that answers Yes to A but No to G, where is your first No?

Ryan
 
James Christian
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For instance: If you divide up a pie into 9 pieces, you could say each piece is "1/9" of the original. Or it might be "0.111(1)" of the originial. Or it might be "0.1 (base 9)" of the original. The fact that in base 10 the piece of pie has a repeating representation while in base 9 it has a terminating representation doesn't change the piece itself; it's still just one ninth of the pie.



Well said!

As regards the decimal representations 0.111(1), 0.222(2), 0.333(3), 0.444(4), 0.555(5), 0.666(6), 0.777(7) and 0.888(8) are valid as there is no other way of representing these figures in decimal without resorting to a fraction.

However, the same cannot be said for 0.999(9) as nobody has any difficulty in reading and understanding 1. In fact the only time we see 0.999(9) mathematically is when we do an operation similar to the following with a calculator:



And this is only due to the fact that the calculator, due its limitations, has lost that certain something that makes 1/3 ever so slightly larger than 0.333 .

But when you ask a human to do the same series of operations, instinctively he/she instantly writes 1 as the result as they remember and account for the loss of precision.

Anyway, even if we are to admit and widely accept that 0.999(9) = 1 it would only stand to uphold the evidence provided thus far that there is no smallest number greater than 0 and hence no greatest number lesser than 1.

How do you feel in this regard?
Is there a smallest number greater than 0?
If so is this the greatest number lesser than 0.1? 0.0999(9)
And this 4.999(9)? Does that equal 5?
And finally what about this? 7.1000(0)1 Is this the smallest number greater than 7.1?
 
Steven Bell
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However, the same cannot be said for 0.999(9) as nobody has any difficulty in reading and understanding 1. In fact the only time we see 0.999(9) mathematically is when we do an operation similar to the following with a calculator:

And this is only due to the fact that the calculator, due its limitations, has lost that certain something that makes 1/3 ever so slightly larger than 0.333 .



1/3 is not larger than .333(3) they are equal. The calculator has made an approximation of 1/3 that is actually less than 1/3 and probably equal to 0.333333333 (to some specific number of digits) rather than the actual .333(3).


Is there a smallest number greater than 0?


No.


If so is this the greatest number lesser than 0.1? 0.0999(9)


No, those two numbers are equal.


And this 4.999(9)? Does that equal 5?


yes


And finally what about this? 7.1000(0)1 Is this the smallest number greater than 7.1?


No, there is no smallest number greater than 7.1 or any other number for that matter.

I think the problem is you are trying to say/show that
1 - .000(0)...1 = .999(9)
This is not the case, the correct thing would be:
1 - .000(0) = .999(9)
Notice there is no trailing 1. 1-0=.999(9) because 1=.999(9)

.999(9) is not the largest number less than one (as it would be if it was the difference between 1 and the smallest number greater than 0, which doesn't exist) it is equal to 1, it is the same number, there is no difference between it and 1.

1 = .999(9)
2-1=.999(9)

This is perfectly correct, but it is simply easier to write 1, and easier to explain.
 
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