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[LOCKED] [PUZZLE?] E Numbers
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thebruce
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Joined: 16 Aug 2004
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Location: Kitchener, Ontario
neon snake wrote:
I can't quite make sense of this sentence.

If it's finite, it proves it's not infinite?
If it's infinite, it proves it's not finite?

But we can never prove it's infinite, so we can't prove it's finite? Unsure what you're trying to get to here!


heh sorry, disproven... if we can't prove it's infinite, then we can't prove it's not finite... seems like wordplay, but the emphasis is on what is being proven or disproven.

If A != B, it proves A=B is wrong.
If A=B, it proves A != B is wrong.

assuming one way or the other, A and B can be calculated, the former or latter statement will br proven. By nature, if we can test that A=B, then we should be able to test if A != B.

If it's finite, it proves it's not infinite?
If it's infinite, it proves it's not finite?

By nature, if we can test that a number is finite, then we can deduce it's not infinite. But we can't, under the same guidelines, prove a number is infinite, so we can't deduce that it's not finite. The statements seem to contradict each other. We simply can't prove whether a number is infinite - it may or it may not be, because we will never know if we can ever come to an end. We can never know the 100% precise value of an infinite number; there will always be an air of uncertainty, even if microscopic, about any result that is calculated which retains that concept of infinity.

We can only ever know whether a number is finite or infinite if we can calculate to its end. Therefore we can never know the precise value of an infinite number. But, again, the infinite aspect of the number can be represented in math, in order that we can work with it - cancel it out, retain it, or round it. The only one of those three that will yield an accurate result (a finite decimal value) is cancelling it out. Rounding it no longer results in a precise value, and retaining it still requires a calculation in order to result in a final accurate value. 1/3, eg, isn't a final accurate value by the definition above, it's still an equation. A fraction is an equation which yields a decimal value. 1/3 is an equation which represents an infinite value (one which we cannot calculation to full precision).
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PostPosted: Tue Dec 28, 2004 5:35 pm
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neon snake
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Quote:
We simply can't prove whether a number is infinite - it may or it may not be because we will never know if we can ever come to an end.


But, we can prove that a number, or more correctly sequence (in this case e) is infinite.

Have a look at the calculation for e again

Quote:
e = 1 + 1/1! + 1/2! + 1/3! +...


Without delving into complicated proofs about why this will never end, it's fairly obvious. Otherwise, we are saying that my earlier '1, 2, 3, 4, ...etc etc' might end at some point.

As for the value of using 'representations' such as e or pi in mathematics, where they are not cancelled out, I'm sure you don't mean that.

Otherwise, you're questioning the value of the equation c=2*pi*r.
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PostPosted: Tue Dec 28, 2004 5:54 pm
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Olorin
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neon snake wrote:
Quote:
e = 1 + 1/1! + 1/2! + 1/3! +...


This is the calculation for e - thanks, Olorin.


No problem, but to avoid being charged with plagiarism,
let me put credits where credit is due, and, specifically, to L. Euler, who got to that forumla much sooner than I did Smile

F.O.R.

PostPosted: Tue Dec 28, 2004 6:06 pm
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thebruce
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neon snake wrote:
But, we can prove that a number, or more correctly sequence (in this case e) is infinite.

Have a look at the calculation for e again

Quote:
e = 1 + 1/1! + 1/2! + 1/3! +...


Without delving into complicated proofs about why this will never end, it's fairly obvious. Otherwise, we are saying that my earlier '1, 2, 3, 4, ...etc etc' might end at some point.

Yes, your sequence itself includes the concept of infinity - 'etc etc', or the '...' in the former equation. Mathematically, that's a valid formula. But, the proof that it's infinite solely lies in your own statement that you say it's infinite.
You could write the equation for e in such a way that is self-contained. But, as the answer is e, the equation must also include the concept of infinity. ie, the fact you have '...' at the end is the factor that 'cancels out' (or balances) the infinite value of e. You can't write an equation to calculate e whose result is a finite value. Obviously Smile.
I never said infinity can't be represented, just that the precise value can never be calculated, and that equations utilizing an infinite value must be balanced - for a precise value, a finite result requires the cancellation of the infinite value, or a result which itself contains an infinite variable, which means the result isn't yet fully calculated.

So, a finite value can only result from a finite equation, where if an infinite variable existed, it was cancelled out. Otherwise, the result can never be calculated to full precision. I don't see how that's a difficult concept.

Quote:
As for the value of using 'representations' such as e or pi in mathematics, where they are not cancelled out, I'm sure you don't mean that.
Otherwise, you're questioning the value of the equation c=2*pi*r.

Not sure what you mean by that... given precise values of c and r, the equation will be reduced to the true, precise value of pi - ie, the value which we cannot calculate to its full precision. So can we truly know the accurate, precise values of c and/or r? Which by nature can't be infinite numbers, otherwise they aren't precise measurements... if you give both c and r finite values, then pi will also be a finite number.
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PostPosted: Tue Dec 28, 2004 6:20 pm
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thebruce
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 		we further calculate the precision of pi by giving c and r further precise values. But as the precision of pi approaches infinity, assuming pi is infinite, the precision of c and/or r will also need to approach infinity.
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PostPosted: Tue Dec 28, 2004 6:24 pm
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yanka
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thebruce wrote:
Quote:
so saying that a fraction cannot really be considered infinite because it cannot be practically calculated to the full extent of its infinity is self-contradictory.

Welcome to paradox and the futility of imagining infinity.

This has nothing to with the ability, or, rather, inability, to imagine infinity, and everything to do with circular logic.
thebruce wrote:
But I never said a fraction cannot be considered infinite.

Hm... perhaps I'm misunderstanding you, but you earlier wrote:
thebruce earlier wrote:
In that sense, any repeating number and any (apparently) non-repeating number may or may not be an infinite number - but we can never truly know if it's infinite unless we never come to the end, which is not a concept we can ever grasp.. .theoretically, a number may be as long as it takes thousands of years to calculate... is it then infinite? nope, cuz it ends. But long before that we have essentially come to the conclusion that it's an infinite number.

I may have interpreted you wrong, but isn't what you're saying here: "a fraction proven to be infinite may not, in fact, be infinite, because it cannot be practically calculated to the full extent of its infinity"??

Furthermore, you also wrote:
thebruce also wrote:
So yes, we can express a value in different ways in math, but when calculated as a decimal number, the only proof we have that it's infinite or repeating is entirely based on how far we can calculate the decimal number.

The essential part bolded by me. Again, I'm understanding that as "a fraction cannot really be considered infinite because it cannot be practically calculated to the full extent of its infinity."

Finally, you said in the post that I replied to:
thebruce wrote:
I was using a fraction as an example of a mathematical representation that may or may not represent an infinite number.

I assumed this to refer to this example of a fraction which "may or may not be infinite":
thebruce wrote:
when it comes to dealing with the concept of an infinite number, even stating a number as infinite is based on an assumption that's based on as far as we've calculated. take the value one third. There's no decimal number we can write that will be 100% accurate. But, with the genius of math, we can express the value with a representation where infinity doesn't need to be calculated. 1/3. And we can work with that, where results may or may not represent an infinite value (assumed) if calculated, whether repeating or not repeating, like 1/7.

With the essentials bolded, again, by me. Are you, or are you not saying that
- a fraction that is considered to be infinitely long (such as 1/3), may actually not be infinitely long ("any repeating number and any (apparently) non-repeating number may or may not be an infinite number")
- it may only be as long as would take thousands of years to calculate (see above)
- since we will never calculate for that long, we are assuming it to be infinite ("long before that we have essentially come to the conclusion that it's an infinite number")
- if we had calculated it for that long, it might actually stop at that point (logical extrapolation)
??

Please correct me if I misquoted/misinterpreted you somewhere; but if I haven't, then what you're saying is blatantly wrong. You demand that something infinitely long be proven as being infinitely long by an infinitely long calculation; in addition, that seems to be the only proof (of that fraction's infinite length) you are willing to accept. This is the circular logic: if we define something "infinitely long" as "something that can never finish being calculated", and the proof of the infinity of that lenght is the infinitely long calculation, then we can never prove that that "something" is infinitely long, because we cannot infinitely calculate it. Aside from being circular, this logic is wrong to boot: the proof we have that a fraction is infinite or repeating is NOT based on how far we can calculate the decimal number. Indeed, we cannot offer "calculation of the decimal number" as proof of a fraction's infinite length. Still we CAN (using that "we" very loosely), of course, prove, that something is infinitely long; we can prove that those 3s will not magically stop repeating at some point without checking each point (the position after the decimal point, in this case) to see whether it is the point where the 3s stopped.
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PostPosted: Tue Dec 28, 2004 7:18 pm
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yanka
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thebruce wrote:
By nature, if we can test that a number is finite, then we can deduce it's not infinite. But we can't, under the same guidelines, prove a number is infinite, so we can't deduce that it's not finite.

We (or, rather, they Smile ) can prove that it's infinite under different guidelines (i.e. not by "testing it".)
thebruce wrote:
We simply can't prove whether a number is infinite

Yes, we can - offhand I remember at least two proofs in this very thread. For some reason you choose not to accept them as such.
thebruce wrote:
- it may or it may not be, because we will never know if we can ever come to an end.

We KNOW that we CANNOT ever come to an end (of an infinite number), by definition. We also know that we don't need to TEST a number to make sure its lenght is infinite in order to PROVE that its lenght is infinite.

thebruce wrote:
We can never know the 100% precise value of an infinite number; there will always be an air of uncertainty, even if microscopic, about any result that is calculated which retains that concept of infinity.

Yes, we can. The 100% precise value of 1/3 is 1/3. There is no 100% accurate decimal representation of that value available, but the precise value is still known... precisely Smile

thebruce wrote:
We can only ever know whether a number is finite or infinite if we can calculate to its end.

Omg, that is just so wrong. I can look at this fraction:
5/10000000000000000000000000000000000000000000000000000000000000000
and immediately know that it is finite, without having to calculate it.
thebruce wrote:
Therefore we can never know the precise value of an infinite number.

[EDIT]
I would argue that the correct statement should be: "We cannot express an infinitely long fraction in decimal form with 100% precision."
I'm taking out the physical example I used for fear it would lead to further confusion and discussion of Planck scales Smile
[/EDIT]
thebruce wrote:
A fraction is an equation which yields a decimal value.

A fraction is not an equation. It is a quotient of two numbers, and is itself a number. It does not have to be expressed in a decimal notation. You are arguing that because the decimal notations of some fractions cannot be written out in their entirety, the values of such fractions are not precisely known, which would only make sense if values were restricted to be represented solely in decimal notation - and they are not.
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PostPosted: Tue Dec 28, 2004 8:35 pm
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RI_Barnica
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yanka wrote:
A fraction is not an equation. It is a quotient of two numbers, and is itself a number. It does not have to be expressed in a decimal notation. You are arguing that because the decimal notations of some fractions cannot be written out in their entirety, the values of such fractions are not precisely known, which would only make sense if values were restricted to be represented solely in decimal notation - and they are not.


Interesting point. We need to remember that a number is not the same as its representation! Althought 1/3 is infinite in base 10 (0.33333...), it is finite in base 3 (0.1). So then, an alternate definition of a trancendental number (beyond the initial definition we were given which was something like " a number which cannot be expressed as a fraction") is one which cannot be expressed in ... what's the word I'm looking for? I want to say decimal form, but that seems to imply base ten. I'm sure someone around here has the mathematical vocab for this... a little help here?

(Damn, blew my resolution already!)
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PostPosted: Tue Dec 28, 2004 11:15 pm
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Chris K
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 		OK, thebruce, care to explain this equation?

e ^ (i * pi) + 1 = 0

That has been mathmatically proven.
The so called 'infinite' numbers come out a normal number.

Also, this series:

1 + 0.8 + 0.64 + 0.512 + 0.4096 ...

is infinte, but it adds up to exactly 5.

The beauty of mathmatics is that we can do stuff like adding up infinte series, and calculus - which involve infinity, even though as humans we can't really interact with infinity.

PostPosted: Wed Dec 29, 2004 9:29 am
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BoBzOmBiE
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 		Ah, its wonderful to see that we have a nice mix of philosophers and mathematicians, and its also nice to see how different their views can be. I myself am a mathematician, but infinite maths has never been my strong point. Als, to the post above, the sequence does not add up to five, as the number tends toward infinity, the sum of the numbers in the sequence tend toward five, its not actualy possible to say that it gets there, its just assumed that it does if the sequence is carried on for an infinite amount of time. But i have no doubt that someone will say this is wrong, since we are dealing with a subject which is more down to the method of teaching and who was doing that teaching, than any form of law.
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PostPosted: Wed Dec 29, 2004 11:08 am
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thebruce
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yanka wrote:
Are you, or are you not saying that
- a fraction that is considered to be infinitely long (such as 1/3), may actually not be infinitely long ("any repeating number and any (apparently) non-repeating number may or may not be an infinite number")

I'm stating that our definition of an infinite number is one which cannot be fully calculated. So 1/3 by definition is the representation of an infinite number 0.333rpt...

Quote:
- it may only be as long as would take thousands of years to calculate (see above)

1/3 is not a good example of that, because we're ->100% positive that we can never reach the end Smile we say 1/3 to represent the value 0.3 repeating. The latter already defines that the 3 will never end, so we know 100% that the number is infinite. The question is, can we ever know the true value of 1/3? But a number which we find non-repeating for as long as we calculate, yes, we may consider it infinite before spending thousands of years to find out it may possibly end.

Quote:
- since we will never calculate for that long, we are assuming it to be infinite ("long before that we have essentially come to the conclusion that it's an infinite number")

sure

Quote:
- if we had calculated it for that long, it might actually stop at that point (logical extrapolation)

If we calculate any number for any period of time, we may or may not come to the end of the calculation. How do we know if a number is infinite? Because at some point we've realized that we can't come to the end. The proof can't include the concept of infinity, because we're trying to prove infinity. Without including some representation of infinity in the proof of an infinite number, the only way to prove a number of technically infinite, is by assuming it is because we can't reach the end of the number.

ie prove 1/3 is infinite. Because the answer is 0.3 repeating? That proof includes the concept of infinity - repeating. Sure, in math that's acceptable. Because we deal with infinity by representing it in some form. But the precise proof that a number is infinity can never exist because it can never be reached.

[quote]You demand that something infinitely long be proven as being infinitely long by an infinitely long calculation; in addition, that seems to be the only proof (of that fraction's infinite length) you are willing to accept. This is the circular logic: if we define something "infinitely long" as "something that can never finish being calculated", and the proof of the infinity of that lenght is the infinitely long calculation, then we can never prove that that "something" is infinitely long, because we cannot infinitely calculate it.[quote]
exactly. it's circular because infinity is not a mathematical certainty with which to calculate. It's a mathematical concept with which we can work to arrive at either a finite (by cancellation) or an equally infinite (including a representation of infinite) value.

Quote:
Still we CAN (using that "we" very loosely), of course, prove, that something is infinitely long; we can prove that those 3s will not magically stop repeating at some point without checking each point (the position after the decimal point, in this case) to see whether it is the point where the 3s stopped.

I'd like to see the proof. And if the proof does not either remove infinity or round it (replace with a finite value), or end up simply proving itself (retain the concept of infinity) then I'll stand corrected.

Quote:
Quote:
We simply can't prove whether a number is infinite

Yes, we can - offhand I remember at least two proofs in this very thread. For some reason you choose not to accept them as such.

Which ones? Without including the concept of infinity in the proof, how can you prove infinity?

Quote:
We KNOW that we CANNOT ever come to an end (of an infinite number), by definition

we define true infinity by never being able to come the end.
So, offer proof that a number is infinite, without including a concept of infinity in the proof, or the presupposed claim that the number is infinite.

Quote:
Yes, we can. The 100% precise value of 1/3 is 1/3. There is no 100% accurate decimal representation of that value available, but the precise value is still known... precisely

Quote:
A fraction is not an equation. It is a quotient of two numbers, and is itself a number

It is a type of number which is a representation of another type of number. A fraction is one number divided by another. The fraction A/B = A divided by B. 1/3 is not proof of an infinite number, because it's an unfinished equation. Any equation which results in a final fraction can still be reduced a step further, to a decimal number.
1/3 is not a precise value, it's an equation - 1 divided by 3. In order to work with a precise value, we use 1/3 because 0.3333333333 is not precise. 1/8 is as precise as 0.125 because we know that 1/8=0.125. So in any equation we can choose to use 1/8 or 0.125 depending on which would be easier to calculate with. However, in order to retain precision in an equation that includes the value of 1/3, we continue to use 1/3, because 0.33333 isn't accurate enough. In the end, an equation using 1/3 with either result in a precise value, or an infinite value, depending on whether the 'infinity' of 1/3 (the precise value which we cannot calculate) is cancelled out - which we know by working with the 1/3 representation of the repeating number. 1/3*3 we know cancels out infinity. 1/3*9 we know cancels out infinity. Those numbers we know we can arrive at a precise, finite value. 1/3*2 we know we can never know the precise value because the concept of inifinity is still present. So in order to come to a precise value, any equation must be written using 1/3*2 or 2/3, instead of the value 0.66666666 or whatnot...

We can never come to a precise value without removing the concept of infinity from the equation by cancellation. Representation (eg 3pi) is still not an exact value. You can't make a doorknob if the answer is 3pi. pi needs to be replaced by a sufficiently accurate value in order to calculate a result.

That's what I'm getting at...

Quote:
We need to remember that a number is not the same as its representation! Althought 1/3 is infinite in base 10 (0.33333...), it is finite in base 3 (0.1)

Then you're adding the rules for which numbers are calculated. base 10 1/3 != base 3 1/3. base 3 1/3 = 0.1 = base 10 1/10. You're simply altering the way values are written. What's base 3 0.1repeating in base 10? *shrug* base 3 1/3 doesn't include the concept of infinity because we can calculate the answer.

Quote:
e ^ (i * pi) + 1 = 0

http://www.math.toronto.edu/mathnet/questionCorner/epii.html

There's some good proof for that statement: e^(pi*i)=-1 which utilizes cos and sin... and, the cancellation is in the cos and sin values, two infinite representations where the known difference is finite, the 'infinity's cancel each other out. In other words, we can't calculate the proof that the above equation = 0, because if replace the e and pi variables with decimal values, the result will not be precise. But, through math we can prove the values work by replacing the finite and infinite values with an equation where the infinite variables are workable - where without replacing those variables with a finite amount, the variables' infinity themselves cancel out, in order to result in a finite value.

Maybe it's bordering on metaphysical, or something like that, I dunno. But infinity itself can only be worked with as an all-or-nothing type of value - a number is either finite or infinite. A finite value can only be achieved by removing any aspect of an infinite value from the equation, which can only happen by cancelling infinity with another infinity. Otherwise, the result still retains some form of infinite value.

We can't prove (arrive at) an infinite value, because the equation would require an infinite value, a value which we cannot arrive at to calculate. We can however, arrive at a value representing an infinite number if the equation includes an infinite representation. A finite equation will result in a finite value. An infinite equation will result in an infinite value.

hehe... man, my head burns. Razz
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PostPosted: Wed Dec 29, 2004 4:22 pm
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neon snake
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Quote:
Representation (eg 3pi) is still not an exact value. You can't make a doorknob if the answer is 3pi. pi needs to be replaced by a sufficiently accurate value in order to calculate a result.

That's what I'm getting at...


To be honest, thebruce, I'm starting to lose track of what you are getting at.

The above statement seems to be saying 'having to round numbers to a particular decimal place is a bit inaccurate, innit?'

Well, yes, obviously. But I'm reasonably sure we've not spent this thread arguing about that.

Any chance you can bullet point the points we've made that you disagree with, in as clear and concise a manner as possible?

Then, we can deal with them one by one.

I make the next point only because I myself made an embarassing error when I started this whole thing, by assuming that 'e' is random. I happily admit that I was completely wrong (doubly embarassing given the signature line I've chosen).

Quote:
then I'll stand corrected.


I hereby ask that, if we can prove our arguments, then you do indeed stand corrected. I say this because it seems in some of your posts that you are simply rejecting proofs, although that may just be me misreading them.

Cheers.
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PostPosted: Wed Dec 29, 2004 4:47 pm
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Chris K
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 		OK. I think the problem here is that you think that an infinite series can't have an exact, finite sum.

Wrong. It absolutely 100% definitely definitely can.

This is the basis of huge chunks of mathmatics, including Calculus and Trigonometry.

You said there was 'a nice proof of e^i*pi that used sin and cos' - AHH!!! Sin & Cos are infinite series!!!

By your reasoning:

1/3 = 0.333...
2/3 = 0.666...
3/3 = 0.999...

When really:

3/3 = 0.999.... = 1

-----------------------------

Just for the record I'll prove that 0.9 recurring equals 1:

x = 0.99999.....
10x = 9.9999....
9x = 9
x = 1

Ta da!!

PostPosted: Wed Dec 29, 2004 7:04 pm
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tanner
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Quote:
x = 0.99999.....
10x = 9.9999....
9x = 9
x = 1


haha

check out an earlier post of mine -- same argument

he didnt believe me then so ill be very surprised if he believes you now.

i've just given up (im sure a lot of you will be pleased to know)

and i hope thebruce does well in all his studies
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PostPosted: Wed Dec 29, 2004 7:20 pm
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yanka
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neon snake wrote:
Any chance you can bullet point the points we've made that you disagree with, in as clear and concise a manner as possible?

If you don't mind, I would rather wait for you to do that^, than reply to your post now only to find out that I responded to a bunch of stuff that you are not saying.

I am, however, relatively certain that you are saying this:
thebruce wrote:
1/3 is not a precise value

so I will reply to it now.

Let me begin by saying that I am boggled by your insistence.

I am not sure how else to express this: a fraction is a value. It is not some sort of vague, incomplete thing that achieves pure precision, completion, or a status of a "number", by being written out in decimal form. It already is that.

Here, let's look at that 0.125 that you used. You say that a "fraction can still be reduced a step further, to a decimal number." 0.125 is a fraction, and it has not been reduced (quite the contrary, as you will see in a moment); it is merely 125/1000 written in a different way. <-- That number can be reduced to 1/8 (by dividing the numerator and the denominator by 125) - not the other way around. In order to write out any fraction in the decimal form, that fraction needs to be operated on (again, not the other way around): it needs to: first be expressed with a denominator that is a positive power of ten; then be written in the decimal notation. Basically, you have to do some work to write 1/8 "decimally": multiply it by 125 to get the denominator to be 1000, or do something else, like long division... You don't have to do all of this work in order to express the value of 1/8 - you can just leave it as is.

What you're doing with one third is this:
- does 3/10 equal 1/3? Nope, not quite
- does 33/100 equal 1/3? Nope, not quite
- does 333/1000 equal 1/3? Nope, not quite
...
and so on forever

Why bother? Why go through all the trouble?? If you want to know the precise value of 1/3, just leave it written as 1/3, and be done with it Smile
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PostPosted: Wed Dec 29, 2004 9:43 pm
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