Can a = b when a and b are different integers?

I apoligize for misquoting you bugz, I wasn't expecting someone else to start aruging, so I thought that it was you:o.

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lol i didn't post that message. you got your names wrong Alphanos!

but exactly, as you yourself pointed out

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then (1+0.9 repeated)/2 could not possibly be greater than 0.9 repeated!
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so what is it equal to? it can't be greater than 1. it can't be less than 0.9 repeating. so its either 1, 0.9 repeating or undefined.

undefined won't make any sense.

so if its 1 or 0.9 repeating, it would indicate that the three numbers are the same, like:

if the average of 4 and x is 4, what is x? you would say 4 wouln't you?

the truth is, if there is indeed a difference between two numbers x and y, their average should be in between them. however as indicated, this would happen iff there isn't a difference between x and y, so x=y, or in other words, 0.9 repeating =1

i understand that conceptual numbers can be represented algebraically or can be used as an approximation, but i don't see how it is relevant to anything we've talked about. btw i am representing those conceptual numbers algebraically, not using an approximation, in case thats what u r saying.

Gandalf:
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but that last digit would be at infinity making it infinitely less than .99 repeated, which is infinitely less than 1. so if .99 repeated is infinitely less than one and .999 .... 8 is infinitely less than .999 repeated then .999 repeated = .999 ... 8 so they both equal 1. If done an infinited amount of times 1 could equal 0. This is with your logic though. I agree that it isn't possible. Even though the difference is infitesimally small it adds up
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you cannot have .999...8 where there is an infinate number of 9's. you can't just say at the end of infinity I will add a 8, because there is no end to infinity!!! That was what I am trying to say. sorry about the bold, i was just illustrating my point, or in NotLKH's words:IMHO, lol!

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First things first, you should understand that Gandalf is not saying that there can be a number with an infinite number of 9s followed by an 9. He's saying that there can't be!!! His entire argument was to show how stupid it is to say that (0.9 repeated+1) / 2 is greater than 0.9 repeated. If you could pretend to evaluate that, I would expect you to get 0.9 repeated with a 5 at the end (thus the greater part). This makes no rational sense, because the number of 9s is INFINITE.

Since you obviously agree with this, this must mean that you agree that (0.9 repeated+1)/2 is not greater than 0.9 repeated. I definately agree that it cannot be less than 0.9 repeated.

Since we have agreed that it can be neither greater than or less than 0.9 repeated, we have two options: either the result IS 0.9 repeated (which clearly does not obey the laws of math), or the question cannot be evaluated. I stated this a while ago.

lol. ok, I understand your position, but i am going to repeat my position again.

.(9) (just gonna stand for .9 repeating to make my life easier) has some "very special" properties, such as it has no square roots, and (.(9)+1)/2 is undefined (not something over 0 -- take the word undefined in the face value).

but it is precisely as I have told you, these two properties does NOT make any sense! for example, no one said you can't divide by 2 (or for that matter, any number that's not 0!) if the number is (1.(9))! or you can't take the sqrt(x) if x is .(9)!

as my proof indicated, x<(x+1)/2<1 if x<1

in terms of an algebraic sense, lets look at two graphs:
y=x, y=(x+1)/2

you have to agree that by solving for the point of intersection,
it will be (1,1).

these two equations are linear functions, which means a line [with no holes].

now how could that be? didn't we say that (.(9)+1)/2 is undefined?

my point is, saying (.(9)+1)/2 is undefined is saying that my original proof that has algebraically proved that x<(x+1)/2<1 if x<1 is wrong!

if you look over my original proof, there is absolutely nothing wrong with what I did or my reasoning, except you claim that 0.(9) does not fit this case.

do you understand of the proof by contradition?

you assume something is true, show that a contradiction exists, and therefore proving your assumption to be false.

what you are doing here however, is claiming something to be true and show that it exempts from properties of real numbers! that is pretty absurd to me.

I did a search for this topic, and came up with every website saying (actually i only visited around 4) .(9)=1

here is their proof:

1/3 = .(3)
2/3 = .(6)
so 1/3+2/3=.(3)+.(6)
so 1=.(9)

in case you claim that 2/3 is not .(6) because you typically see .(6) as .666666...6667 on a calculator. i want to say that as you told me your self, .(6) is a conceptual number, and .666...6667 is a close approxmation to .(6).

in case you are still not convinced,

1/3+1/3+1/3=.(3)+.(3)+.(3)
3*(1/3)=3*0.(3)
1=.(9)

now the only way you are getting out of this proof is to say 1/3 is not .(3) is that what you are saying?

and by the way, you seem to have believe that .(9) is not 1. can you offer any proof of any other source that says the same thing (like a professional website, hopefully university website)

this site was one of the 4-5 sites i visited, it tells a very interesting story ;) :
http://reflectiveeclectic.org/tales_of_truth.htm

this is a sweet proof:
http://mathforum.org/dr.math/faq/faq.0.9999.html

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Originally posted by bugzpodder
you cannot have .999...8 where there is an infinate number of 9's. you can't just say at the end of infinity I will add a 8, because there is no end to infinity!!! That was what I am trying to say. sorry about the bold, i was just illustrating my point, or in NotLKH's words:IMHO, lol!

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i am not saying that the 8 is at any defined decimal place it is just at infinity. Maybe you aren't getting this specific reference however when i say .999 ... 8 i mean a number infitesimally smaller than .999 .... An infitesimally small number is possible (the difference between .99 ... and 1).

you can't have a number infinitely small -- when it is actually "infinitely small" it is 0! btw it doesn't make sense to have a 8 at "un-defined" decimal place as you implied. infinity is not a number so you can't have infinity minus 1 nines and a 8. my point is that their is no such thing as .9999....9998 where there is an infinity number of 9s.

the difference between .(9) and 1 is 0. there is no difference. that explains why the number you implied does not exist.

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Originally posted by Gandalf_Grey_


i am not saying that the 8 is at any defined decimal place it is just at infinity. Maybe you aren't getting this specific reference however when i say .999 ... 8 i mean a number infitesimally smaller than .999 .... An infitesimally small number is possible (the difference between .99 ... and 1).

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By saying .999..., then the ... symbolically means the established pattern of that which precedes the ... should be understood to continue, with no deviation, forever. Now, the ... in your statement: ".999...8" cannot mean the same thing as the ... in ".999...", because you are requiring a termination in the established pattern. Now, all you have proven, IMHO, isn't that .999... doesn't exist, but that .999...8 doesn't exist.

You are trying to prove or disprove something by creating a
seperate concept, and use it without proving or disproving the
existence of that second concept. You create some association
between the two concepts to build a statement that is
recognizably false, which you then use as verification that the
first concept does not exist. However, what the falsity of that
statement actually proves is that something used to build that
statement does not exist, but it does not point out which concept
is false.


Certainly, if .999... is nonsense, and does not exist, then .999...8
does not exist.

However, .999... can exist, while .999...8 doesn't, which would
return the same results.

So, Prove to me that .999...8 exists upon the condition
that .999... exists, then you can use it as a keystone to build your
arguement. Then, if your arguement produces a statement that is
recognizably false then you will have proven that .999... does not
exist.
-Lou

I'm curious. These proofs are nice and simple, but I wonder if they do hold.

BugZ, you had a link to a page (Dr Math one) that uses a limit in their proof, but does the limit of a function actually ever reach that value? Just because you say that 1/x approaches infinity as x -> 0 from the right, does this mean that AT infinity, x = 0?

I guess my real problem at the moment in completely accepting the fact, is can you say that .(3) + .(6) = .(9) ? Is it an exact answer, or an approximation? No matter how far towards infinity you go (in decimal places), that n^th digit will still not allow the number to be exactly one. The limit approaches one, yes, but only that. Approaches.

I really do like the little proof of 1 = 3(1/3) = 3(.(3)) = .(9), but again I'm wondering if math is being applied in the wrong way. If 1 <> .(9) (IF), then either 1/3 <> .(3) or 3(.(3)) <> .(9) (I wonder if, again, there's an approximation occuring that we're overlooking.)

Destined

yes i can assure you they do hold.

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but does the limit of a function actually ever reach that value?

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we are evaluating a sum of a infinite series, namely 9/10+9/100+9/1000+...

because without the x grows out of bounds, it won't be 0.(9)! because there is an infinite number of 9's so you need to have x to grow out bounds thing.

just like e are defined.

e=lim (x->oo) of (1+1/x)^(x)

does that answer your questions (since your questions are all what i've explained)?

here is a post about limits and such: they all came from the same website i gave above

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Rather than saying "giving infinity a value," it's perhaps a bit
clearer to say, "giving the concept of a limit of an infinite sequence
of numbers a value."

.9 is not 1; neither is .999, nor .9999999999. In fact if you stop the
expansion of 9s at any finite point, the fraction you have (like .9999
= 9999/10000) is never equal to 1. But each time you add a 9, the
error is less. In fact, with each 9, the error is ten times smaller.

You can show (using calculus or other methods) that with a large
enough number of 9s in the expansion, you can get arbitrarily close to
1, and here's the key:

THERE IS NO OTHER NUMBER THAT THE SEQUENCE GETS ARBITRARILY CLOSE TO.

Thus, if you are going to assign a value to .9999... (going on
forever), the only sensible value is 1.

There is nothing special about .999... The idea that 1/3 = .3333...
is the same. None of .3, .33, .333333, etc. is exactly equal to 1/3,
but with each 3 added, the fraction is closer than the previous
approximation. In addition, 1/3 is the ONLY number that the series
gets arbitrarily close to.

And it doesn't limit itself to single repeated decimals. When we say:

1/7 = .142857142857142857...

none of the finite parts of the decimal is equal to 1/7; it's just
that the more you add, the closer you get to 1/7, and in addition, 1/7
is the UNIQUE number that they all get closer to.

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this next story is funny:

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you're _really_ thinking of

0.999...999

which is not the same thing. You're absolutely right that 0.999...999
is a little below 1, but 0.999999... doesn't fall short of 1 _until_
you stop expanding it. But you never stop expanding it, so it never
falls short of 1.

Suppose someone gives you $1000, but says: "Now, don't spend it all,
because I'm going to go off and find the largest integer, and after I
find it I'm going to want you to give me $1 back." How much money has
he really given you?

On the one hand, you might say: "He's given me $999, because he's
going to come back later and get $1."

But on the other hand, you might say: "He's given me $1000, because
he's _never_ going to come back!"

It's only when you realize that in this instance, 'later' is the same
as 'never', that you can see that you get to keep the whole $1000. In
the same way, it's only when you really understand that the expansion
of 0.999999... _never_ ends that you realize that it's not really 'a
little below 1' at all.

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this one makes sense:

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Another way to approach the question is to subtract.

1.0000000....
- .9999999....
--------------
0.0000000....

Sure looks equal to me. What about the "1" at the end, I hear you ask?
Well, I'll write it as soon as I finish writing infinitely many 0s.
Any decimal place you name (say, the four billion three hundred
twenty-eight million two hundred seven thousand four hundred
ninety-fifth) has a 0 in it. A number with a 0 in every decimal place
is certainly 0.

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Originally posted by bugzpodder
we are evaluating a sum of a infinite series

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Hehe.. silly me. :p Yes, now I see what you mean.

I guess the confusion is between a purely-infinite series (0.(9)) and a finite series. The finite series will never reach 1, but the infinite series will. I guess this is another feature of infinity that makes one gasp in its awe. :D

I think the link I was talking about should have the statement that .(9) IS an infinite series.

Knowing this, you can't apply the 0.9999...8 method, as it's a finite number.

Destined

lol, here is an article about 1.999...98. close enough to .9999...98 which is what we were talking about:

http://mathforum.org/library/drmath/view/53387.html

http://www.math.toronto.edu/mathnet/...econd1eq2.html

this is a very interesting proof of 1=2 (of course, it is wrong)

i never caught the error. (i had to try almost every step to find the correct one) y don't you guys take a look.

other such "proofs" are in the middle of this page: http://www.math.toronto.edu/mathnet/falseProofs/

i find it to be rewarding

Is it the first step? I know it's before Step 4, since 1/i = -i, not +i.

For step 1, they say 1/-1 = -1/1. This can be rearranged as (1)(1) = (-1)(-1)

From here on, we get into the arguments that we've been going over for quite some time.

Or, as I am sometimes, am I wrong?

[edit]: Yup... I'm wrong, but not too far off, really. ;)

Destined

well in case you don't know, you can click on the step number and it will tell you on which step an error occured.

*sigh* You don't seem to understand my point of view. I think we'll just have to agree to disagree. I frankly don't care how many websites you can find that agree with you, what I'm interested in is the logic. I fully understand why it is that you think your logic works, but I find it contradicted by my logic regarding the infinite nature of (0.9). You seem to not like the concept of infinity and attempt to define or approximate it, but you really can't do that.

1!=(0.9) because even though there is an infinite number of nines, there is an infinitely small difference between 1 and (0.9). Yes, this is insignificant in 99.999% of the cases where the comparison is made, but this is the case where it does matter. You cannot prove that 1=(0.9) by saying that you don't care about the infintessimally small difference.

On a side note, (0.3)!=1/3; (0.3) is our approximation. Fractions cannot be converted to decimals when they have an infinite number of decimal places.

We cannot use in evaluated math numbers that contain an infinite number of digits. This IS a contradiction in math, and I would think you would know that contradictions exist in math and many other subject areas. Perhaps you've heard of Godel's Incompleteness Theorem. If not, a brief overview can be found here. Godel's Incompleteness Theorem proves that in all areas of knowledge, there are problems that are unsolvable without additional knowledge not contained within the boundaries you are working in. While I have no specific examples of others using this in this particular case, I have not looked since I think logical argument is more important than the number of people who agree with you anyway. I believe that this case of numbers not being usable in math is an application of this theorem.

I will attempt, however erroneously it may be, that the value of .999... doesn't exist at all!

1/9 = .111...

(1/9)*(9/1) = 1

.999...=.111...*9

.999...=(1/9)*(9/9)=1

This is yet another occurence where the English Language fails us. I'm not sure that it can be explained perfectly in any language. We can agree that .999... is definitely .111...*9. But it is actually 1. So .999... does not exist, because infinity is a concept, not a value or measure.

good point Alphanos. Godels incmpleteness theorum has a surprising tie to computer science. In CPU (june 2002) Pete Loshins article "The Nature of Computer Security: An interdisciplinary Approach" outlines Godels theorum relative to computer security. If anyone wants it i could scan it and email the jpegs to them

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Originally posted by snakeeyes1000
I will attempt, however erroneously it may be, that the value of .999... doesn't exist at all!

1/9 = .111...

(1/9)*(9/1) = 1

.999...=.111...*9

.999...=(1/9)*(9/9)=1

This is yet another occurence where the English Language fails us. I'm not sure that it can be explained perfectly in any language. We can agree that .999... is definitely .111...*9. But it is actually 1. So .999... does not exist, because infinity is a concept, not a value or measure.

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I agree .999 ... doesn't exist, it [EDIT] Has a has an infinite number of decimal places[/Edit] , and since infinity is just a concept niether exist outside the realm of human thought. No human can comprehend infinity. Alphanos and i were explained the notions of infinity by a teacher who went to a conference on infinity. The infinity which humans think we can comprehend is what is called the countable infinity. That is just a real number that happens to be exessivly large. True infinity though cannot even begin to be comprehended so i guess we just label it as non existant. Sort of like the universal perspective vortex (i think thats the name of it) in Douglas Adams Hitchhikers Trilogy. I must refer to fiction here mainly because it gives a good example about how the human mind cannot grasp the nature of infinity. Anyways the universal perspective vortex would allow a human to grasp how insignificant they were relative to the universe in all of its infinity. In the story it was regarded as the worst possible way to die. :p.

I am not really making a point in this post other than just agreeing with bugzpodder and rembling on about infinity :p

I never said infinity doesn't exist, I said .999... doesn't exist.

I am saying that infinity doesn't exist. You cannot write down infinity because it can only exist inside the realm of human thought. The sideways 8 is just a symbol suxh as pi which details something that no human can fully comprehend. When i say infinity doesn't exist i mean it is not something any person can comprehend to its fullest.

Ok, obviously you disagree with the proofs because you think infinitely small doesn't equal to 0. or in other words:
lim (x->0) x !=0 (now you see, it is x aproaches 0, not x=0 -- in your words, there is an infinitely small difference between x->0 and x=0)

and you also don't believe that 1/3=.333...
because of some wierd reason thats to me is completely could not believe (like 1/3 cannot be translated into a fraction because it is an approximation even though .333... stands for .3 repeating).

and also you believe that infinity exists (which i do so too, but i believe that it is a concept and not a value/number)

now show me a proof of 1 and 2. does anyone ELSE believe you? give me a professional source (hopefully a university site) where it clearly indicates that either lim (x->0) x != 0 or that
1/3=.3333... I want this because I don't believe you are a professional mathematician that specializes in number theory. well you don't have to be one, unless you show me another person's proof or your own proof that may have been published in a professionally recognized publication then I'll believe you.
in short, you offered me no proof while you dismissed my proofs (which has been largely recognized as a valid proof) as rubbish without a proper reason.

By the way the incompleteness theorem just indicated that such a case may exist. as I said, no one has offered me any proper reason as to why .999... != 1, other than .999... is only an approximation even though it clearly is not.

why would .999 repeating be an approx.? That's like saying "professional environmental safety and management engineer" is an approximation for janitor. Infinity is a concept, math is a concept, red is a concept, etc. Why don't we just get rid of fractions? I never understood their importance:D

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Originally posted by snakeeyes1000
Why don't we just get rid of fractions? I never understood their importance:D

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We have numbers like pi and e, which can't be written as a fraction, but then we have fractions which can't be accurately written as decimals (like 1/7 I think). [edit] Well, they might be able to be written accurately, but not exact [/edit]

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Ok, obviously you disagree with the proofs because you think infinitely small doesn't equal to 0. or in other words:
lim (x->0) x !=0 (now you see, it is x aproaches 0, not x=0 -- in your words, there is an infinitely small difference between x->0 and x=0)

and you also don't believe that 1/3=.333...
because of some wierd reason thats to me is completely could not believe (like 1/3 cannot be translated into a fraction because it is an approximation even though .333... stands for .3 repeating).

and also you believe that infinity exists (which i do so too, but i believe that it is a concept and not a value/number)

now show me a proof of 1 and 2. does anyone ELSE believe you? give me a professional source (hopefully a university site) where it clearly indicates that either lim (x->0) x != 0 or that
1/3=.3333... I want this because I don't believe you are a professional mathematician that specializes in number theory. well you don't have to be one, unless you show me another person's proof or your own proof that may have been published in a professionally recognized publication then I'll believe you.
in short, you offered me no proof while you dismissed my proofs (which has been largely recognized as a valid proof) as rubbish without a proper reason.

By the way the incompleteness theorem just indicated that such a case may exist. as I said, no one has offered me any proper reason as to why .999... != 1, other than .999... is only an approximation even though it clearly is not.

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I think we must agree to disagree. No, I am not a professional mathematician, and neither have I looked for any kind of professional proofs for what I'm talking about. I don't intend to simply because this debate really isn't that important to me, and I'd rather do some work on a programming project I'm working on;). Al of what I have said on the topic is based on my personal opinion and logic. That doesn't necessarily make it invalid though:).

I am no longer going to debate the issue, although I would like to clarify my stance on several of the issues that you misinterpreted.

I think that infinity is a concept, but not one that cannot be used in such terms as "infinitely large" or "infinitely smaller". You are correct in that I think that evenly an "infinitely small" value is distinct and separate from 0, and that 1/3 != (0.3). IMO, (0.3) is an approximation we use for 1/3, mainly because 3/3 != (0.9). (0.3) is infinitely close to the true value of 1/3 however.

I was not attempting to dismiss your proofs, but simply state that I find them to be faulty. That is my opinion.

I think that the case of these numbers that have an infinite number of decimal places, such as (0.3) or (0.9), is an example of Godel's Incompleteness Theorem.

I think that this last part is the part of my viewpoint that has been most confusing. With numbers, such as pi, (0.9), or (0.3), that have an infinite number of decimal places, I believe that there are only two ways in which they can be used: conceptual/algebraically or approximately. You can use these numbers in ratios or equations where they cancel out, or in cases where you want a result like 6*pi, without error. You could say (9*(0.9)) / (3*(0.9)) = 3, and that is fine. You could not, however, evaluate 6*pi to a single real value without the second usage of such numbers, approximation. In this case, you take an approximation of the infinite number you're going to use. The number of decimal places of the approximation depends on how much accuracy you need. You could evaluate 6*pi by saying that pi = 3.1415926, because you don't need anymore accuracy than that. Alternatively, if you needed LOTS of accuracy, you could approximate pi to 1000 decimal places, and get enormous accuracy. But even if you used 100 trillion decimal places, it still would not be exactly correct, because the actual value of pi is not a real number. Exactly what it is I do not know, but it cannot be contained in a real number.

Approximating numbers such as pi or (0.9) in nearly all cases is certainly sufficient. You could use 1000 decimal places or 1 million, and have whatever level of accuracy you need. The only problem occurs when you need infinite accuracy. As an example, take the error estimates used in scientific results. You could have 5.78+/- 0.05 as a number you were working with, do math with it, and carry through the error estimates to get the final error estimate. If you had 3 eggs or something, though, then there is no error, because 3 is an absolute counted value. pi, being a mathematical constant, has no error estimate; it is devoid of error. This, however, is only applicable to the true non-real value of pi. The value that Windows calculator gives for pi is 3.1415926535897932384626433832795. If you approximate this to 3.1415926, you are essentially introducing an error of +/- 0.0000001. If you approximate pi to 3.141592653589793238462643383279, you are introducing an error of only +/-0.000000000000000000000000000001. If you approximate pi to its first 1 million decimal places, any error would be nearly non-existant, and it would probably be accurate enough to use in astronomical calculations on stuff millions of light years away.

The key word, however, is nearly. In the extremely few cases where infinite accuracy is desired, even the tiniest, most infintessimal amount or error is infinitely greater than the desired accuracy of the result. If your error is +/- 1/100 trillion, even that is far, far, far greater than 1/infinity. The true, non-real value of (0.9) is (1 - 1/infinity). If you need accuracy to within 1/infinity, then 1/100 trillion is a huge, vast amount of error.

It is because of this that I find (1+(0.9))/2 to be a faulty proof, because the error you introduce when approximating (0.9) is far greater than the desired accuracy of the result. This proof is only valid when the absolute value of (0.9) is used. Unfortunately, (0.9) is not a real number, and it is simply not possible for us to evaluate such a question. Trying to do so would be the same as trying to count to infinity. The most powerful computer creater 1 million years from now would work at the task forever and never succeed.

I think that (1/infinity) != 0. You may think otherwise. If you do agree that 0 != (1/infinity), then 1 != (0.9). In any case, I hope that we can at least agree to disagree with understanding of and respect for each others' viewpoints:).

BTW, I've found this debate to be very interesting and enjoyable. I apoligize if in some posts it seemed as though I was being rude or angry. It was not intentional:(. I was simply too interested in the debate:).

Lots of interesting points brought up. A lot of them show people trying to use concepts they only have a wooly grasp of, but it's a start.

On the subject question: Can a = b when a and b are different integers?
Depends on your definition of different integers. I would define it as something like:
a element of N
b element of N
a != b
From this definition clearly a = b is a contradiction.
However, people saying that in computer memory a = b when a = -1 and b = 6--- is NOT contradicted by this proof, as a, b are not elements of N in this case.

If we use modulo arithmetic (similar to the computer memory approach), 1 = 5 (mod 4) or more generally a = xn + a (mod x) where n is some integer.

The main argument seems to be over whether 1 = 0.(9) [Nice way to notate it on a computer]
Many arguments to counter this have used the idea of infinity. However, they all seem to have got it wrong. If you are saying that there is a difference (significant or not) between 0.(9) and 1, then you are wrong. Talking about the "infinitieth" place is absolute rubbish, since infinity is not a number. However, it is more than just a concept. It is well defined. Because it doesn't appear on the real number line does not make it just a concept that can be ignored.
Infinity is used in this case as a limit. And this is how.

Code:

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1 - 0.9 = 0.1
1 - 0.99 = 0.01
...
1 - 0.99999999 = 0.00000001
...
    n
1 - E 9/(10^n) = 1/(10^n)
    1

---

i.e., take 0.999.... with a finite sequence of n 9's, the answer is 0.0000...0001 with n-1 0's after the decimal point.

Now as n tends to infinity (note, you can't say when n is equal to infinity, that doesn't make sense), the answer tends to zero. Now if you know how limits work, we can see that we have a continuous differentiable function here, which means the limits are the values at all points and from both directions essentially (loosely worded, I know, but it gets the point across). Therefore, if you have an infinite sequence of 9's (i.e. 0.(9)), then the answer is an infinite sequence of 0's with a 1 in the last place. Now, where is the last place in an infinitie sequence? There isn't one. Therefore there isn't a place for the 1. Therefore 1 - 0.(9) = 0, and since we're using real numbers, we can use standard arithmetic rules to find that 1 = 0.(9).

Of course, this proof will be no more valid than any other offered to you so far.

One final point to take exception to: Pi is not a real number.
Do you actually know what a real number is? I think you're either using "real" when you don't realise there is a mathematical definition of a real number, or you're grossly mistaken about what the definition is. It lies on the number line, which makes it real. It's not rational, certainly. It's not complex, for sure. But it most definitely is Real.

And what makes you so sure that 1 is any more valid that Pi as an "accurate" number? 1 doesn't exist any more than Pi does. "1" is defined to stand for the value of x, such that a.x = a, where a is any number. "0" is defined to stand for the value of x where a.x = x for any a. If you can point me to 1 in the real world (not one thing, but to just one - the value, not the digit) then I'll concede the point.

I reckon 0 and 1 are the two most annoying numbers. They have so many similar properties (and probably even more unique properties).

a.x = x and a.x = a surely work for 0 and 1, but how do you go on from there? Say for two, would you say: a.x = a + a? That of course simplifies, but then you are using the number 2 to 'define' the number 2 (it's like trying to explain the word 'do' to a foreigner without using the word 'do'). Really, for two, you say a.x = 2.a. If that's the case, then for 1, it is really a.x = 1.a. We don't show the coefficient (when it is 1) because there's no need too. The number is still there, just not shown.

Ummm, yes, but that isn't the way you'd define 2, 3, 4, ...
It's only 0 and 1 that would need to be defined like that. 2 would be 1+1, 3 would be 2+1 etc. No need to get yourself in a mess like that method would surely get you into.

what the heck is a.x=a? don't understand it! :confused: someone explain?

It's an equation with two variables, x and a.
a is defined to be any number.
x is a specific value that we call "1".
the "." is a multiplication symbol.
I'll assume you know what the "=" means ;)

So, I guess if we say PI = 3.14, then PI is 42!

Cool!
:D

Usually the dot (or period) is placed in the middle of the line, not the bottom, so sort of like a bullet point. It's just too lengthy to look up the character code (or unicode value) for the dot that means times. I assume it will be under the math section of the unicode values.

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Originally posted by NotLKH
So, I guess if we say PI = 3.14, then PI is 42!

Cool!
:D

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In that case, could we say PI is the meaning of life? LOL :D.

lol i'd use a*x or ax, but never a.x lol:p

...and certainly not axx - that would definitely be confusing!

If 2=3 then 1=8 and 493=6984. In effect if any of this was correct then there would only be one true number in the entire universe :D Let's face it, if 2=3 then there would be no need for the number 3 in the first place as it would already have an equivalent (2). So every number would = 2 and so every number would be 2 and so 2 would be the only unique number in existence. I guess this argument has far reaching implications. If 2=3 then maths wouldn't exist because every number would be 2 and so manipulation of this number would always result in the equivalent...we wouldn't need calculators, we wouldn't need maths - we wouldn't need this forum. I guess what I'm trying to say here is that; if 2=3, I'm a donkey’s hairy behind :cool:

OK, just one confusing thing used:
many people are writing the C++ "!=" for 'does not equal'
please can you use the VB "<>" , for one main reason:
"0.(9) != 1" can be read as
"0.(9)! = 1" which is also true, but it's using the Factorial sign. This gets confusing when people write "a != b", where at first glance it seems they might mean "a! = b"
(N.B, 0! = 1 as well)

Anyway, an interesting question:
1 - 0.(9) = what???

1/infinity. I said that already:).


The flaw with Evil Giraffe's attempt to prove that they are equal is that he is approximating the answer. When you begin subtracting (0.9) from 1, yes, you get an endless number of 0s. But the fact that you cannot actually reach the theoretical 1 in the 'last place' does not mean that the theoretical 1 is meaningless. It is theoretical, not real. Not all meaningful numbers must be real.

By stopping because you realize you will never be able to finish the question, you are getting the answer for a different question! You are instead getting the result of 1 - 0.9999999999999999999999999999999999999999 (which BTW is NOT zero, so you're approximating that too). (0.9) is not a real number, but a conceptual value, and attempts to use it in real math will always fail. This does not mean you can pretend to use an approximation as its true value with the reasoning that its true value does not make sense!

1 - (0.9) = 1/infinity. 0 is, of course, the closest real approximation of that value, but is not the correct answer.

BTW, yes I can use 1/infinity, because I'm not doing anything remotely related to limits here. This is a single, non-changing conceptual value. If infinity was NOT the denominator, then (0.9) would have a finite number of digits.

BTW #2: The next time anyone asks you what pi is, say that it is the result of 6 * 9;).

One thing comes to mind:

"what a couple of dumbasses"

hahahaha

Oh yeah, and for those of you who say 1=2(or the like). You need to take 3rd grade math again. :rolleyes:

Back in 6th Form Physics, we had to write an essay on anything to do with what we'd covered so far. I did Nuclear Fusion and Fission, but that's besides the point. My friend did "How do bumble bees fly?". It may sound like a dumb subject, but a while ago, some physician (or is it physicist) proved bumble bees can't fly, which is absurd because, well, they DO fly. He got full marks for his essay, using the 'proof' that they don't fly to prove they do.

The moral of the story is, just because you think it's dumb, doesn't mean it is. So you shouldn't exactly call these people dumbasses.

Obviously to us all, 1 <> 2, it's getting to that point which is clever.

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Originally posted by Alphanos
The flaw with Evil Giraffe's attempt to prove that they are equal is that he is approximating the answer. When you begin subtracting (0.9) from 1, yes, you get an endless number of 0s. But the fact that you cannot actually reach the theoretical 1 in the 'last place' does not mean that the theoretical 1 is meaningless. It is theoretical, not real. Not all meaningful numbers must be real.

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It's not actually theoretical even. It just doesn't exist. Please go and read up on the idea of infinity.

Quote:

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Originally posted by Alphanos
By stopping because you realize you will never be able to finish the question, you are getting the answer for a different question! You are instead getting the result of 1 - 0.9999999999999999999999999999999999999999 (which BTW is NOT zero, so you're approximating that too).

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I never said we were stopping before finishing. My point was that this pattern continued forever. I wasn't stopping, I had already calculated it!

Quote:

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Originally posted by Alphanos
(0.9) is not a real number, but a conceptual value, and attempts to use it in real math will always fail. This does not mean you can pretend to use an approximation as its true value with the reasoning that its true value does not make sense!

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Um, I'll say it again. Go and look up what is meant by a "real" number, and explain why you think 0.(9) isn't one. (a real number that is. Not why you think it isn't "1" :D)
I'm not sure about your point of "real" maths. And, as I mentioned before, it's not an approximation, it is a real number, and the true value does make sense

Quote:

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Originally posted by Alphanos
1 - (0.9) = 1/infinity. 0 is, of course, the closest real approximation of that value, but is not the correct answer.

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So 0.(9) isn't a "real" number but 1/infinity is? 1/infinity is undefined. Using that as a value doesn't make sense.

Quote:

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Originally posted by Alphanos
BTW, yes I can use 1/infinity, because I'm not doing anything remotely related to limits here. This is a single, non-changing conceptual value. If infinity was NOT the denominator, then (0.9) would have a finite number of digits.

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You're not dealing with limits, so you think you can use infinity as a number? It doesn't work like that. Infinity is the cardinality of the Natural Numbers, not an actual number, therefore you can't use it in arithmetic like you have there. You can only throw it in when you are dealing with limits.

Okay, another argument:
0.(9) is the geometric progression with a=0.9 and r=0.1
Sum of an infinite G.P. is a/(1-r) = 0.9/(1-0.1) = 0.9/0.9 = 1

A geometric progression is aka a geometric series right? Otherwise I'm completely lost! :)

Yep. Geometric Series and Geometric Progression are just different names for the same thing