" it's the only way that provides useful results"

Really? It seems the wrong method is the only method to describe the truth? Interesting. Still, I doubt the obsessive part to eliminate the residue error necessary or not.

"it can be proven that a set (we call them natural numbers) exists such that an "inductive property" covers all of them.  This was Peano's 5th postulate."

Excuse me for a moment, on what basis does the ZCF and Peano's 5th postulate stand? Sheer logic? Or some unintuitive assumptions plus logic? Ironically, the assumptions themselves might be illogical. Hence, the whole rational may just start on an arbitary basis.

Why it needs to go so far as to distort a number as set of numbers in ZCF to get your "only" way to gain some results? Haven't you seen the absursity in it to introduce unnecessary clumsy assumptions similar to a white elephant?

And what is so wrong to go intuitive? After all, intuition observes the truth as best as possible. While arbitary assumption and definition coincide with the truth at best.

The link in 922 by thedude also tells few physicians believe in infinity and infinitesimal idealism. And they came up with results while disbelieving your only method. I guess at least they disagree with you, Ricky.

first we should know that 0.999...... is not a number.
Note

[math0.99\1-\frac{1}{10^2}math]
so

Let

when we say that 0.999......=1,we just mean that

Here is a intitutive proof:

we can see that as n goes bigger and bigger,0.999...... goes more and more close to 1,but it's not accurate in mathematics.I will give a precise proof.
obviously 0.999......<1,if 0.999...=A where A between 0 and 1,then we can easily find some N such that

a contridition derived.

moxiuming, please fully understand the concept of infinity......

0.999999.... is represented as, you can even consider it to be a expression which holds the value 1
 

Last edited by Dragonshade (2008-05-02 01:33:47)

hmm this is a vey interesting subject, indead!

As Ricky said if you have 1/3=0.333...  & you multiply both sides by 3 then you would get 1=0.999...  But then doesn't that make it true that 1 is not equal to 0.999...  ?  Wow amazing how great the span of math is, it is similar to space, how we cannot grasp it being Infinite

Hmm yeah... I guess I was thinking about the problems where you have to determine if an equation is true such as 2x-1=2, x=1 which would be false... I am just wondering how 0.999... can be equal to 1 if they are not the same, because in theory 0.999... would get infinitely closer to 1 but would never actually reach 1. Such as in Physics when someone touches someone else, in theory we are really never touching them because the atoms electron clouds would prevent the nucleui from thouching each other, but we do get an electrical signal to the brain signifying touch... Another example is when an oblect moves halfway to another object then halfway again, then halfway again to infintity but the object can in theory always move another half closer to the other object... 0.25(0.5)=0.125; 0.125(0.5)=0.0625 then it would move on to infinity... I guess in theory 0.999... is not equal to 1, but in reality we set it equal to 1 to make computing easier, just like in theory we never actually touch each other but in reality we feel as if we do because of what our brain is telling us...

Last edited by MooseofDoom (2008-08-14 05:56:40)

Such as in Physics when someone touches someone else, in theory we are really never touching them because the atoms electron clouds would prevent the nucleui from thouching each other, but we do get an electrical signal to the brain signifying touch...

This is incorrect.  You are trying to use it as an example of things that get "infinitely" close, but it is not.  The two nuclei always have a positive, finite, distance apart.

Another example is when an oblect moves halfway to another object then halfway again, then halfway again to infintity but the object can in theory always move another half closer to the other object... 0.25(0.5)=0.125; 0.125(0.5)=0.0625 then it would move on to infinity...

As mathsyperson said, this is not what happens with 0.999...   Nothing is "moving", and nothing is "changing".  It just... is.

I guess in theory 0.999... is not equal to 1, but in reality we set it equal to 1 to make computing easier, just like in theory we never actually touch each other but in reality we feel as if we do because of what our brain is telling us...

No.  0.999... = 1 in theory.  Two real numbers are not equal when they are a finite distance apart.  This is actually, in part, how they are defined.  0.999... is not a finite distance from 1, and thus, it equals 1.

Pretty much. The 1 appears at the end of the zeroes, but in the infinite case the zeroes don't have an end and so the 1 can't be there.

Wow this is a crazy concept to grasp...

mathsyperson wrote:

Pretty much. The 1 appears at the end of the zeroes, but in the infinite case the zeroes don't have an end and so the 1 can't be there.

To add to that, if you were to ever put a 1 at any point in the chain of zeroes, you would thereby be ending the chain of zeroes, and so the number of zeroes would be finite; so any argument along the lines of 'an infinite number of zeroes, then a one' makes no sence.

Don't forget that you can have
a river with banks on both
sides and the river could run
forever.  And hence you could
have 7water7, where 7 are the
banks, and the water could
fill up forever.  But maybe forever
is not long enough to reach
infinity.

Hi guys. I just submitted one article to philosophy of science. Basically I argue arbitary fraction cannot exist.

Hi. New member here. I have been discussing this problem with some students of mine and have reached a bit of a wall. I have a way of thinking about this and I hope someone can point me in right direction.

We started with the following "proof".

let x = 0.99999999.......

10x = 9.9999999........
10x - x = 9.999999....... - 0.999999......
9x = 9
x = 1

My problem is with the following line:

9.999999....... - 0.999999...... = 9

The kids are happy to accept this and gave the following response (among others).


"Consider 9.999... - 0.999....

If this is subtracted digit by digit it become clear that the answer is 9:

First we have 9-0 so we have 9 'units'.

Then 9-9 so 0 tenths; then 9-9 so 0 hundredeths....

As it is an infinte decimal this will continue forever so the answer will be:

9.00.... = 9"

Now my response to this was to write down each term for the subtraction as follows:

9.9999.... = 9 + 0.9 + 0.09 + 0.009 + ....
0.99999... = 0.9 + 0.09 + 0.009 + 0.0009 + ....

So far so good. If we then combine these then we get

9.9999... - 0.99999... = 9 + 0.9 - 0.9 + 0.09 - 0.09 + ....

It therefore appears that each term apart from the 9 will cancel giving the answer 9 (exactly). However, I have a problem with this and it is as follows:

If we group the terms together in a different way then I think there is a different solution.

9.999.... - 0.9999.... = 9 x [(1-0.1) + (0.1 -0.01) + (0.01 - 0.001) +....]


9.999.... - 0.9999.... = 9 x [(0.9) + (0.09) + (0.009) + (0.0009)....]

                                          = 9 x 9 (0.1 + 0.01 + 0.001 + 0.001 + ...)

                                          = 81 x (0.1 + 0.01 + 0.001 + 0.001 + ...)
                                          = 8.1 + 0.81 + 0.081 +0.0081 + ...
                                          = 8.999...

Therefore the alternative line in the original proof should be

9x = 9.999... - 0.9999.... = 8.9999...

which gives the trivial answer that x = 0.9999.... which is where we started. It has not been shown that 0.9999.... = 1.

My reason for grouping the terms in this way can be understood if we again consider the series for each value.

9.9999.... = 9 x sum (n=1 to n= ∞ )[10^-(n-1)]
0.99999.... = 9 x sum (n= 1 to n= ∞ )[10^-(n)]

Apologies for the notation but I hope that this is clear.

Now if these expressions are grouped together we get

9.999... - 0.9999... = 9 x sum (from n = 1 to n = ∞ ) [(10^ -(n-1)) - (10^ -n) ]

which indeed gives the expression that does not allow 0.9999... to equal 1.

(   9.999.... - 0.9999.... = 9 x [(1-0.1) + (0.1 -0.01) + (0.01 - 0.001) +....]    )


My logic is that even though there are an infinite number of different terms in each original series the statement (from n = 1 to n = ∞ ) also suggests there should be the same number of terms in each expression. For the expression to be written in a form that allows all the non-9 terms to cancel the series for 9.9999... must contain one extra term than the series for 0.9999... If this is the case then the two series cannot be combined to give the expression

9.999... - 0.9999... = 9 x sum (from n = 1 to n = ∞ ) [(10^ -(n-1)) - (10^ -n) ]                                   

and thus,

9.999... - 0.999... = 8.999

and not 1.

So finally here is my problem. If the symbol ∞ is defined in the system to mean infinite then this surely must have a definite meaning. My thinking is that the symbol ∞ cannot mean infinity and (infinity + 1) in the same axiomatic system. I am happy to accept that infinity + 1 is still infinite but is it possible that ∞ and ∞ + 1 are actually different numbers. I am aware that Cantor defined different classes of transfinite numbers but know very little about his methods and am wondering if anyone can elaborate. If these numbers are different then all the terms other than 9 will never cancel and it is surely true that trying to prove 0.9999.... = 1 (by this proof at least) is a fallacy.

Apologies for the overly long message and thanks in advance.

If these numbers are different then all the terms other than 9 will never cancel and it is surely true that trying to prove 0.9999.... = 1 (by this proof at least) is a fallacy.

If you keep taking math classes, you will eventually deal with series and convergence in a rigorous setting.  If you have taken calculus classes, you may have already seen some theorems about series, though probably not proved them.  In any case, there is a theorem:

If a series of terms b_n is such that

Then the any rearrangement of the series will converge to the same number.

Now "rearrangement" is rigorously defined, but I am attempting to avoid getting lost in the definitions, and it pretty much means exactly what you did in your post.  In other words, both ways you summed the terms converge to the same number.  In essence, you proved that 8.9999 = 9.

This may be a bit tricky logic to work with, but hopefully this explanation makes sense.  By the above theorem (assuming that you accept it, and you certainly don't have to, I can prove it if you would like) we have that your first way of summing to get 9 is valid.  Therefore, 0.999... = 1 by the original proof.  Now, by the same theorem, the way you summed your the terms the 2nd time, they have to agree with the first way.  Thus, we conclude that 8.999... = 9.

In any case, if you still don't accept it, then here is another version using the same technique, but a whole lot cleaner:

1/3 = 0.333...
1/3 * 3 = 1
0.333... * 3 = 0.999...

As for the stuff on infinity, it will take quite a bit of study to come to a good understanding.  We consider infinity to be the same size as infinity + 1 based on the following theorem:

If A and B are two sets containing a finite number of elements, then |A| = |B| (the number of elements in A and B) if and only if there exists a function from A onto B that is 1-1 and onto.

The existence of this function with special properties (if you're not familiar, either ask or wikipedia) can only occur when two sets of finite size have the same number of elements.  And having this map will mean that the sets have the same number of elements.  We say that these two statements are "equivalent", and what we mean is that they represent the same idea.

The great thing about this "definition" of size is that it is easily extended to the infinite set case:

If A and B are two sets (of possibly infinite elements), we say that |A| = |B| means there exists a function from A onto B that is 1-1 and onto.

Now to show you that "infinity" = "infinity + 1", let's take the natural numbers and find a 1-1 map that is also onto and maps into the natural numbers as well as the number 0.

It shouldn't be too hard to prove that this map is both 1-1 and onto.  Just ask if you want to see it.