Proof that 2=1...

Ok. If you promise not to ever bring it up again. I forgot about the whole thing until you brought it up.

:)
 
Absane said:
Also, it is an axiom. I quote this from one of my many overpriced mathematics books:

The real numbers can be equated with the set of all base 10 decimal representations. That is, every real number can be written in a form like 338.1898..., where the decimal might or might not terminate, and might or might not fall into a pattern of repetition.

Furthermore, every decimal form you can construct represents a real number. Strangely, though, there might be more than one decimal representation for a certain real number.
Click to expand...

Your book doesn't seem to be taking decimal representations as an 'axiom', and there's no need to do that. It looks like they have defined the real numbers in some way prior to this quote, then are stating some facts about decimal representations that can be derived from the properties of the reals (as defined earlier).

You can define the reals as being the decimal expansions if you like, defining the appropriate operations additions, multiplication and so on in terms of these decimals (of course all these different defintions of the reals are shown to be equivalent). It's still not an axiom that these real numbers have decimal representations, it's a definition.
 
shmoe said:
Your book doesn't seem to be taking decimal representations as an 'axiom', and there's no need to do that. It looks like they have defined the real numbers in some way prior to this quote, then are stating some facts about decimal representations that can be derived from the properties of the reals (as defined earlier).

You can define the reals as being the decimal expansions if you like, defining the appropriate operations additions, multiplication and so on in terms of these decimals (of course all these different defintions of the reals are shown to be equivalent). It's still not an axiom that these real numbers have decimal representations, it's a definition.
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When I get a hold of my book later on, I will check it out.

All I remember are the basic "field axioms" and a few that are called axioms (but are really provable from the field axioms).

It treats numbers as either N, W, Z, Q, or R. However, the only time it actually uses decimal representation for ANYTHING is for cardinality proof that R is uncountable. Everything else is eithetr p/q or just "a real number."

But I'll check it out for you later. I am off to the gym :)
 
Well I looked in my book. Of all the books I own, I cannot find a bridge between decimal representations and q/r. All I can find is really is "it just is."

Not good enough for me. Everyone KNOWS how to divide, but what is the theory behind it? The closest I can get is "the division algorithm." Why can we say that 3.12 = 3*10^0 + 1*10^-1 + 2*10^-2?

Is it just definition of the base ten system? If so, I can live with that.
 
Absane said:
Well I looked in my book. Of all the books I own, I cannot find a bridge between decimal representations and q/r. All I can find is really is "it just is."
Click to expand...

Get new books, or just work out the details yourself.

Absane said:
Not good enough for me. Everyone KNOWS how to divide, but what is the theory behind it? The closest I can get is "the division algorithm." Why can we say that 3.12 = 3*10^0 + 1*10^-1 + 2*10^-2?

Is it just definition of the base ten system? If so, I can live with that.
Click to expand...

That is how 3.12 and decimals are defined. When you don't have a finite number of digits a limit is involved, or a supremum if you prefer (no difference here)

A small amount of work is involved in showing that every real number has such a representation (not necessarily unique!). You can prove all the properties you are used to, like the 'division algorithm' method for finding a decimal representation of a rational number p/q will actually give a representation of p/q, and the usual arithmetic you've learned in grade school.
 
where did the whole 1/3 conversation come from??

the whole thing was about finding the problem, and yes, it was when you divide by (A-B), which since A=B, A-B=0.

funny attempt though
 
Arcane said:
Given: A=B

Multiply both sides by A: A^2=AB

Now subtract both sides by B^2: A^2 - B^2 = AB - B^2

Factor both sides: (A+B)(A-B) = B(A-B)

Divide both sides by (A-B): A+B = B

But we know that A=B so Substitute A for B: B+B = 2B = B

so: 2B = B

Divide by B: 2 = 1

THATS THE PROOF

Can anyone see where the fault is?? >:p

Arc
Click to expand...

the fault is that A and B aren't numerical so that A is therefore A or B and vice versa so that A=/1 and then B=/1 and therefore once again 2A or 2B =/ 2 and so on? :D :bugeye: :D
 
Regarding: "1/3 = 0.3333...."

For anyone who is wondering - if you study elementary calculus then this "identity" (I will explain the scare quotes in a minute) will be explained. Here is what it means.

(a) The limit of the sequence 0.3, 0.33, 0.333,... is 1/3.

Equivalently,

(b) the limit of the sequence 3/10, 33/100, 333/1000,... is 1/3.

Equivalently,

(c) take a number e. *However* small e is, there is a point in the sequence 3/10, 33/100, 333/1000,... such that, for any element (k) of the sequence after that point, the difference between k and 1/3 is less than e.

That is, however small a distance you choose, you can make the sequence go (and remain) closer to 1/3 than that distance. That is what mathematicians mean by saying that 0.333... = 1/3.

(I used scare quotes earlier because (c) evidently does not have the logical form of an identity statement, so saying "0.3... = 1/3" misrepresents the situation.)
 
Funky... I just want to let you know we are trying to drop the whole "0.333.. = 1/3" bit. It's a conversation that started on another forum and found its way to Sciforums. It has been argued from every angle and we get nowhere. So, we are going to pretend like it is not an issue.

:)
 
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