0.9 repeater = 1

[lionking]

Best zombie thread yet.

 

[gnome]

oldest too, I think?

Sadly, whenever this one rises from the grave, it's because there was a lack of braaaains.

 

[Jorghnassen]

Teh multiplication proof.

[latex]$$x=0.9999....$$ $$10x = 9.9999...$$ $$10x - x = 9$$ $$ 9x = 9$$ $$x=1$$[/latex]

Q.E.D.

Infinity and infinitesimals are funny concepts.

 

[Modified]

No one has yet addressed one important question: "Repeater" rather than "repeating"? Is that an Australian thing?

 

[Fnord]

If "Almost = Identical", then "Pi = Three".

 

[ponderingturtle]

0.999... only equals one if you understand math.

See that is the problem, you assume people understand math.

 

[atomx]

First question.

You agree that infinity exisits right? I'll assume yes, due to the repeating, and the acceptance that you believe the equals theory.

Therefore you have to accept and infinite number of 0's between the decimal and 1 when you take 0.9rep away from 1.

Funny things happen when you involve infinity and a difference.

Concepts and the human mind are wonderful things. Don't just accept things.

 

[Marquis de Carabas]

First question.

You agree that infinity exisits right? I'll assume yes, due to the repeating, and the acceptance that you believe the equals theory.

Therefore you have to accept and infinite number of 0's between the decimal and 1 when you take 0.9rep away from 1.

Funny things happen when you involve infinity and a difference.

Funny things happen when you involve infinity and a lack of understanding.

 

[jsfisher]

First question.

You agree that infinity exisits right? I'll assume yes, due to the repeating, and the acceptance that you believe the equals theory.

Therefore you have to accept and infinite number of 0's between the decimal and 1 when you take 0.9rep away from 1.

Funny things happen when you involve infinity and a difference.

Concepts and the human mind are wonderful things. Don't just accept things.

What funny thing are you expecting to happen with 1.000... - 0.999... ?

ETA: The "equals theory"?

 

[Ocelot]

First question.

You agree that infinity exisits right? I'll assume yes, due to the repeating, and the acceptance that you believe the equals theory.

Therefore you have to accept and infinite number of 0's between the decimal and 1 when you take 0.9rep away from 1.

Funny things happen when you involve infinity and a difference.

Concepts and the human mind are wonderful things. Don't just accept things.

The 1 that I've highlighted doesn't exist.

 

[Rasmus]

The 1 that I've highlighted doesn't exist.

now I understand what was meant.

Question:

Why is it any more difficult to accept an endless strings of threes than it is to accept an endless string of zeroes?

 

[Ocelot]

now I understand what was meant.

Question:

Why is it any more difficult to accept an endless strings of threes than it is to accept an endless string of zeroes?

I have no problem accepting an endless string of threes.
I have no problem accepting an endless string of zeros.
I do have a problem accepting an endless string of zero's with a one after the end of the endless string. That's a logical contradiction.

 

[Rasmus]

I have no problem accepting an endless string of threes.
I have no problem accepting an endless string of zeros.

I didn't mean you. And yes, I know I was replying to your post. My bad.

I do have a problem accepting an endless string of zero's with a one after the end of the endless string. That's a logical contradiction.

Well, yes.

A lot of people here seem to expect a zero terminating the endless strings of threes or nines. I just meant to point out that it makes no sense to assume that a zero would terminate the row anyway. (Yes, I know it doesn't terminate at all.)

 

[jsfisher]

The 1 that I've highlighted doesn't exist.

Oh! So atomx believes 1 - 0.999... = 0.000...0001. I see.


Ok, question for atomx:

If you divide your 0.000...0001 by 2 then add it to 0.999..., what do you get?

That would be 0.000...00005 + 0.999..., right? What do you get, and is it bigger or smaller than 1?

ETA: Heck, is 0.000...00005 bigger or smaller than 0.000...0001?

 

[bjornart]

Teh multiplication proof.

[latex]$$x=0.9999....$$ $$10x = 9.9999...$$ $$10x - x = 9$$ $$ 9x = 9$$ $$x=1$$[/latex]

Q.E.D.

Infinity and infinitesimals are funny concepts.

How bout you point out the flaws in this proof, atomx? Not that your attempts so far to disprove 1 = 0.999... are correct in any way, I just wonder if you have a way to warp this one, or if you're willing to accept that your understanding of the definition of repeating decimals gives us inconsistent math.

 

[ravdin]

It's funny this would be contentious. Most people have no problem accepting that 0.333... is equal to 1/3.

 

[Marquis de Carabas]

It's funny this would be contentious. Most people have no problem accepting that 0.333... is equal to 1/3.

That doesn't violate the assumption that most people have drilled into them that 0.whatever is necessarily less than one. I can see where the initial contention comes from; I just don't understand the stubborn refusal to accept it once clear and simple demonstrations are offered.

 

[jsfisher]

It's funny this would be contentious. Most people have no problem accepting that 0.333... is equal to 1/3.

Not really. I think the point that trips them up is the idea that a number can have more than one decimal representation. 0.999... jumps out as an unexpected exception.

Truth is many numbers have two representations. 1/4, for example. It has the very familiar 0.25, and it also has 0.24999....

 

[Vorticity]

What is usually left out of threads on this topic just about everywhere is the following:

The assertion that 0.9~ = 1 is not some independent truth that exists "out there" in the real world. Its truth or falsehood stems from the axioms of the number system we happen to be dealing with at the time. In fact, one can construct axiomatic number systems such that 0.9~ does NOT equal 1.

If we are working within the system of the standard real numbers, however, then it can be rigorously proven that 0.9~ = 1. This comes about because the following definition is part and parcel of what it means to be working in the field of the real numbers:

Definition:
The value of a real number x with an infinite decimal expansion is equal to the limit of the sequence of its partial expansions.

This statement is a definition, it's not something you can prove. What does this definition say in this case? Our number, x, is

x = 0.9~ = 0.99999...

What is the "sequence of its partial expansions"? That refers to the following sequence of numbers:

0.9, 0.99, 0.999, 0.9999, etc.

The limit of this sequence can be shown (via some real analysis I'm not getting into here) to be 1. Thus 0.9~ = 1, by definition. No further proof is required.

 

[This is The End]

Some (most?) of the proofs I see on the first page of this thread assume 1/3 = .3 repeating... as far as I remember from math (a long time ago, mind you) that is not so. .3 repeating is only an approximation of 1/3, though it is a very good one.

You remember wrong.

You forget that I may have been taught wrong, in a backwater mudhole.

(Try dividing 1 by 3 and see what you get.)

How much time are you going to give me? It would take forever and a day.


I'm still pretty sure about the second part of my post you didn't include though:

Basically, what I'm saying is, you can't use .3 repeating = 1/3 in a proof of .9 repeating = 1. That's using the exact same assumption of what you are trying to prove.

Just to make clear, I don't see a problem with the multiplication proof that has been quoted a few times already in this thread (the, 10x | x = .999 | etc., one ).