bob bundy wrote:

You do a division instead

Hi;

He will accept 1 / 3 doing the division = .3333333... But he will not accept the multiplication, I am forcing him to.

Hi

That is because you do not have enough digits of precision.

Exactly! .3333333333333333 < .33333333333...

As you multiply you are going to have to admit that .9999999 eventually will become .99999999...
This is just notation. You cannot think that 1 / 3 = .333333333... and that .999999999... does not exist.

Now you are talking. .99999999... = 1

Here is a question none of them could deal with. If .99999999... does not equal one then what does it equal? It is obvious it keeps approaching one but like Zeno they do not believe it will ever arrive.

It just means more nines, saves us from writing all of them. You can not banish it. It is a definition.

Of course I can.

3 * .33333333... = .99999999...

Why won't a number exist?

did I ever say that 1 and 2 does not exist?

Just the other day I decided to eat 0.99999..... apples.  It was delicious! mmmmm

Of course I didn't really eat it all because 1/3 was core and I left that.

So I only ate 0.66666666....... apples.

Point to ponder.  Should it be 0.66666..... apples or 0.66666666....... apple.

Now that's a much more interesting debate than whether the 0.3333333..... core and the bit I ate, 0.666666..... apple,  makes altogether

0.99999999....... apple.

One thing is certain.  It existed!

Bob

0.999... is only equal to 1 because we humans made rules stating it is. I'd like to see you prove 0.999... = 1 without those rules.

For example. Take a common non-calculus proof.

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

That explodes on the first line. 0.333... is not exactly equal to 1/3. It is the closest Base 10 number to 1/3.  1/3 is in fact: 0.333... + 1/∞*3. This proof was made to trick people into believing 0.999... = 1 without calculus proofs.
Now lets rewrite that proof.

1/3 = 0.333... + 1/∞*3
0.333... * 3 = 0.999...
1/3 * 3 = 3(0.333... + 1/∞*3) = 0.999... + 1/∞
∴ 0.999... + 1/∞ = 1

So the big question is: How much is an infinitesimal?

Calculus tells us that the limit of 1/x as x goes to infinity is 0, but that's calculus. It was practically made for answering this. Can you prove that  "1/∞ = 0" without those rules?

Almost all people say 0.000...1 (1/∞) is 0 because you'd never write that 1 down. Why would this be? There are infinite numbers between 0 and 1, but 1 still exists. This is a concept, an idea, not a physical object. To apply physical-object ideas to a concept would be reification.

I will say this again.
0.999... is equal to 1 ONLY BECAUSE we humans made extra rules saying it is. It is not equal to 1 without those rules.

Last edited by PKBeam (2013-03-22 18:37:55)

Reppy was so cool.

bobbym wrote:

Hi;

1/3 = 0.333... + 1/∞*3

I am afraid that is not correct.

1/3 = .333333333... easily proved by long division.

OK then, how?

I don't know how to put this in a way you can understand, but it's generally accepted that 0.999... + 1/∞ = 1, people just choose to believe 1/∞ = 0. So we can use this equation to prove the value of 1/3.

0.999... + 1/∞ = 1
(0.999... + 1/∞)/3 = 1/3 <- Divide both sides by 3       
0.333... + (1/∞ * 1/3) = 1/3
0.333... + 1/(∞*3) = 1/3 <- Simplify (1/∞ * 1/3)

So to sum this up in words:
A third is equal to point three repeating plus a third of an infinitesimal, which sounds absurd, but it's true. Base 10 isn't perfect cause it's a round number. There's always a flaw. But it's too late to change number systems easily, so we just have to deal with these flaws.

PKBeam wrote:

but it's generally accepted that 0.999... + 1/∞ = 1

Not by number theorists.  The rules (axioms) for number theory specifically do not allow infinity to be treated as a number.  It doesn't obey the rules of arithmetic so 1/∞ is meaningless.

You can make up a mathematical system which uses the infinity symbol:

eg.

∞ + 1 = ∞
∞ + ∞ = ∞

and so on,

but this is NOT number theory and is usually avoided by mathematicians.

In the context of the discussion about 0.999999 recurring and 1, it is not appropriate to use ∞ as if it were a number at all.

Throughout I have said you can define numbers to mean what you want so you might think that allows someone to define ∞ as a number.

But you have also to show your definitions pass the 'consistency test'.  ∞ as a number does not.

Background reading: such as

http://www.calvin.edu/~rpruim/courses/m … nt-pp4.pdf

Note axiom 7.

Bob

Last edited by bob bundy (2013-03-24 19:02:46)

hi ncik

Excellent points.  And totally relevent.  If mathematicians want to give a meaning to 0.9999..... then it is essential to consider what properties it would have.  Once 'you' have a water-tight definition it is 'safe' to go ahead and define what it means, knowing that there are no chinks where someone can drive through an objection.

Bob

hi Maiya

Haven't heard from you for a while, how are you?

I think you can have two numbers that are the same like this