[1 paradox]

bobbym · 2013-03-06 09:44:01

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.

anonimnystefy · 2013-03-06 10:48:18

Actually, 0.999999 will eventually become 1. Everywhere you write 0.999..., 1is also a valid figure.

bobbym · 2013-03-06 10:53:02

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.

anonimnystefy · 2013-03-06 11:00:24

If it existed, it would be 1. But, I still banish it from maths. It is needless.

bobbym · 2013-03-06 11:03:09

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

anonimnystefy · 2013-03-06 11:34:17

It is not a definition. Can you prove the existence of 0.999...? No!

bobbym · 2013-03-06 11:35:26

Of course I can.

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

Bob · 2013-03-06 20:04:56

Stefy wrote:

It is not a definition. Can you prove the existence of 0.999...? No!

Whatever do you mean by 'existence'? All numbers are defined and all have equal 'existence', whatever that is.

You cannot pick and choose. If somebody uses it, then it exists.

Bob

Agnishom · 2013-03-07 03:41:56

Why won't a number exist?

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

bobbym · 2013-03-07 09:11:37

Hi;

Those are easy to understand. Because I can immediately see 2 apples or 1 pear.

In this case all the confusion will disappear when we all agree that .9999999... = 1.

Bob · 2013-03-07 10:46:15

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

bobbym · 2013-03-07 11:47:20

One thing is certain. It existed!

You will have a hard time proving that now.

PKBeam · 2013-03-21 19:21:30

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 (2013-03-21 19:37:55)

bobbym · 2013-03-21 22:06:53

Hi;

1/3 = 0.333... + 1/∞*3

I am afraid that is not correct.

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

BECAUSE we humans made extra rules saying it is

The aliens also think .99999999... = 1.

I am not supposed to release this to the public because it is still A.T.S. but here is an authentic picture of the leader of the reptilian mother ship giving a lecture at the Nevada test site on June 3, 1941.

Using this advanced alien knowledge we were able to defeat the Axis powers in the coming war ( WWII ). I was proud to have been the one that asked Reppy, " that is what he was affectionately called by his friends" what .999999... really equaled.

I remember it as if it were yesterday because Albert Einstein was sitting next to me, I remember because he had eaten garlic and he patted me on the back for my good question.

Last edited by bobbym (2013-03-21 22:15:36)

MathsIsFun · 2013-03-21 22:24:30

Reppy was so cool.

Bob · 2013-03-22 20:31:21

hi

I agree with PKBeam (welcome by the way, PKBeam)

But all numbers have to obey rules like that.

For example, there is a rule that allows you to convert 0.5 into 1/2 and another to convert back.

We have a rule to make 4^(1/2) = 2.

We have a rule to make 1 + 1 = 2.

Er, hang on Bob, everyone knows that last one; you don't need a rule for it.

Well actually, yes you do. We are so used to it as an absolute truth that we forget that there are rules governing this.

You see walk one mile plus walk one mile is not necessarily two miles.

One pile of sand added to one pile of sand is one pile of sand.

The adding rules are there to define the results and when you can use them.

So there is NOTHING remarkable about the rules that make 0.99999....recurring = 1. They're just another application of the usual thing that maths does:

Look at part of the real world; find some maths that describes it and allows useful predictions about new situations; check the maths is consistent; write it up in a paper; sit back and look smug.

Bob

Last edited by Bob (2013-03-22 20:34:26)

PKBeam · 2013-03-23 00:35:50

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?

bobbym · 2013-03-23 00:49:22

If you do schoolboy division you would say 3 goes into 1 with 1 left etc...

We keep getting another three tacked on

.3
.33
.333
.3333
.33333

So we write .33333... the three dots just means continue on. Looking at it like that you can see that 1 / 3 is just shorthand for .3333333333...

This is the way people in computational math would tend to look at it. We tend to view it as just notation.

Last edited by bobbym (2013-03-23 01:03:10)

PKBeam · 2013-03-23 12:20:24

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.

bobbym · 2013-03-23 12:44:21

I don't know how to put this in a way you can understand, but it's generally accepted that 0.999... + 1/∞ = 1

That is not generally accepted because of a couple of things, for one 1 / ∞ ≠ 0.

We could say

but that is not the same thing.

This is also a little funny

0.999... + 1/∞ = 1

you can not use infinity as if you were using the number 5 or 56. Infinity is not a number. You can not do arithmetic with it. It would lead to many contradictions.

What is true is

1 / 3 = .3333333...

you need nothing else notation wise. Now just drop the 1/∞ from your thoughts and you are there. We both can then agree

1 / 3 = .33333333...

3 ( 1 / 3 ) = 3 ( .3333333....) = .99999999...

1 = .99999999....

Now we can start to do some mathematics! We agree with everyone else about the rules of the game.

Base 10 isn't perfect cause it's a round number.

Base 10 is a positional number system that uses powers of ten and the digits 0 - 9 to represent numbers. It is a not a round number nor is it flawed.

Last edited by bobbym (2013-03-23 16:49:03)

Bob · 2013-03-23 19:58:40

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 (2013-03-23 20:02:46)

ncik · 2013-04-04 14:52:11

I don't know if somebody already said this, but saying that .999...≠1 implies that there is a number between .999... and 1 that is distinct from both of them. There is not.
Also, the sum from n=1 to infinity of 9(1/10)^n = 1. But that's beside the point and not a very intuitive proof.

Bob · 2013-04-04 18:52:56

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

Maiya · 2013-04-05 23:28:48

Hello guys;
i totally agree with Bobbym...
since infinity is no real number and since its value is not known,arithmetic of infinity cannot be performed as we do in case of other numbers.....

And 0.9999999999999.....can never be equal to unity.....
limit of .999999..... could be 1
but saying that .99999999...itself is equal to 1 would be absurd since both of those numbers have their own existence which cannot be equated......

Bob · 2013-04-06 00:07:50

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

Bob

Math Is Fun Forum

#51 2013-03-06 09:44:01

Re: [1 paradox]

#52 2013-03-06 10:48:18

Re: [1 paradox]

#53 2013-03-06 10:53:02

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#54 2013-03-06 11:00:24

Re: [1 paradox]

#55 2013-03-06 11:03:09

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#56 2013-03-06 11:34:17

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#57 2013-03-06 11:35:26

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#58 2013-03-06 20:04:56

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#59 2013-03-07 03:41:56

Re: [1 paradox]

#60 2013-03-07 09:11:37

Re: [1 paradox]

#61 2013-03-07 10:46:15

Re: [1 paradox]

#62 2013-03-07 11:47:20

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#63 2013-03-21 19:21:30

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#64 2013-03-21 22:06:53

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#65 2013-03-21 22:24:30

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#69 2013-03-23 12:20:24

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