Hi Calligar;

Welcome to the forum. That is exactly the point.

3.333333333... = 3 + 1 / 3

I have seen mathematicians that hate set theory. Mathematicians that hate continuous math. Mathematicians who hate pure math, ones who hate applied math and computers. Ones who hate Topology and Logic, almost every type. But I have never seen anyone advancing while he/she held on to the idea that .9999999... ≠ 1

hi rya

yes that is correct,but my point is that we have multiple ways of representing the same number.

Hi;

First of all this has been argued many times before by the theoretical type of mathematicians and they can explain it better than I can.

When you say .333333333... = 1 / 3 what you are really saying is

That equals 1 / 3. Same thing with .9999999... If you really can not accept that then think of math like a formalist does. That it is a game and games have rules. We do not debate why the knight moves as it does in chess.

Rule 1) .999999999... = 1

You can now accept this and move on.

however I realise this idea will likely be debated and/or dismissed by assuming I'm automatically wrong.

Although that sometimes happens in mathematics it is much rarer than in other fields like biology or psychology for instance. In math we prove things, there are dozens of proofs about this.  I have seen many arguments that
.9999999999... ≠ 1 but I have never seen any proof of that.

What about 9.999999999999...89999999...  See the problem, Ricky would have said this number is
impossible because you can't append on the end of the 9.99999999999999... , but why not append
after infinite digits, if you can imagine it, it should be true, so therefore, there are many numbers
that all seem to be equal to one another, but have different digits after the infinite decimal place,
which most agree doesn't exist, but I say it does, otherwise the outerspace would not go in two directions
or more in 3-D, it would just go one way.  In order to have shapes in the real world you need infinity to
have ending points and boundaries where new shapes can exist in the same room.  For example, all
of the furniture in your room you may think is just 2 feet or 3 feet wide, but imagine again if there is
no smallest unit of measure, if this could be true then there everything we see that has shapes and
therefore a beginning and an end if measured, is also infinite in size when you shink yourself to a
very small size an look at it.  There is no end to how small things can be and no end to the fact that
these infinite shapes have contours and therefore 9.999....643...999.... can exist!!  (this is not the belief
of most math people, however: (disclaimer))

Hi John;

I do not think you can do that. When you write:

.99999999999... the three dots means continue on forever with nines. If you put an 8 later than the original statement is violated.

hi

I’ve been watching this discussion for some time, without comment, but feel it’s time to throw in my thoughts.

Numbers, all numbers, are just abstract concepts.  You cannot hold a ‘three’ in your hand.  You can have three apples or write a squiggle on the page to represent three, but three doesn’t exist except in the mind of a mathematically inclined person.

We’ve spotted that you can have consistent rules for these abstract concepts.  For example, if I’ve got three apples and someone gives me another four apples then I’ve got seven apples.  If I’ve got three goats and I buy four more goats, then, when I count them up, I find I’ve got seven goats.

So, in certain circumstances, it’s ok to say 3 + 4 = 7. 

But, notice this rule doesn’t always work. If I walk 3 miles and then walk 4 miles, I'm not necessarily 7 miles away from where I started.

Someone once critcised Isaac Azimov for using an "imaginary" number (square root of minus one.)  But Azimov responded by arguing that all numbers are imaginary (ie. made up figments of our imagination).  When the critic protested that he knew some numbers were completely ‘real’, like a half for example, Azimov pointed to the blackboard (this tale is an old one) and asked him to hand over half a stick of chalk! 

The point is, you make the rules for the situation.  If the rule works then it’s worth knowing.  Otherwise it’s likely to be consigned to the great rubbish bin of ideas that didn't get off the drawing board.

So, for instance, when we make the rule for powers:

we find it works ok with all the other stuff we know about powers.

Then when someone poses the question “I wonder if we can find a sensible meaning for:

the answer is, yes, we can.  It makes good sense to let it have the value ‘1’ because this is consistent with the rule of powers:

Now to get back to those infinitely recurring decimals.  They are just another abstract form of number.  I know I cannot write all the digits, but I can still think about how the number behaves as if I could.  I know that if I try to work out 1 divided by 3, I’ll keep getting a remainder, and so if I keep on dividing I’ll end up with 0.33333333r.

So, it’s entirely consistent to treat 1/3 and 0.3333333333r as if they were the same.  It does lead on, as others have shown, to results like ½ = 0.5 = 0.49999999999r but so what?  The rules that allow this are quite consistent and allow us to teat all terminating decimals as if they were recurring.  eg. 0.37 = 0.36999999999999r.  That then allows us to explore the concept of number density as in ‘the reals’ are more dense on the number line than the ‘rationals’.

And there’s nothing wrong with exploring mathematically interesting concepts.  Number theory, and especially all the theorems involving primes, were once thought to be the province of the most pure of pure mathematicians.  But now this theory is essential for internet trading.

And consider all those uses for complex numbers.  Not so ‘imaginary’ now, are they?

So don’t worry if some mathematical ideas seem a little strange when you first meet them.  You just need to let your imagination run with the ideas and be prepared to see where it all leads.

Infinity, underground maps, non Euclidean geometry .... all are possible in the world of mathematics.  And there’s definitely a continent where

If you’ve read this far, thank you for your patience. 

Bob

Last edited by bob bundy (2011-10-11 10:23:08)

Very nicely explained, bob, well done.

bob bundy wrote:

but three doesn’t exist except in the mind of a mathematically inclined person.

It need not be a person.  Certain animals have a concept of numbers and counting.
And numbers, (counting numbers, for instance) have their own existence independent
of human thought.  Before there were humans to think about numbers,
there were always places where there was one thing, two things, etc.

bob bundy wrote:

So, in certain circumstances, it’s ok to say 3 + 4 = 7. 

But, notice this rule doesn’t always work. If I walk 3 miles and then walk 4 miles,
I'm not necessarily 7 miles away from where I started.

This is not a correct example of "a rule that doesn't always work,"
as you put it.  You're comparing apples and oranges.  Those movements equal
a total of 7 miles walked.  The rule works.  Mixing in where someone ended
up relative to the starting point is changing the subject.
 

bob bundy wrote:

So, for instance, when we make the rule for powers:

we find it works ok with all the other stuff we know about powers.

Then when someone poses the question “I wonder if we can find a sensible meaning for:

the answer is, yes, we can.  It makes good sense to let it have the value ‘1’ because this
is consistent with the rule of powers:

You have made a blanket statement.  It is not so where x = 0.


Under that method,

So 0^0  could  equal 1,

to reconsideryouranswer

This is not a correct example of "a rule that doesn't always work,"
as you put it.  You're comparing apples and oranges.  Those movements equal
a total of 7 miles walked.  The rule works.  Mixing in where someone ended
up relative to the starting point is changing the subject.

I was trying to simplify the idea that the rules for, say, vectors, are not the rules for arithmetic.  You have supported my argument by demonstrating that one needs to be clear of the circumstances before applying a blanket rule .  And ...

You have made a blanket statement.  It is not so where x = 0.


Under that method,

<bits of code lost here>


So 0^0  could  equal 1,

<bits of code lost here>

Here again you have shown that mathematicians may select their own rules to fit what they want to use them for.

You could make an axiom that 1 = 2.  Plus all the usual rules.

It wouldn't be long before you found that the whole number system collapses down to a single number, 1, and a single binary operation, say, times.

Thus you would end up with 1 x 1 = 1.  True but somewhat limited in application.

That's my point.  You can make up the rules and explore where it leads.  If it has use then others will want to use it too.

It need not be a person.  Certain animals have a concept of numbers and counting.
And numbers, (counting numbers, for instance) have their own existence independent
of human thought.  Before there were humans to think about numbers,
there were always places where there was one thing, two things, etc.

That's very interesting.  If it's true, I don't think it changes my argument; just substitute 'some animals including humans' for 'person'.  But which animals did you have in mind?

If all humans (and the animals of the above paragraph) were to disappear from the Universe there would still be one thing and two things etc.  Yes, but where is the concept of 'three'.  You need an intelligence to conceive a 'three'.  I still claim it has no independent existence.

Bob

Last edited by bob bundy (2011-10-11 18:52:49)

Okay, I have spoken to my brother about this in great detail (whom I consider a math genius).  He was explaining to me that this has been argued numerous times before in history, and ultimately from what it is considered now, I was wrong about this.  On top of that, he corrected me about infinitesimals saying that was the accepted way how to represent that, but that doesn't make what I say related to one, as infinity is mostly only used in calculus and higher.  As an example, he showed me an indefinite integral problem.  Now, my brother did state something interesting.  Logically, I can argue this, because he understands the way I'm looking at this, and can understand where I am coming from.  However, that according to the rules that already exist in math to prevent misunderstandings and things that people don't understand well enough to be definite on, that I am wrong, and that in order to be right, the rules would need to be changed on that.  But just as Bobbym had said in an earlier post, there were in fact rules in place that my brother actually showed and explained to me why it was like that.  So to conclude all of this, I admit defeat.  I was told I was wrong here, only to be proven it later by my brother.  It's like my brother said, I can argue this logically, but that wouldn't make it correct in math.  So, let me say this: 3.¯3+6.¯6=10 because 9.¯9 DOES equal 10, unlike what I was arguing before.

Au101 wrote:

(I'm certainly thinking of my younger self when I say this)

This reminds me of "By his bootstraps" and "The man who folded himself"...

I give it a 9.99999999999...