Ok lets do this 1/3 = Infinite 0.33333333333333333333...................

Anthony, we are so close.  I just ask you to do one further thing.  We both agree to the above.  Just multiply both sides by 3 and tell me what you get.

Quote Ricky!
Ok lets do this 1/3 = Infinite 0.33333333333333333333...................

Anthony, we are so close.  I just ask you to do one further thing.  We both agree to the above.  Just multiply both sides by 3 and tell me what you get.

The above Number has nothing to do with Infinite 0.9!
Any Calculations you put forward must be based around the number we are working on! ie Infinite 0.9

Its like me saying show me if you can divde 4 by 3 ? and make a whole number!

oh and by the way can you show me that is 7 - 4 = 3 its nothing to do with the above! I just want to see your answer!

I'm back

Ricky wrote:

George, we need to define what we mean by "decimal expansion" for the question, "Does 0.999... = 1?" to ever make sense.  You find my definition arbitrary, so can you please provide one that isn't?  In fact, all definitions are arbitrary.  That's why they are definitions and not theorems.

My definition is for defining the question, not the solution.  By intuition alone, I believe that:

1/3 = 0.333...
1/9 = 0.111...
square root of 2 = 1.41421...

I made a definition supporting this view.  My definition treats the digits of a decimal expansion as a sequence of numbers, and it is the limit of this sequence which we treat as the number itself.  With this definition, I claim that every real number has at least one decimal expansion.

You do not believe that such infinite decimal expansions exist.  As a result, not all real numbers, or even all rational numbers, have a decimal expansion.  You may not see this as a problem.  I do.

Now let me state the following very carefully because you have not been understanding my words.

Mathematical spaces are formed by definitions.  Every set of axioms can form a mathematical space, and each are just as valid as others.  Your rejection of my definition is just as valid of my acceptance of it.  Neither of us can say that one is right and one is wrong.  However, there is a trend in mathematics for definitions that get us somewhere.  When we find a set of definitions which can solve interesting and useful problems, people tend to flock to those definitions.  It does not mean they are any more valid, only popular.

The definition I use are the popular ones.  But I have also provided reasons for my definition.  I want a decimal expansion for every real number.  Frankly, I can't understand why you don't.  But that doesn't really matter in the end.

Now, as to your answer to the question, "Does 0.999... = 1", it should not be, "No."  Rather, is should just be, "The question does not make sense with my definitions of the real numbers and decimal expansion."  Those are two entirely different answers, and if you don't accept infinite decimal expansions, then how could you ever answer the question with anything other than "It doesn't make sense"?  And that is a perfectly valid answer.  For example, I have proved elsewhere that if you accept the real numbers as a field, then division by 0 just doesn't make sense.  And I find that as a valid conclusion.

But I'm curious.  Do you see any problems with my definition?  You have stated that you see problems with my use of infinity.  I use the same concept of infinity as used in the limit of an infinite sequence, and in fact, I set up the digits of a decimal expansion as an infinite sequence.  Does this make sense?

To be honest, I am rather insulted that you claimed that I was trying to, "suppress the minority's opinion by the majority's."  I come to this forum to help others, have fun, and because there are some rather smart people here worth talking to about math.  And I don't take claims that I'm trying to suppress someones opinion lightly.  I am not having this conversation/debate for my health, but because it interests me.  Do you really think I'm trying to suppress your opinion by having an open debate on a public forum?

"Most people accept this definition by intuition alone.  You're the first person I've ever seen which did not."-Ricky
Since you have explained this sentence enough,  I will no longer indicate that you were trying to judge me as "absurd".

I may also note you a phsycologic fact that people tend to stick to what they were told during their childhood with prejudices without enough questions, making it difficult for a heratic opionion to spread. For example, after my post exploring the inconsistency of infinity, still a lot of people tried to amend the "infinity" concept rather than to throw it away. Also some one sticked to 1-0.999...=0.00..1 where 0.00...1 is nothing or invalid, while in fact 0.00..9 is invalid according to the same logic.

On the popularity issue, I shall argue that infinite expansions never rivals sufficient but finite expansions when the real thing is clear. Pi is great, but it is more accurate to say that all "circles" we can encounter coincide with Only a finite digits of Pi. So my alternative to replace an infinite expansion is just sufficient expansions, or their group. Actually is my alternative so far that all the people of the world along with all the computers of the world can practice?

At last, a defination cannot be ARBITORY, it should be logicly consistent with itself and should mirror the reality. And these are far more important than intuition alone-people used to have many intuitions, including one that the sun rotates around the earth.

To Anthony:
You cannot agree that 1/3=0.333... for it is easy to use the "proof" with multiplying them with 3.
Instead, you should point out no one has calculated 1/3 out yet.(As I insisted in the previous thread, which has already been closed by Ricky)
Or simply further, 0.333... is inconsistent with itself. (As I insisted in this thread)

Why is this?  How do you know a measurement, any measurement can be described in a finite amount of digits?  How do you know that it isn't just an artifact of our measurement technique which produces finite digits?

Even if the thing we are measuring is continuous (height, for example), because of our limited technology, we may only measure discrete distances.  You seem to be claiming you can accurately measure reality with no error.  Certainly, that is not the case.
---I mean matters-wood, plastics, paper, to name a few. They are made of an integer amount of particles, disabling them to form any real circles.


First off, show me how my definition is not logically consistent or does not mirror reality.  Actually, first, you can show me where infinity occurs in reality.  Or do you think infinity should not be part of mathematics too?  But I believe definitions are still arbitrary.  It is arbitrary is the same way that the word "chair" is arbitrary.

--- I mean 0.999... with infinite digits is not, at least the form. So with a line formed by points.Secondly, I need not show you where is infinity. Instead, you need to show me how can a line be formed by "infinite" points in reality.
The English word "chair" may be arbitary, but the concept of "chair" is based on a repetitive object designed to  hold an math with a surface and usually 4 legs beneath the surface.

The Biggest mistake Mathematicians make! Is that they believe an Infinite Endless number some how is! Or ends up a whole number!
The example below is the purest Math, with no assumptions, no beliefs, no forecasts, and no calculations other than what is the truth in front of one’s eyes!!

A = 1
B = Infinite 0.9
C = 3

A + B + C = 4.99999999999999…………………………………………(NOT (5) ?

x = 4.999...
10x = 49.999...
10x-x = 49.999...-4.999...
9x = 45
x = 5

4.999... = 5

p.s I forgot to say!

"I'ts impossible to actually multiply an Infinite number! because to be able to multiply a number! first a person has to know the size of the number!"

"I'ts impossible to actually multiply an Infinite number! because to be able to multiply a number! first a person has to know the size of the number!"

Contradiction.  If you agree that 1/3 is a real number, and that 1/3 = 0.3333..., then it must be that you are able to multiply it.  This is because the real numbers are a field under addition and multiplication.

Let’s have a look at why it’s impossible to Mutilply, Subtract, Divide Etc any Infinite Numbers!

For this I will use the Infinite 4.9 put forward as an example.

A = Infinite 4.9 (stage one) 1 decimal place.

B = Infinite 4.99 (stage two) 2 decimal places.

C = Infinite 4.9 (Actual Number) Endless decimal places!

10 x A = 49.9 we know this is a true Number because we know the size of the Number i.e. 1 decimal place!

10 x B = 49.99 we know this is a true Number because we know the size of the Number i.e. 2 decimal place!

10 x C = 49.9 ? It’s impossible to know this because we don’t know the size of the number and more important how many decimal places!

The only possible answer would be based on the facts! C started 0.1 less than 1.
In the calculations no other Numbers have been Add to C (or can be!).
No other Numbers have been multiplied,(or can be) the Number remains the Infinite!

So an Answer would be Infinite 4.9 = (0.1 < 5) this way we don’t need to know how many .9’s there are!

ok anthony, lets put it this way.

what do you do when you multiply a number by 10? you move the decimal place one place to the right.

so if we have 4.999999...

times it by 10, move the decimal place one place to the right

we have 4.(9) * 10 = 49.(9)

If you try to multiply 4.999... one digit at a time you won't ever get an answer, but if you step back and see the *idea* then you will find that you can.

For example

4.9 × 3 = 14.7
4.99 × 3 = 14.97
4.999 × 3 = 14.997

So, the pattern is that 4.999... = 14.999...

And what about that last 7? You will *never* reach it. Infinity is endless.

George, with my definition on what an decimal expansion is, it does in fact make sense.