The Foundations of a Non-Naive Mathematics

[hello3719]

1/3 is not just an arithmetical operation of 1 divided by 3, but it is also the Q member 1/3.

The same we can say about pi, for example:

If the diameter is 1 then pi is pi/1.

well the Q set existence is based on the fact that the division operation is admitted in the system using the integers thus giving 1/3 a unique identity.
(saying "1/3 is not just an arithmetical operation of 1 divided by 3, but it is also the Q member 1/3." means nothing at all)

 

[hello3719]

The same we can say about pi, for example:

If the diameter is 1 then pi is pi/1.

That's the best you can do ?

 

[Lama]

Q are called rational numbers because these numbers are defined by the ratio that can be found between at least two integers, where the irrational numbers cannot be defined in this way, so as you see '/' is also used for a ratio.

Therefore (for example) 1/3 means the ration between 1 and 3, which is the rational number 1/3.

 

[hello3719]

But because some fractals can be found in infinitely many different scale levels, then from the quantitative point of view they never fully represent 1/3 or pi.

Who said that we want to use fractals to represent number theory concepts ?
The concepts we are talking about use some more basic defintions than the fractals' one.

 

[terrabyte]

LOL. Exactly .333... CAN BE EXPRESSED as a ratio of integers. Seems you have a flaw in your logic.

not so.

.333... is an incomplete expression. the irrational computational result of dividing 1 by 3.

even if you have INFINITE digits with value 3, you STILL do not have the number that EQUALS 1/3. I can add 1 more digit to whatever your number is and come up with a number that is Closer or More Accurately Expressed as 1/3 than your number.

.333... is NOT 1/3

there is no flaw in my logic. just a failure in realization on your part.

 

[hello3719]

Q are called rational numbers because these numbers are defined by the ratio that can be found between at least two integers, where the irrational numbers cannot be defined in this way, so as you see '/' is also used for a ratio.

ratio and division is the exact same thing in mathematics, so why are you saying "also" ?

 

[Lama]

Who said that we want to use fractals to represent number theory concepts ?
The concepts we are talking about use some more basic defintions than the fractals' one.

If you use the base value expansion methods then you use a part of a fractal, whether you want it or not.

In this case you cannot ignore anymore the properties of a fractal.

 

[hello3719]

not so.

.333... is an incomplete expression. the irrational computational result of dividing 1 by 3.

even if you have INFINITE digits with value 3, you STILL do not have the number that EQUALS 1/3. I can add 1 more digit to whatever your number is and come up with a number that is Closer or More Accurately Expressed as 1/3 than your number.

.333... is NOT 1/3

there is no flaw in my logic. just a failure in realization on your part.

OMG. Now this proves that you DON'T know what infinity means neither what rational and irrational means.

"the irrational computational result of dividing 1 by 3." this doesn't make sense at all. I will say it one more time, irrational simply AND ONLY characterizes a number THAT CAN'T be expressed as a ratio of 2 integers. It is a simple as this.

 

[terrabyte]

yes. and I've said it a few times as well. .333... IS NOT properly expressed by 1/3

step 1. take the number 6
step 2. Divide it by 3.

step 3. take the number 6
step 4. multiply it by .333... (i don't care how many digits you expand it out to)

compare results.

notice anything? :|

 

[Lama]

ratio and division is the exact same thing in mathematics, so why are saying "also" ?

Because 1/3 is also known as the constant symbol of the number that is produced by this division operation.

 

[hello3719]

If you use the base value expansion methods then you use a part of a fractal, whether you want it or not.

In this case you cannot ignore anymore the properties of a fractal.

refering to http://mathworld.wolfram.com/Fractal.html

"A fractal is an object or quantity that displays self-similarity, in a somewhat technical sense, on all scales. The object need not exhibit exactly the same structure at all scales, but the same "type" of structures must appear on all scales"

Did you know that "scales" are consequences of the arithmetic operation of division AND NOT VICE-VERSA. This means that you can stop using fractals to explain your ideas and just use more basic concepts.

 

[hello3719]

yes. and I've said it a few times as well. .333... IS NOT properly expressed by 1/3

step 1. take the number 6
step 2. Divide it by 3.

step 3. take the number 6
step 4. multiply it by .333... (i don't care how many digits you expand it out to)

compare results.

notice anything? :|

k we have a problem of communication it seems. To clear things out,
When i write .333... i mean there is an INFINITY of 3's. so by definition of infinity i can conclude that 6 * .333... is equal to 2. So what's the problem ?

 

[Lama]

Did you know that "scales" are consequences of the arithmetic operation of division AND NOT VICE-VERSA. This means that you can stop using fractals to explain your ideas and just use more basic concepts.

Not correct, we can define scales also by mutiplication.

 

[hello3719]

Because 1/3 is also known as the constant symbol of the number that is produced by this division operation.

Logical flaws as always.
1/3 means in english " we are dividing 1 by 3"
so it doesn't have a double identity, like i said before don't compare the result with operation.

 

[zeronem]

not so.

.333... is an incomplete expression. the irrational computational result of dividing 1 by 3.

Like I said again, an irrational number has no repeated pattern of integers in decimal notation of a ratio.

1/3 has a repeating pattern of 3's, which makes it rational. 1/7 has a repeating pattern of .142857... which makes it rational. However Pi, being the ratio of the circle's circumference to diamter, is an irrational number because it has no repeating pattern of integers in decimal notation.

$$\frac {1}{3} = .\overline {3}$$


$$\frac {1}{7} = .\overline {142857}$$

 

[hello3719]

Not correct, we can define scales also by mutiplication.

well sure we can, since division is just going backwards in the multiplication.
Still a scale ISN'T as basic of a definition as is division or multiplication nor , as a consequence, is fractal.
You didn't prove anything.

 

[Lama]

Logical flaws as always.
1/3 means in english " we are dividing 1 by 3"
so it doesn't have a double identity, like i said before don't compare the result with operation.

Let us make it simpler.

I choose @ as the result of 1/3 so 1/3 = @.

But If you use the base value expansion method, then you use a part of a fractal, whether you want it or not.

In this case you cannot ignore anymore the properties of a fractal.

And these properties can be cleary understood by:

-----------------------------------------------------------------

If we use a structural point of view in this case, then 0.9999... is a single path of a base 10 fractal ( http://www.geocities.com/complementarytheory/9999.pdf pages 3,4 ), that exists upon infinitely many scale levels that cannot reach 1.

Also we can say that 0.999... = 0.9+0.09+0.009+0.0009+... and we can clearly see that this infinitely long addition cannot reach 1.

Therefore 0.999... < 1.

Another example:

Please look at this beautiful Koch Fractal members.cox.net/fractalenc/fr6g6s.577m2.html[/URL]

Now let us say the there is a 1-1 map between each fractal level of 0.9999... to each different blue level of Koch Fractal.

0.9999... = 1 if and only if we cannot find anymore a 1-1 map between some 0.000...9 to some Koch Fractal blue level.

Since Koch Fractal can be found in infinitely many blue levels and each blue level has a 1-1 map with some 0.000...9 fractal level, then we can conclude that 0.999... < 1.

Also we can say that 0.999... = 1 if and only if the outer contour of this multi-leveled Koch Fractal can be a smooth curve with no sharp edges.

It is clear that the outer contour line is not a smooth contour in any arbitrary examined scale level.

Therefore 0.999... < 1.

From this model you also can understand what is a "leap".

In short, any transition between a non smooth curve to a smooth curve, cannot be done but by a phase transition leap that also can be described by a smooth_XOR_no-smooth connection.

This model is better than any "abstract" mathematical definition, which leads us to "prove" that 0.9999... = 1.

Also by this "proof" we simply ignore infinitely many information forms that can be found in 0.9999... fractal.

Now think how many information forms are ignored by this trivial and sterile approach of standard Math.

 

[Lama]

well sure we can, since division is just going backwards in the multiplication.
Still a scale ISN'T as basic of a definition as is division or multiplication nor , as a consequence, is fractal.
You didn't prove anything.

By these two operations we get the same result, which is a fractal.

Therefore the fractal property is the invariant and these operations are only tools to explore it.

To cealrly see and understand this fractal please look at http://www.geocities.com/complementarytheory/No-Naive-Math.pdf page 5.

 

[sparkster]

yes. and I've said it a few times as well. .333... IS NOT properly expressed by 1/3

step 1. take the number 6
step 2. Divide it by 3.

step 3. take the number 6
step 4. multiply it by .333... (i don't care how many digits you expand it out to)

compare results.

notice anything? :|

step 1. Take the number 1.
step 2. Divide it by 3.
step 3. Keep dividing
...
...
...
step 4,653. Keep dividing
...
...
...
step 3039209823752820. Keep dividing.

Notice anything?

 

[sparkster]

Your "rigorous" proof depends on excluded-middle black_XOR_white reasoning.

therefore you cannot deal with the complexity of 0.999... case.

Only Included-middle reasoning can deal with the complexity of this fractal.

In short, your "rigorous" proof is nothing but the image of your trivial black_XOR_white reasoning method.

I used a proof by contradiction which you attempted (and failed) to do. I used "<" which you've tried to use again and again.

You reply with meaningless gibberish.

Will you actually address what is posted rather than ramble in you private littel world of non-math.

 

[sparkster]

A cute answer:

1/3 is not a fractal where 0.333... is a single path of a fractal.

They are not the same number exactly as 0.999... and 1 are not the same number.

When your logical system is based on an Included-Middle reasoning, the internal structural properteis of any mathematical element, cannot be ignored anymore.

You've already established that you can't do analysis. Surely you can do long division and see how inane this statement is.

 

[sparkster]

not so.

.333... is an incomplete expression. the irrational computational result of dividing 1 by 3.

So you're just going to use a unique definition of irrational number? That's fine, but don't expect anyone who actually does math to understand you (or take you seriously).

even if you have INFINITE digits with value 3, you STILL do not have the number that EQUALS 1/3. I can add 1 more digit to whatever your number is and come up with a number that is Closer or More Accurately Expressed as 1/3 than your number.

This is just silly. If there are an infinite number of 3's you can't add anymore to the end.

.333... is NOT 1/3

there is no flaw in my logic. just a failure in realization on your part.

You're right, there's no flaw in your logic, you just don't understand what irrational and rational numbers are.

 

[Hurkyl]

I can add 1 more digit to whatever your number is and come up with a number that is Closer or More Accurately Expressed as 1/3 than your number.

If there's a 3 in every position to the right of the decimal place, then where, pray tell, are you going to add one more digit?

 

[terrabyte]

i'm going to add 1 more digit wherever you "stop" at of course.

otherwise how do you expect to convey the "meaning" of your number to me?

1. we derive numerical meaning from the differences in number digits. 1 is clearly different from 2. 1.1 is slightly different from 1.2. the differences in these digits allow quantities to have meaning.
2. the quantity .333... has no meaning until it is brought into the realm of known quantities. hence it has to be definable within the boundaries of numbers within proximity to it AND be distiguishable as such. this may seem like a pithy statement with no meaning but hold on...
3. the number .333... for this number to be exactly 1/3 it must have infinite digits of 3. Infinite in the sense that they're unending, not in the sense that they're greater than all numbers.
4. define a number that is slightly less than this number. will .333...2 work? not really, that number is closer to 1/3 than .333... or maybe .333... where the number of digits 3 is (Infinity-1) <getting into cardinality with that but meh whatever>.
5. as you can see, because of the nature of digits stringing out to infinity it is impossible within the current system to define the number in relation to other numbers within proximity.

 

[terrabyte]

which is one more reason why it's irrational, not merely in the sense that it cannot truly be expressed as a ratio of two integers, but because the quantity is unplacable on the number line.

 

[sparkster]

i'm going to add 1 more digit wherever you "stop" at of course.

otherwise how do you expect to convey the "meaning" of your number to me?

1. we derive numerical meaning from the differences in number digits. 1 is clearly different from 2. 1.1 is slightly different from 1.2. the differences in these digits allow quantities to have meaning.
2. the quantity .333... has no meaning until it is brought into the realm of known quantities. hence it has to be definable within the boundaries of numbers within proximity to it AND be distiguishable as such. this may seem like a pithy statement with no meaning but hold on...
3. the number .333... for this number to be exactly 1/3 it must have infinite digits of 3. Infinite in the sense that they're unending, not in the sense that they're greater than all numbers.
4. define a number that is slightly less than this number. will .333...2 work? not really, that number is closer to 1/3 than .333... or maybe .333... where the number of digits 3 is (Infinity-1) <getting into cardinality with that but meh whatever>.
5. as you can see, because of the nature of digits stringing out to infinity it is impossible within the current system to define the number in relation to other numbers within proximity.

You really don't understand math do you? That's just sad.

 

[Hurkyl]

i'm going to add 1 more digit wherever you "stop" at of course.

Stop what?

3. the number .333... for this number to be exactly 1/3 it must have infinite digits of 3. Infinite in the sense that they're unending, not in the sense that they're greater than all numbers.

There is no (right) end to .333...

 

[terrabyte]

thus it will NEVER equal 1/3

 

[Hurkyl]

thus it will NEVER equal 1/3

Can you exhibit a number between 0.333... and 1/3?

 

[terrabyte]

.333... is not a rational number.

can you exhibit a number slightly greater than or less than .333... ?

 

[Hurkyl]

Yes, lots.
0.3 < 0.333... < 0.4
0.33 < 0.333... < 0.34
0.333 < 0.333... < 0.334
0.3333 < 0.333... < 0.3334
...

Pick any upper bound for "slightly", and I can find one of these inequalities such that the smaller and larger are slightly different than 0.333...

What is the point of this exercise?

 

[terrabyte]

k expand those out to infinity. infinite precision, so-to-speak

all of those numbers cease to have distinction

 

[terrabyte]

.333...2 and .333... and .333...4 are all the same number.

there is no designation or incrementation to distiguish their properties of being different numbers because of the current definition limitations of "Infinity"

that's the purpose of the exercise

 

[sparkster]

.333...2 and .333... and .333...4 are all the same number.

there is no designation or incrementation to distiguish their properties of being different numbers because of the current definition limitations of "Infinity"

that's the purpose of the exercise

.333...2 and .333...4 aren't numbers. They're products of your misunderstandings. If "..." is means inifinitly many, you can't have infinitely many 3's and then add a 2 or a 4.

 

[Hurkyl]

.333...2 and .333... and .333...4 are all the same number.

No... one of those is a number; the other two are gibberish.


Anyways, you've evaded my response. None of the approximations I intend to make have an infinite number of nonzero terms, however if you pick any positive value as an allowable tolerance for error, one of my approximations will be within this tolerance.


It seems you are trying to ask me to tell you what the "next" decimal number is, but there is no such thing. (If there was a "next" number, then what happens if I take their midpoint?)


And I'll ask again, what does all of this have to do with whether 1/3 = 0.333...?