Cont: Deeper than primes - Continuation 2

[doronshadmi]

Well, not without a few additional details--those pesky things that cause you so much angst--but once all pieced together, sure 0.999... is identical to 1.

Please explicitly show those pesky things that once all pieced together, clearly show that 0.999...₁₀ is identical to 1.

And again, 0.999... is undefined without base > 9.

 

[doronshadmi]

I always like how Doronshadmi likes to drag in other number systems.

For example, 0.111...₂ or 0.111...₃ are two different numbers, as clearly seen by the following diagram:

So, exactly as 0.111... without base is undefined, so is the case about 0.999...

 

[Rum]

Can a relative mathematical illiterate interrupt with a change of tack and a question please? Occurred to me when reading and trying to understand some of the above.

Is it possible to create a spreadsheet which identifies prime numbers?

 

[jsfisher]

Can a relative mathematical illiterate interrupt with a change of tack and a question please? Occurred to me when reading and trying to understand some of the above.

Is it possible to create a spreadsheet which identifies prime numbers?

All prime numbers? No, there are infinitely many such things. However, within the limitations of the spreadsheet application, it is possible to implement the Sieve of Eratosthenes^WP to identify primes in a simple, straightforward way.

 

[Little 10 Toes]

Link for Rum:

http://www.excelexchange.com/prime_number_test.html

 

[doronshadmi]

Can a relative mathematical illiterate interrupt with a change of tack and a question please? Occurred to me when reading and trying to understand some of the above.

Is it possible to create a spreadsheet which identifies prime numbers?

Currently the best is http://en.wikipedia.org/wiki/Formula_for_primes.

 

[doronshadmi]

No, that's not any more difficult.

There is more than one limit if a given infinite sequence does not have any term that is identical to a given limit.

This reasoning can't be deduced unless infinity is used in order to construct a given number, exactly as done in:

http://www.internationalskeptics.com/forums/showpost.php?p=10652939&postcount=550

http://www.internationalskeptics.com/forums/showpost.php?p=10653953&postcount=556

http://www.internationalskeptics.com/forums/showpost.php?p=10654703&postcount=558

and more comprehensively explained in http://www.internationalskeptics.com/forums/showpost.php?p=10652239&postcount=539.

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

I am still waiting to jsfisher's detailed reply to http://www.internationalskeptics.com/forums/showpost.php?p=10655265&postcount=561.

 

[Rum]

Thanks guys. Yes, I am aware there are (probably) an infinite number of primes. Was hoping for a test or generator - amply provided. Many thanks.

 

[jsfisher]

There is more than one limit if a given infinite sequence does not have any term that is identical to a given limit.

Mathematics has its way of doing things; you have yours. The latter has no influence on the former.

 

[jsfisher]

Thanks guys. Yes, I am aware there are (probably) an infinite number of primes. Was hoping for a test or generator - amply provided. Many thanks.

The proof that the number of primes is infinite is very simple. If you are interested:

This is a proof by contradiction. We will begin by assuming the opposite of what we wish to prove and show how that leads to a contradiction: Assume the number of primes is finite. There must then be a largest prime number. Call it P.

Let Q be the product of all the integers between 1 and P (inclusive). (So, Q = 1 x 2 x 3 x ... x P).

Now, consider the number Q+1. Because of how Q was constructed, dividing Q+1 by any prime number will have a remainder of 1. So, Q+1 must be prime number.

It is clear that Q+1 is greater than P, so if Q+1 is prime and greater than P, then our original assumption is contradicted. There can be no largest prime number, and so the number of primes is infinite.

 

[doronshadmi]

Mathematics has its way of doing things; you have yours. The latter has no influence on the former.

Mathematics is not a private property of any authority, so not me not you or any one else can categorically determine its way of doing things.

Please reply to http://www.internationalskeptics.com/forums/showpost.php?p=10655265&postcount=561.

 

[doronshadmi]

There can be no largest prime number, and so the number of primes is infinite.

Also rum there can't be more than |N| prime numbers, which is the number of all natural numbers, known also as the smallest infinite cardinal number.

For more details please look at http://en.wikipedia.org/wiki/Aleph_number.

I use the difference between |N| and |R| cardinality in order to construct numbers like 0.000...1₁₀ where .000... is at |N| size and ...1₁₀ is at |R| size (which is the cardinality of the smallest uncountable amount of elements, that is observed as one element from the point of view of the |N| countable amount of elements).

 

[jsfisher]

Mathematics is not a private property of any authority, so not me not you or any one else can categorically determine its way of doing things.

Once again, you are invited to build whatever mathematical system you'd like to demonstrate your superior approach. And as previously stated, your new system would not make any existing system wrong, only different. However, since you have heretofore been unable to define anything, it is unlikely you'll ever exhibit such a system.

Please reply to http://www.internationalskeptics.com/forums/showpost.php?p=10655265&postcount=561.

It is the same question you have asked before then rejected the answer because you don't like how Mathematics works. No need for me to again provide you the opportunity to re-express your incredulity.

 

[doronshadmi]

Once again, you are invited to build whatever mathematical system you'd like to demonstrate your superior approach. And as previously stated, your new system would not make any existing system wrong, only different. However, since you have heretofore been unable to define anything, it is unlikely you'll ever exhibit such a system.

The results of the definition of convergences are simply some particular case of the results of my definition of limits (defined and explained in http://www.internationalskeptics.com/forums/showpost.php?p=10656499&postcount=567), exactly because the definition of convergences forces the finite on the infinite, and therefore does not use the difference between transfinite cardinals on order to construct numbers.

It is the same question you have asked before then rejected the answer because you don't like how Mathematics works. No need for me to again provide you the opportunity to re-express your incredulity.

I reject your answer, because it does not provide any proof that, for example, 0.999...₁₀ is identical to 1.

Please be aware of the fact that my definition does not reject the fact that 1 is one of the possible limits in case of the infinite sequence (0.9₁₀, 0.99₁₀, 0.999₁₀, ...), so your attitude about Math development is simply the narrow one without any reason that supports its narrow view in the discussed case of limits.

 

[Little 10 Toes]

Please explicitly show those pesky things that once all pieced together, clearly show that 0.999...₁₀ is identical to 1.

And again, 0.999... is undefined without base > 9.

Please tell me which step confuses you:

1. X = 1.0
2. X / 3 = 1 / 3
3. 1 / 3 = 0.333...
4. X / 3 = 0.333...
5. 3 * (X / 3) = 0.333... * 3
6. 1.0 = X = .999...

 

[jsfisher]

I reject your answer

Yes. We know.

 

[Little 10 Toes]

Also rum there can't be more than |N| prime numbers, which is the number of all natural numbers, known also as the smallest infinite cardinal number.

For more details please look at http://en.wikipedia.org/wiki/Aleph_number.

I use the difference between |N| and |R| cardinality in order to construct numbers like 0.000...1₁₀ where .000... is at |N| size and ...1₁₀ is at |R| size (which is the cardinality of the smallest uncountable amount of elements, that is observed as one element from the point of view of the |N| countable amount of elements).

Please define "size". Please define "observed ... from the point of view".

Edit: Please tell me how you define "size". Please tell me how you define "observed ... from the point of view".

 

[Little 10 Toes]

The results of the definition of convergences are simply some particular case of the results of my definition of limits (defined and explained in http://www.internationalskeptics.com/forums/showpost.php?p=10656499&postcount=567), exactly because the definition of convergences forces the finite on the infinite, and therefore does not use the difference between transfinite cardinals on order to construct numbers.


I reject your answer, because it does not provide any proof that, for example, 0.999...₁₀ is identical to 1.

Please be aware of the fact that my definition does not reject the fact that 1 is one of the possible limits in case of the infinite sequence (0.9₁₀, 0.99₁₀, 0.999₁₀, ...), so your attitude about Math development is simply the narrow one without any reason that supports its narrow view in the discussed case of limits.

post history

 

[doronshadmi]

Please tell me which step confuses you:

1. X = 1.0
2. X / 3 = 1 / 3
3. 1 / 3 = 0.333...
4. X / 3 = 0.333...
5. 3 * (X / 3) = 0.333... * 3
6. 1.0 = X = .999...

You are confused in step 3), and an equivalent confusion is seen in http://www.internationalskeptics.com/forums/showpost.php?p=10329336&postcount=88.

Moreover your notations are undefined since you are not using any base (you simply ignore http://www.internationalskeptics.com/forums/showpost.php?p=10655285&postcount=562).

 

[doronshadmi]

Edit: Please tell me how you define "size". Please tell me how you define "observed ... from the point of view".

I use the definitions and reasoning of transfinite cardinality in both cases, which are not used by the definition of convergence.