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Topic: comparing Peano axioms of Naturals with Continued Fractions; why pi
needs 3 zeroes in a row #76.33 Math-Professor-text 8th ed.: TRUE CALCULUS

Replies: 8   Last Post: Dec 9, 2013 7:57 PM

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 plutonium.archimedes@gmail.com Posts: 18,572 Registered: 3/31/08
comparing Peano axioms of Naturals with Continued Fractions; why pi
needs 3 zeroes in a row #76.33 Math-Professor-text 8th ed.: TRUE CALCULUS

Posted: Dec 3, 2013 12:52 AM

Alright, what I am basically looking for is the necessity of pi having 3 zero digits in a row before the Infinity borderline is reached. Here is pi to the 10^-603 place value

3.14159 26535....132000
?2.71828 18284....117301?
1.61803 39887....764861

Now the Peano axioms are very much similar to the Continued Fraction of generating numbers and representing those numbers.

The Peano axioms concern themselves only with the Counting Numbers or Counting Rational Whole Numbers. Continued Fractions envelopes all Rationals.

The Peano axioms start off with 2 given numbers (although Peano and his successors never admitted their mistake of logic that you cannot have the Naturals unless you state 0 and 1 exists, both together) and that Peano made a huge mistake in thinking he could do the axioms by stating only 1 exists or that only 0 exists, for then in his successor axiom we have the hidden assumption that 1 exists. The logical mistake of Peano and his successors was that in order to start a axiom program, you need the existence of 2 numbers at the start so as to gain a metric and that metric of adding 1 in the Successor axiom is reinforced by the existence of 0 and 1 with their metric distance of 1 spacing between them so as to generate the next number of 1+1= 2. So that was a huge error in the Peano axioms.
Now we see the Peano axioms with the successor as a generator of producing all the Counting Numbers by the adding of 1 to the previous number to gain the next number.

We see the generator in Continued Fractions, only it is a reverse generator where we start with a fraction and decompose it into its constituent parts of integers. Such as for example 45/16 = {2; 1, 4, 3} as a continued fraction, or as 2.8125 as a decimal.
How does the Continued Fraction generate the rational number? How does Peano axioms generate the Counting Numbers? Well in Peano the successive adding of 1 generates all the Counting Numbers. In Continued Fractions we start off with a fraction that is a rectangle composed of squares. This is seen on this website:

http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/cfINTRO.html

So that as we start with a large fraction, we circle around and downwards into smaller and smaller squares inside of rectangles. In Peano we go higher by adding 1, in Continued Fractions, we go lower by cycling downwards into squares.

So in a sense, fractions or rationals have an axiom system very much like the Peano axioms are to the Counting Numbers.

Now we return to our beginnings of the Peano axioms and its crucial axiom (flawed by Peano) that we need at least **two counting numbers to exist as given** in order to produce all the other counting numbers. We can say that both 0 and 1 exists and they provide a measure or distance apart that provides us with a distance of 1 so that the third counting number is 1+1= 2.

Now for Continued Fractions and taking one of the most simple Continued Fractions of phi, the golden ratio of 1.61..... we see it is {1; 1, 1, 1, ...} So that means if we look at the golden ratio logarithmic spiral it ends up with the innermost two squares of 1 by 1 and 1 by 1 square contiguous to a 2 by 2 square forming the most primitive of three squares in Continued Fractions.

The 1 by 1 and 1 by 1 and the 2 by 2 three squares form the basis of Continued Fractions, just as the existence of 0 and 1 forms the basis of the Counting Numbers.

So, now, here, I have to link why pi at infinity with those 3 zero digits in a row
is the necessity for having all Continued Fractions of 1, 1, 2. Of course the next squares are 3 then 5, then 8 etc etc.

So, why does pi need three zero digits before infinity relates to why the basis of Continued Fractions is the 1 and 1 and 2 squares.

--
Drexel's Math Forum has done an excellent search engine for author posts as seen here:
http://mathforum.org/kb/profile.jspa?userID=499986

Now, the only decent search for AP posts on Google Newsgroups, is a search for plutonium.archimedes@gmail.com for it brings up posts that are mostly authored by me and it brings up only about 250 posts. Whereas Drexel brings up nearly 8,000 AP posts. Old Google under Advanced Search
for author, could bring up 20,000 of my authored posts but Google is deteriorating in quality of its searches, likely because AP likes an author search and Google does not want to appear as satisfying to anything that AP likes. If AP likes something, Google is quick to change or alter it.

So the only search engine today doing author searches is Drexel. Spacebanter is starting to do author archive lists. But Google is going in the opposite direction of making author archived posts almost impossible to retrieve.

All the other types of Google searches of AP are just top heavy in hate-spam posts due to search-engine-bombing practices by thousands of hatemongers who have nothing constructive to do in their lives but attack other people.

Now one person claims that Google's deteriorating quality in searches of science newsgroups is all due to "indexing". Well, that is a silly excuse in my opinion, because there is no indexing involved when one simply asks for a author search. No indexing involved if one wants only the pure raw complete list of all posts by a single author. And Google is called the best search engine of our times, yet I have to go to Drexel to see 8,000 of my posts of which I had posted 22,000 to 36,000 posts from 1993 to 2013. It is a shame that Drexel can display 8,000 while Google has a difficult time of displaying 250 of my authored posts. Where the premiere search engine of Google is outclassed by Drexel and even by Spacebanter.

Archimedes Plutonium