Dear Alex,
You write that a number r is rational if it can be written a fraction r = p/q where both p and q are integers.
Does that mean that phi - = "The golden section", "The golden ratio", "The divine proportion" - should be considered rational, as the number phi = r obviously is based upon r = p/q?

Also, if ?2 likewise can be written as r = p/q, as for example r = 47321/33461 = ?2, (where 33461 = prime and 47321 = prime 599 x prime 79), should this not also be seen as a rational number - Especially since in mathematics rational means "ratio like"?? Note that the ?2 behaves in exactly the same manner as phi, i.e. as it approaches infinity it becomes what it is!

Please enlighten me if I somehow got this wrong!
Kind regards
Klaus Kastberg.

>Dear Alex,
>You write that a number r
>is rational if it can
>be written a fraction r
>= p/q where both p
>and q are integers.
>Does that mean that phi -
>= "The golden section", "The
>golden ratio", "The divine proportion"
>- should be considered rational,
>as the number phi =
>r obviously is based upon
>r = p/q?

By "based" you mean what? Can any of them be written as p/q?

>Also, if ?2 likewise can be
>written as r = p/q,
>as for example r =
>47321/33461 = ?2, (where 33461
>= prime and 47321 =
>prime 599 x prime 79),

47321/33461 is a good approximation to http://www.cut-the-knot.com/gifs/sqrt.gif";]2. But the two are not equal. Any real number can be approximate with arbitrary precision by rational numbers. Does it mean that every real number is rational. No, of course not.

>should this not also be
>seen as a rational number
>- Especially since in mathematics
>rational means "ratio like"??

Rational means being represented as a ratio, not "ratio like."

>Note
>that the ?2 behaves in
>exactly the same manner as
>phi, i.e. as it approaches
>infinity it becomes what it
>is!

You should be more careful using words in your arguments. [img src="http://www.cut-the-knot.com/gifs/sqrt.gif2 never approaches infinity. In fact it's a constant.

>Please enlighten me if I somehow
>got this wrong!

You may try getting help on the sci.math newsgroup. You may be able to get a more enlighten argumentation.

We know that 'phi' exist in nature as 'reality', i.e. that this ratio and proportion was existant long before anybody thought
about us human beings - who, incidentally, also to some extent are 'based' upon this ratio in our build. (Ref. Dürer,Da Vinci etc.).
We know that the ratio's 8/5 - 55/34 - 377/233 - 3571/2207 - etc.
gives phi. We know that when the fractions progressively increase in size, phi becomes more and more; because when a graph is drawn and the sums of the various fractions are plotted along a x-axis, a 'wavelength' or 'oscillation' is formed which more and more
approaches the x-axis (although never meets it) as the size of the fractions progressively approaches infinity. This simply means that the ratio 'phi' cannot possibly consist of an endless series of decimals, but must forever alternate near its 'shortest' point of ratio, which in this case would be near
1.618034..., and alternating on both sides of ..034.. e.g.
..039999...n, and ..0340000...1n. Therefore one can confidently say that phi is still phi regardless whether it manifest itself through the smallest or 'largest' fraction, i.e. its lowest or highest oscillation as plotted!
Hence phi is a 'live' breathing ratio and proportion, and therefore not 'irrational'??

Exactly the same scenario applies to the square roots.

47321/33461 should not be seen as mere approximation to ?2, as
in fact 47321/33461 becomes (is) ?2, in the same way that fraction e.g. 9512/6726 becomes (is) ?2 and fractions 114243/80782 - 5017856/3548160 - 17132032/12114176 and e.g.
1550614528/1096450048 becomes (is) ?2, etc. etc.

Likewise the square root of 3, as the fraction e.g. 56813/32801
becomes (is) ?3 in the same way that e.g. 367245/212029 and
3040493371/1755429667 becomes (is) ?3, etc.etc.

Thus it would be in error to assume that square roots consist of endless non-repeating decimal expansions, as this is obviously not the case!
So, it can be said again that the square roots behave in the same
manner as phi, i.e. they are 'alive' - or in other words - their
progressive fractions (or each term) is continually the sum of two preceding ones! This their property of being at the same time
additive and geometrical is a characteristic they share with phi
and its ròle ...in the growth of living organisms. Ergo, the square root of 2 therefore, is not a constant!! (Ref. phi =
1 + ?5/2)!

As I'm not a mathematician its not for me to give proper equations to the square roots and their natural progressions - or
harmoniously ordered or rhytmically repeated proportions or recurrences! This must be left to someone without my sad handicap!

The term "ratio like" I got from one of your colleagues at:
http://www.utm.edu/research/primes/glossary/rational number.html
- ...sorry!

The questions remain: Should phi be considered an irrational number, even though it is 'perfectly' rational? And why should square roots equally be considered irrationel, when they too are
'perfectly' rationel? In fact, why should any whole, natural, real number be considered irrational, when all natural numbers,
whether odds, evens or primes, consist of reciprocal fractions
which all are based upon decimal expansions with repeating equal
length blocks - or repeating root-stems - the only exemptions
being the multiples of 2 and 5?? I.e. all reciprocal numbers
(decimal expansions), mirror or reflect in various modes either 1/2, 1/3, 1/7, 1/9, 1/11 or 1/17 configuarations!

Forget about enlighten me this time, but your comments or views
to the above would nevertheless be greatly appreciated.
With kind regards
Klaus Kastberg

>We know that 'phi' exist in
>nature as 'reality',

Does anything exist without being observed is a philosophical issue which I am sure has been and is being debated till this very day.

>i.e. that
>this ratio and proportion was
>existant long before anybody thought

But it's not a (rational) ratio nor a (rational) proportion, its name notwithstanding.

>about us human beings - who,
>incidentally, also to some extent
>are 'based' upon this ratio
>in our build. (Ref. Dürer,Da
>Vinci etc.).

Let them be based on that ratio, which is OK with me. Are they equal to it?

>We know that the ratio's 8/5
>- 55/34 - 377/233 -
>3571/2207 - etc.
>gives phi.

I do not undertsand this. 8/5 is a ratio. 55/34 is a ratio, and so are 377/233 and 3571/2207. And the sequence thus constructed converges to phi. That's OK. But this does not make phi a ratio.

>We know that when
>the fractions progressively increase in
>size, phi becomes more and
>more;

phi does not become more and more. For, it's a constant.

>because when a graph
>is drawn and the sums
>of the various fractions are
>plotted along a x-axis, a
>'wavelength' or 'oscillation' is formed
>which more and more
>approaches the x-axis (although never meets
>it) as the size of
>the fractions progressively approaches infinity.

Do please post the above to the sci.math newsgroup. Listen to what other people have to say.

>This simply means that the
>ratio 'phi' cannot possibly consist
>of an endless series of
>decimals,

Nonetheless, as any irrational number phi has a decimal represenation which is neither finite nor periodic.

>but must forever alternate
>near its 'shortest' point of
>ratio, which in this case
>would be near
>1.618034..., and alternating on both sides
>of ..034.. e.g.
>..039999...n, and ..0340000...1n. Therefore one can
>confidently say that phi is
>still phi regardless whether it
>manifest itself through the smallest
>or 'largest' fraction, i.e. its
>lowest or highest oscillation as
>plotted!

Do please post the above to the sci.math newsgroup. Listen to what other people have to say.

>Hence phi is a 'live' breathing
>ratio and proportion, and therefore
>not 'irrational'??

Does it also breeding?

>Exactly the same scenario applies to
>the square roots.

I understand what you mean, but

Do please post the above to the sci.math newsgroup. Listen to what other people have to say.

>47321/33461 should not be seen as
>mere approximation to ?2, as

I wish you all the best.

Do please post the above to the sci.math newsgroup. Listen to what other people have to say.

>As I'm not a mathematician

You do not have to state the obvious.

>its
>not for me to give
>proper equations to the square
>roots and their natural progressions

You seem however quite determined about
your viewpoints.

Etc.

>Does anything exist without being observed is a philosopical
>issue...

Now you got me confused! Are you here saying that nothing existed before mankinds appearance on earth? What! - not even earth itself... just because there was nobody here to observe it!
Then reason and logic didn't exist before man either? Well what
about the perfectly reasonable and logical relationship between the sun and the moon prior to mans arrival... should that be considered non-existent also!! Oh dear - what thoughts.

>But it's not a (rational) ratio nor a (rational) proportion, its
>name notwithstanding

Who says? I mean, there exist great confusion as to what constitutes a rational number. Some say it can be written as the ratio of two integers p/q. Others say that you can only have a rational if a square root of an integer is an integer itself! Which means for example that integers from 3, 5, and 7 are irrationals, but integers from 9, 25 or 49 are rationals, doesn't
it? But then again, it also means that integer 3 is rational because it can be written 3 = 3/1? And we are told that rationals are those with terminating continued fraction expansions, and those with repeating decimal expansions. We are told that almost all real numbers are irrational, and that the decimal expansion to these numbers do not repeat in equal length blocks!
But they actually do.

If you base rationality and irrationality on how decimal expansions behave, then you will run into all kinds of trouble.
E.g., do you call Prime 499 one thing because it has one kind of decimal expansion, and do you call Primes 239 and 271 something else because their decimal expansions behave differently?
This would smell of prejudice and lack of understanding - would
it not!
All natural numbers(n) of value (i.e. born either directly or
indirectly from zero and one or unity) are interrelated such that
all integers(n) (Primes or otherwise), all roots(n) or fractions
of any kind(n), doubles in value by going through unity as shown
in the simple formula of fraction (n)/(1/n) = 2n. Because of this
common bond and interrelatedness, all natural numbers of value(n)
should hence be termed rational numbers. And all numbers which are man-made non-naturals and therefore without value, and as such are deemed meaningless, as is the case with the present pi(<pi>) and the present e (base of 'natural' logarithms), only these numbers should be called irrational, simply because they carry within them the general meaning of this word as well!

The true value of pi(<pi>) is 22/7 or 3 1/7 (=rational). Yes,
Archimedes was quite right that long ago, and this can now be proven with very little effort. But one suspect that mathematicians are not really ready for this yet perhaps!

As a curiosity only though, the reader can here be given a little
foretaste to this fact:
If we use the Fibonacci series based upon the integers 1, 3, 4, 7, 11, 18, 29,.. etc., we find that the 16th number of this series turn out to be 2207. If we use the ratio of phi (1,618034..), we find likewise that phi to the power of 16 gives us exactly the sum of 2207.
2207 is a Prime number.
Should we add together all the Fibonacci numbers above from 1 to 2207, we get the sum of 5775. 5775 multiplied by .04 gives 231,
which again equals 3 x 77. 2 x 77 equals 22 x 7. If we check the
natural Fibonacci series of 1, 2, 3, 5, 8, 13, 21, 34, 55, 89,..
and add these 10 numbers together, also gives 231. From the same
series as 2207 we find that the 10th number gives us 123.

This of course proves nothing, but the reader must surely admit,
that this new pi of 22/7 do feel a bit nicer and a bit better to
look at, than the present one... especially when it seems, in this harmonious way, to be directly associated with the ABC of
numbers!

>Phi does not become more and more. For, it's a constant

Yes it does become more and more. It becomes endlessly more and more closer to what it in essence already is! The same happens to
square roots. And yes, it's also a constant, but that doesn't
contradict anything?

>Do please post the above to the sci.math newsgroup.
>Listen to what other people have to say

Who! And why?
I'm quite happy listening to what you have to say Mr. Bogomolny.

>Does it also breeding?

Well actually it does, but indirectly only of course, through
being present in the blueprint of living organisms such as the logarithmic spiral of Nautilus Pompilius, the Gnomonic Growth of the Triton Tritonis, marine animals, seed distribution in cactus plants, or sunflowers, or the human body etc. etc.

It is never a good idea to use mean scorn, ridicule and sarcasm,
as by established laws this will always return to the originator.
(Same as hatred and love does)!

Kind regards
Klaus Kastberg.

distribution of leaves around stems

>>Do please post the above to
>>the sci.math newsgroup.
>>Listen to what other people have to say

>Who! And why?
>I'm quite happy listening to what
>you have to say Mr. Bogomolny.

I've nothing to add. We seem to use different definitions of what a ratio or a rational number is. I believe mine is the one commonly used in mathematics. On the sci.math group that is visited by many people from all walks of life, there may be different views - some even closer to yours. Why not to try?

>I've nothing to add. We seem to use different definitions of
>what a ratio or a rational number is

For the benefit hopefully, perhaps, to some other visitor to this
forum:

1.5 = 1 1/2 = 3/2 = 3 : 2 ratio

1,414285714... = 1 29/70 = 99/70 = 99 : 70 ratio.

1,414201183... = 1 70/169 = 239/169 = 239 : 169 ratio.

1,414213562... = (?2) = e.g. 1 13860/33461 = 47321/33461 =
47321 : 33461 ratio.

1,414225941... = 1 99/239 = 338/239 = 338 : 239 ratio.

1,414141414... = 1 41/99 = 140/99 = 140 : 99 ratio.

1.4 1 2/5 = 7/5 = 7 : 5 ratio.

1.333333333... = 1 1/3 = 4/3 = 4 : 3 ratio.

99/70 x 140/99 = 2. 239/169 x 338/239 = 2.
47321/33461 x 47321/33461 = 2. (= 1 x 1 = 1 : 1 ratio.

Why should the square root be treated different to the others?
And why make life so difficult?

Perhaps I should check out the sci.math after all, what do you think?
Kind regards
Klaus Kastberg