Re: Irrational numbers questions

Jesse F. Hughes wrote:

Han de Bruijn <Han.deBruijn@xxxxxxxxxxxxxx> writes:

Quoted from this page: Let's associate with any irreducible fraction
p/q the number w(p/q) = 1/pq - its simplicity (: by Pierre Lamothe).
There is a theorem that the sum of simplicities of all fractions in any
row (with 2^n elements if n = row #) of the Stern-Brocot tree equals 1.

Obvious facts are: the simplicity of 1/2 is 1/2. The simplicity of 5/12
is 1/60. The simplicity of sqrt(2) is .. 0. In general: the simplicity
of any irrational number is 0 .

The obvious fact is that the simplicity of any irrational number is
undefined.

No. But I did some cut and paste in the wrong order. It's defined in the
sequel, where it reads:

Any real number can always be enclosed between two fractions, say m1/n1
and m2/n2 in the Stern-Brocot tree: m1/n2 <= r <= m2/n2. We cannot just
define the simplicity of a real number. What we can do, however, is to
establish an upper bound for it. The lower this upper bound, the less
"simple" the real number is, hence the more "irrational" it looks like.

Any of the three RS-definitions can be used for this purpose. Let some
irrational r be enclosed by two neighbouring fractions in the S-B tree:
m1/n1 < r < m2/n2 . Then finding another (more accurate) nesting for r
results in either L = m1/n1 < r < (m1+m2)/(n1+n2) = U
or L = (m1+m2)/(n1+n2) < r < m2/n2 = U .
The simplicities of the composed fractions (m1+m2)/(n1+n2) are always
_smaller_ than the simplicities of the constituents: m1/n1 and m2/n2 .

It is expected that the simplicity of r is somewhere in between the RS
of L and U. We play a safe game by considering the upper bound of RS(L)
and RS(U) and attaching it to r : RS(r) < maximum( RS(L) , RS(U) ) .

Jesse F. Hughes wrote:

You really are quite incapable of basic deductive reasoning. And yet
you claim that if Han doesn't get it, it can't be mathematics. Funny
stuff.

Oh, well, 'sci.math' as usual ..

Han de Bruijn

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