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For Infinite Continued Fractions ~ again the reason I'm interested in this is it shows continuity on the
number line
and shows a rational ~ it shows a pattern to the continued fraction which is not
otherwise available in the decimal representation.
Makes it a little bit more interesting. Again you have a pattern for say
the square root of 2
and also in a more complicated version you can get a
patterned for pi ( ı ) This
particular illustration I got from
Wikipedia. I think might have the ~ this is the Dedekind cut article on Wikipedia.
And here's the illustration. And it turns out that the numbers they use
~ the numbers they use here are
a continued fraction ~ are a continued fraction. There are other methods
of demonstrating this but the
continued fraction has the illustration which I can use this grid app for.
The first thing you notice is that this is really not quite to
scale. If you actually used a scale seven-fifths would be point four and point five would be three
halves. It would have to be all sort of
conglomerated in there. A little too close to
work. So here's the continued fraction again the way it works is it provides a
sequence
of rational approximations each which can be represented as a rectangle.
(Which) when divided will provide you with a closer and closer approximation
to the square root of 2. And the Dedekind cut
just indicates that's where ~ that's a (an) irrational number.
other methods ~
did (do) exist for this, the Babylonian Method and
the Newtonian method that I'm aware of. I'm not very proficient in either one.
But I don't have any visual
demonstration of those other than the algebraic solution.
And here we go. The square root of 2 approximately equals one. This starts the process.
You rewrite this as one plus one half. You get three halves.
Now we're working top to bottom on this. Now we've got
one plus one over (two plus one half) which is
five-halves, two-fifths
five-fifths plus two-fifths is seven-fifths. (The process) converts the last mixed fraction to an improper fraction
and inverts it. We've
done this before. Here, this is where I'm going to stop.
A
continued fraction
as a nested sequence of intervals
leading to a point on the number line which we call an irrational
number.
It works something like this. And it goes on forever.
This is where it was first stated as far as I can see.
Richard Dedekind, 'Continuity and Irrational Numbers' and he said " ~ if we knew
for certain that space was discontinuous
there would be nothing to prevent us in case we so desired
from filling up its gaps in thought" - and according to
Courant Robins you could use Dedekind cuts, convergence, which was
Cantor's terminology and/or nested intervals.
He states they are more or less equivalent for this purpose of proving
continuity of the number line.
That is not entirely
undisputed these are the two best articles i've seen
on arguing against the continuity of the number line.
And this is the best philosophy I have found,
" ~ the existence of Math does not depend on a satisfactory answer ~
constructive intuition always remains the vital element ~ "
Another ~ method of demonstrating it. You take the
square, revolve the
diagonal around, Now you've got a square
plus a remainder rectangle. One is the whole number representation the
remainder is the
fractional representation. You divide that.
Now you get two squares. The width goes into the length two times with the remainder
rectangle.
In the case of this ~ this continued fraction this
square root, you always get the same dimensions from
the remainder rectangle from here on in. So it goes on
like that forever. OK. Now here the sides of the
rectangles are all going to be multiples that unit value.
You can sort of see three by two.
Seven by five.
Seventeen by twelve.
Here we go. By placing the
rectangle in the ~ vertical direction
you can demonstrate the tangent which is something about like this.
The tangent being the opposite over the adjacent.
This angle three by two. Three by two.
And seven by five.
Seventeen by twelve.
Et cetrera.