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The Euclidean Algorithm and Continued Fractions
For the Euclidean Algorithm
you've got a greatest common denominator. This is the normal way it's introduced in
school these days. Euclid used the subtraction method.
This is going to be a visual method. Where the
two numbers are the width and length of a rectangle.
Subtract the width of the rectangle from the length of the rectangle.
You get twelve minus five
arithmetically. Visually you get this.
Here's the divising ~ the method of division.
And you continue the process, you get another residual rectangle five by two.
Five by two et cetera. Another residual rectangle. You continue the process till you
get down to a square.
Now you've got one is the greatest common denominator. The same thing here, you get
it down to 0 and it's the prior remainder.
Now the quotients here are twos
And in the continued fraction you see the twos down here. The way I would introduce that
k-12
you have two divisor symbols here. Take the second from the last.
This one and underneath that you've got a
mixed number convert that to an improper fraction, in this case five over two.
Invert that; it's one over (five over two) so it becomes
two over five. Now you've got two plus two over five. Convert that to a
rational. And that would be; in this case in order to add them: ten-fifths.
So ten-fifths plus two-fifths equals twelve-fifths.
OK a few more examples. Now this would be another method of ~
dividing out the Euclidean A lgorithm. Eight goes into fourteen one time remainder six. Six goes
into
eight one time remainder two. Two goes into six three times remainder zero. That makes
two the greatest common denominator as reflected in the
rectangular grid by the smallest square being a two by two
square. The continued fraction itself would look something like this:
one plus one-third which would be four-thirds.
One over four-thirds would be three-fourths and that would be
four-fourths plus three-fourths is seven-fourths which is proportional to
fourteen-eighths. And ~ notice the ~
greatest common denominator two by two
Two by two; two times seven,
two times four is fourteen-eighths which is the size of the
grid. Next, one more.
The greatest common denominator three which is the size of the smallest square.
Continued fraction is let's see, six-fifths one over six-fifths is five-sixths
plus six-sixths is eleven-sixths,
times the three by three. Three (times three) is thirty-three
Three times six is eighteen. And that's the size of the
grid. Let's see, there should be one more example
Do it out here. Let's do this one.
We'll use an approximation (to the square root of two) of about one point four one decimal.
Which becomes one hundred forty-one over one hundred. Here's the
long one (long fraction). Here's the visual representation. The geometric representation as you can see
each
~ Let me zoom in on this.
A little bit more. As you can see, each
~ square is a multiple (rational multiple) of the previous one
and
"show all" - Now we go into this, one and a half down here becomes three-halves.
One over three-halves becomes two-thirds. One plus two-thirds would be;
let's see three-thirds plus two-thirds: five-thirds. One over five-thirds
becomes three-fifths. Again inversions ~
~ mixed numbers and inversions are fairly basic to k-12 so
this would be ~ within range of that
age group. ~ Three plus three-fifths would-be
fifteen-fifths plus three-fifths: eighteen-fifths. One over eighteen-fifths is
five-eighteenths. Two plus five-eighteenths, this would be thirty-six-eighteenths
plus five-eighteens which is forty-one. What you notice is; up there it's getting,
approaching something we can recognize. There we go forty-one eighteenths (41/18)
Inverted eighteen forty-ones (18/41). Two is
let's see eighty-two forty firsts (82/41) plus eighteen is
one hundred forty-firsts (100/41). There we go inverted is
forty-one one hundredths (41/100). One is a hundred-hundredths.
Now you've got that (141/100) and there it's proved!
OK, and that's what I wanted to have the app do.
You can also show a finite continued fraction algebraically.
And this illustration I got from the Euclidean ~
Euclidean Algorithm article in Wikipedia. And all's I did was add in the letter
names and the numbers
so you can see the relationships a little bit clearer. There's two types
of units here.
One is the square units which would represent the area. The other is the linear
units
and in each case the side of the smallest square
indicates the unit. I'd like to show why I would like the continued fractions in the
first place. ~ For something like an irrational number like the square root
of two
a continued fraction will show a regular pattern.
It evaluates to a sequence of rational
approximations which approximate closer and closer
to the decimal representation of
the square root of 2.
You can also show pi ( ı ) with a more complex variation on the ~
continued fraction. You just have to know that the regular pattern exists.
It's the odd number squared over a sequence of one plus
two plus two plus two - forever. And it sort of makes ~ Just the idea that a rational
approximation
exists makes pi more interesting.
At least it did so for me. I'll go into
infinite continued fractions in the next video.