Introduction
Chapter 1 is devoted to the definition of the term continued fraction and proving of statements about the its connection to the common fraction. Also there are proved statements about the features of the continued fractions. Examples on finding roots of continued fractions equations are given.
Chapter 2 considers the usage of continued fractions in other natural sciences. They were used to find the differences among Julian, Gregorian, Omar Khayyam’s and Midler’s calendars.
It is shown that it is possible to use continued fractions in physics to find the method of connection electric conductors to each other. Also the are given examples on usage of continued fractions in botany, mechanics and architecture.
Chapter І
Continued fractions and their usage
§1. Scheme of the continued fractions
Such fractions as shown here
are
called “continued fractions”. In such fractions a fraction with
numerator 1 is added to the
number
,
a fraction with numerator 1 is added to the number
of
that fraction, and
a
fraction with numerator 1 is added to
the number
of
a fraction of that fraction and so on.
Here the
numbers
are whole numbers (number
can
be equal to 0).
Fractions
with the meaning
in
continued fractions are called making fractions.
If makers are infinitely many, they are called infinite fractions,
and if makers can be counted,
they are called finite fractions.
The fraction shown above can be written in this form
( ).
Examples: this fraction we can be written in this form
Theorem1. Every finite continued fraction can be expressed in a form of common fraction with the value equal to the it.
Proof. Actually every fraction consists of adding and dividing operations, that is why if its making fractions are countable, it is possible every time to solve and finish adding and dividing operations shown among them. The solvation of those operations is certainly can be in a form of a common fraction.
for example: we have such fraction
Then
2)
3)
4)
5)
This finite continued fraction can be expressed in a form of common fraction with the value equal to the it.
Theorem 2(converse theorem): Every common fraction with positive value can be expressed in a form of continued fraction with the value equal to it.
Proof.
Let’s take a fraction with meaning
.
If we take whole numbers from it, we will get equation
,
where a – whole multiple, value of t would be equal to the value of dividing x by y. If there is no whole numbers in the fraction , the condition a = 0 is suitable and it is equation t = x.
Now
if we divide both members of the fraction
into
t,
so
there will be equations
.
Here
number
is a whole multiple as a result of dividing y
by
t
and
is
the difference.
Now
if
we divide both members of the fraction
by
,
so
we will get the equation:
Here the number is a whole multiple from dividing t by .
If
we use this fraction
line continueously for the
following made fractions, it is proved that in the end given fraction
will make an expression in the above form. Hereof the theorem is solved.
In
the above approach when we divide x by y, y by the first difference
t, the first difference by the second difference, numbers
are
the resulting whole multiples.
That
is why they are also called multiples of the continued fractions.
The mentioned approach
is also used when finding the biggest new multiples of two numbers of
value x
and
y.
Example:
It
is necessary to convert the fraction
into
the infinite fraction.
In
this case it has the following expression:
