§3. Some ways to use the continued fractions
If the numerator and the segment of the given fraction are very large numbers, sometimes it is easier to take fractions with simpler numbers convergent to its real meaning. So it is necessary to change the given fraction for the continued fraction and to use instead of it a value of one of other convergents. For example, if the real value between the circle and its diameter is between the fractions 3,1415926 and 3,1415927, if we express the required multiple π (or otherwise pi), to find the values of π easy to write and convergent to the real value, first of all let’s convert the previous fractions into the form of the continued fraction.
Then
If
we take only mutually common multiples of two continued fractions, we
will find that π is equal to this fraction:
.
So the first convergents of π would be the following.
15 |
1 |
|
|||
3 |
22 |
333 |
355 |
||
1 |
7 |
106 |
113 |
||
A
scholar Archimed found the convergent value of π is
,
hence when we take it instead of π our error will
be less than
.
If
we take
instead
of π
,
our error will be greater than this fraction:
As the conergent fractions of Archimed and Meshiy are at the even places, values of both actually are greater than π.
Finding a square root
Let’s
take a problem
to find a square root of 13.
For it let’s use the following project: 1) the greatest square root
entering into 13 is 3, because
if we take 4,
it is
and
its square power is greater the given number.
And now let’s consider this equation to be appropriate:
(3.1)
Hereof through the expression
(3.2)
the following equation is suitable:
(3.3)
As the value of the equation (3.3) is equal with adding one fraction,
(3.4)
we can make the expression shown above, hence through the expression
we make this equation:
(3.5)
And now observing that the meaning of the equation (5) is equal to the value of adding one fraction to 1:
(3.6)
we will have this expression. Consequently, through the expression
(3.7)
this expression results:
(3.8)
If we try to convert the equation (3.8) into the following form according to the above project
(3.9)
then it is
and expression
is found. After converting the equation into this form:
,
as a result we will find this equation:
(3.10)
In the last equation it is performed the following expression:
(3.11)
If we again convert the equation into such a form:
(3.12)
then we will find this equation
(3.13)
If
we try to compare equations (3.13) and (3.2),
we know that
or
.
Therefore
through the equations (3.1),
(3.4), (3.6), (3.8), (3.9), (3.12) we
can
write this fraction:
(3.14)
Hence
now the square root of
is
converted into the form of the unlimited and repetitive fraction.
Here the multiple (1,1,1,1,6) recurs limitless. Certainly, every
convergent fractions of the continued fraction (3.14) are the
convergent roots of
.