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.

1) 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 .

Дата добавления: 2015-11-14