
How do you find the square root of .
Answer
461.4k+ views
Hint: In this question, we used continued fraction expansion. And fraction is that in mathematics, a fraction is an expression obtained through an iterative process of representing verity because the sum of its integer part and therefore the reciprocal of another number, then writing this other number because the sum of its integer part and another reciprocal, and so on.
It is a finite continued fraction, where n is a non-negative integer, is an integer, and is a positive integer, for .
It is generally assumed that the numerator of all of the fraction is . If the arbitrary values and functions are utilized in place of one or more of the numerator or the integer in the denominators, the resulting expression may be a generalized continued fraction.
Complete step by step solution:
The number is .
The factor of has no square factors, so can’t be simplified.
Then,
It is an irrational approximation; I will find a continued fraction expansion for then truncate it.
To find the simple continued fraction expansion of , we use the following algorithm.
This algorithm stops when , making the end of the repeating part of the continued fraction.
Then, the continued fraction expansion is.
Next, in the question the value of and , since .
So by using the continued fraction expansion:
Now we will calculate :
Now we will calculate
Now we will further simplify it as,
Now we will calculate
Having reached a value which is twice the primary value , this is often the top of the repeating pattern of the fraction, and that we have:
The first economical approximation for is then:
Then, we again used the repeated value.
Therefor the closer value of square root of is:
Note: As we know that continued fraction is just another way of writing fraction. They have some interesting connections with a jigsaw puzzle problem about splitting a rectangle into squares etc. it is the simple method for finding the square root of a number which has no square factor.
It is a finite continued fraction, where n is a non-negative integer,
It is generally assumed that the numerator of all of the fraction is
Complete step by step solution:
The number is
The factor of
Then,
It is an irrational approximation; I will find a continued fraction expansion for
To find the simple continued fraction expansion of
This algorithm stops when
Then, the continued fraction expansion is.
Next, in the question the value of
So by using the continued fraction expansion:
Now we will calculate
Now we will calculate
Now we will further simplify it as,
Now we will calculate
Having reached a value
The first economical approximation for
Then, we again used the repeated value.
Therefor the closer value of square root of
Note: As we know that continued fraction is just another way of writing fraction. They have some interesting connections with a jigsaw puzzle problem about splitting a rectangle into squares etc. it is the simple method for finding the square root of a number which has no square factor.
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