Question

In Newton-Raphson’s method write the formula for finding the square root of the number N.

Answer 1

Hint: First of all, assume that the guessed number ‘x’ is the square root of the number ‘N’. We know the property that the square root of a number is that number which is multiplied twice to itself to get the original number. So, \[N={{x}^{2}}\] . Now, add the term \[{{x}^{2}}\] in LHS and RHS of the expression \[N={{x}^{2}}\] . Now, divide by x in the expression \[N+{{x}^{2}}=2{{x}^{2}}\] . After this, again divide by 2. Now, solve it further and get the formula.

Complete step-by-step solution:
According to the question, we have to get the formula for finding the square root of the number N.
We know the property that the square root of a number is that number which is multiplied twice to itself to get the original number ………………………………..(1)
Let us assume that the guessed number ‘x’ is the square root of the number ‘N’.
The original number = N ………………………………………(2)
The square root of the original number = x ……………………………………..(3)
Now, using the property shown in equation (1), we can say that when the number x is multiplied to itself then we get the original number. So,
\[N={{x}^{2}}\] ……………………………..(4)
Now, on adding the term \[{{x}^{2}}\] in LHS and RHS of equation (4), we get
\[\Rightarrow N+{{x}^{2}}={{x}^{2}}+{{x}^{2}}\]
\[\Rightarrow N+{{x}^{2}}=2{{x}^{2}}\] ………………………………………..(5)
Now, on dividing by the term x in the LHS and RHS of the equation (5), we get
\[\Rightarrow \dfrac{N}{x}+x=\dfrac{2{{x}^{2}}}{x}\]
\[\Rightarrow \dfrac{N}{x}+x=2x\] …………………………………….(6)
Now, on dividing by 2 in the LHS and RHS of the equation (6), we get
\[\begin{align}
  & \Rightarrow \dfrac{\left( \dfrac{N}{x}+x \right)}{2}=\dfrac{2x}{2} \\
 & \Rightarrow \dfrac{\left( \dfrac{N}{x}+x \right)}{2}=x \\
 & \Rightarrow \dfrac{1}{2}\left( \dfrac{N}{x}+x \right)=x \\
\end{align}\]
Now, the square root of a number N is can be calculated using the formula \[\dfrac{1}{2}\left( \dfrac{N}{x}+x \right)=x\] . In this formula, N is the original number and the number x is any guessed square root.
Therefore, the square root of the number N using Newton-Raphson’s method is \[\dfrac{1}{2}\left( \dfrac{N}{x}+x \right)=x\] where the number x is any guessed square root.

Note: We can also check this formula for the number 4.
Let us find the square root of the number 4 using Newton-Raphson’s method.
Let us guess that 2.5 is the square root of the number 4.
The square root of the number N using Newton-Raphson’s method is \[\dfrac{1}{2}\left( \dfrac{N}{x}+x \right)=x\] …………………………………..(1)
Now, on putting \[N=4,x=2.5\] in LHS and RHS of equation (1), we get
In LHS, we get
\[\dfrac{1}{2}\left( \dfrac{4}{2.5}+2.5 \right)=2.05\] …………………………………….(2)
In RHS, we get
\[x=2.5\] ……………………………………..(3)
From equation (2) and equation (3), we can say that \[LHS\ne RHS\] .
So, 2.5 is not the square root of the number 4.
Let us guess that 2 is the square root of the number 4.
Now, on putting \[N=4,x=2\] in LHS and RHS of equation (1), we get
In LHS, we get
\[\dfrac{1}{2}\left( \dfrac{4}{2}+2 \right)=2\] …………………………………….(4)
In RHS, we get
\[x=2\] ……………………………………..(5)
From equation (4) and equation (5), we can say that \[LHS=RHS\] .
So, 2 the square root of the number 4.
We also know that the square root of the number 4 is 2 and also from Newton-Raphson’s method, we get the square root of 4 equal to 2.
Therefore, the square root of the number N using Newton-Raphson’s method is \[\dfrac{1}{2}\left( \dfrac{N}{x}+x \right)=x\] where the number x is any guessed square root.