Question

Solve the square root of 3 bisection method​

Answer 1

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Approximation of a square root using bisection method, is a root finding method implementing Bolzano’s theorem for continuous functions which implies, If a function f(x) is continuous on an interval [a, b] and f(a).f(b)<0, then a value c∈(a, b) exist for which f(c)=0. Here the function, we can use for the approximation of the square root of 3 is f(x)=x^2–3. Then the positive root of the function will be √3.

Or For a better understanding, we can state it as a method of finding square root of a number by binary search. For binary search, we start with two initial values, one lesser than the value of the number, for which we are approximating root. And another value greater than the value of the number. For a easier understanding, we’ll be denoting the lesser value by L and greater value by G, then x=(L+G)/2, here x being the square root of 3.

And for binary search, we’ll be following two conditions,

If x^2<3, then L=x

If x^2>3, then G=x

In the above table, we can see that after 11 iterations, the value of x^2 is approximately equal to the number 3 (Upto 3 decimal places), giving us the value of x as 1.732177735 (1.73, only upto 2 decimal places).

Now, if we consider the function f(x)=x^2–3, It will be equal to 0 for x=√3. And according to Bolzano’s theorem, √3 will have a value somewhere between the lesser and greater value we choose for the approximation, here which turned out as 1.732177735. And if we round off that value to 2 decimal places, we get 1.73, So, 1.73 is the approximation of square of 3, correct to 2 decimal places, using the Bisection method

Answer 2

Answer:

√3=1.732 is your answer

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