Question

1.a.)Use the assumed Babylonian square root algorithm (also known as Archimedes’ method) √ a 2 ± b ≈ a ± b/2a to show that √ 3 ≈ 1; 45 by beginning with the value a = 2. Find a three-sexagesimal-place approximation to the reciprocal of 1; 45 and use it to calculate a three-sexagesimal-place approximation to √ 3.

1.b)An iterative procedure for closer approximations to the square root of a number that is not a square was obtained by Heron of Alexandria (ca. 75 CE). In his work Metrica he merely states a rule that amounts to the following in modern notation: If A is a non square number, and a^2 is the nearest perfect square to it, so that A = a^2 ± b, then approximations to √ A can be obtained using the recursive formula:

x0 = a

xn = 1/2 ( Xn−1 + A/( Xn−1)), n ≥ 1

(i) Use Heron’s method to find approximations through n = 3 to √ 720 and √ 63.

(ii) Show that Heron’s approximation x1 is equivalent to the Babylonian’s square root algorithm.

Answer 1

SOLUTION:

Given That data Use the assumed Babylonian square root algorithm of the text to show that v3 1:45 by beginning with the value 2. Find a three-sexagesimal-place approximation to the reciprocal of 1:45 and use it to calculate a three-sexagesimal-place approximation to V3

So the answer is given below