1.a.)Use the assumed Babylonian square root algorithm (alsoknown as Archimedes’ method) √ a 2 ± b ≈ a ± b/2ato show that √ 3 ≈ 1; 45 by beginning with thevalue a = 2. Find a three-sexagesimal-place approximation to thereciprocal of 1; 45 and use it to calculate athree-sexagesimal-place approximation to √ 3.
1.b)An iterative procedure for closer approximations to thesquare root of a number that is not a square was obtained by Heronof Alexandria (ca. 75 CE). In his work Metrica he merely states arule that amounts to the following in modern notation: If A is anon square number, and a^2 is the nearest perfect square to it, sothat A = a^2 ± b, then approximations to √ A can be obtained usingthe 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 theBabylonian’s square root algorithm.
