Question

7. Approximating Square Roots. Heron of Alexandria came up with the following iterative procedure for approximating square roots. Suppose you want to find and you have that A = a + b where al is the perfect square nearest A. Then you start by averaging a and A. Call this number ay. This is your first approximation. To find the next approximation, as you average aj and A. You can repeat this indefinitely a. Using this method, find the first, second, and third approximations to 231 b. What value does the Archimedean/Egyptian/Babylonian approximation give for 231? (This is the approximation given in class.)

please solve this question.
(history of math)

Answer 1

(a) Find three Aproximation of √23) soin perfect square 231 = (15²+ 6 Where (15) 2 is the nearest A ie 225 First Approximation, a= a + 4 a = 15 second mation 42 - apf 231 T5.2 t 15.2 1S. 198 684 21

third Approximation, Az= A a2 2 + 2 23 15.198684 21 = 15-19868421+ 2 is. 198684 15 (6) At n should be Ancient time was Sa noire root of ñ = Viaj²tr a atr 2x at Where a Is nearest square bon so $231 = reis) 2+ 6 15 + 6 2x 15+1 h 15 + 0.19 3 5 48 387 15-19 354838. ~