How do you find the square root of 203?
1 Answer
Use a Newton Raphson method to find:
#sqrt(203) ~~ 6497/456 ~~ 14.247807#
Explanation:
It is an irrational number between
#14^2 = 196 < 203 < 225 = 15^2#
As such, it cannot be represented in the form
We can find rational approximations using a Newton Raphson method:
To approximate the square root of
#a_(i+1) = (a_i^2+n)/(2a_i)#
I prefer to re-formulate this slightly using separate integers
#p_(i+1) = p_i^2+n q_i^2#
#q_(i+1) = 2 p_i q_i#
If the resulting pair of integers have a common factor, then divide by that before the next iteration.
So for our example, let
#{ (p_0 = 29), (q_0 = 2) :}#
#{ (p_1 = p_0^2+ n q_0^2 = 29^2 + 203*2^2 = 841+812 = 1653), (q_1 = 2 p_0 q_0 = 2*29*2 = 116) :}#
Note that both
#{ (p_(1a) = 1653/29 = 57), (q_(1a) = 116/29 = 4) :}#
Next iteration:
#{ (p_2 = p_(1a)^2 + n q_(1a)^2 = 57^2+203*4^2 = 3249+3248 = 6497), (q_2 = 2 p_(1a) q_(1a) = 2*57*4 = 456) :}#
If we stop here, we get:
#sqrt(203) ~~ 6497/456 ~~ 14.247807#
Each iteration roughly doubles the number of significant figures in the approximation.