How do you find the square root of 193?
1 Answer
We can find approximations to it using a Newton Raphson method.
Explanation:
We can find approximations to it using a kind of Newton Raphson method.
Given a number
#a_(i+1) = (a_i^2 + n)/(2a_i)#
I like to reformulate this slightly using integers
#p_(i+1) = p_i^2+n q_i^2#
#q_(i+1) = 2p_i q_i#
If the resulting
Let
Then:
#p_1 = p_0^2+n q_0^2 = 14^2+193*1^2 = 196+193 = 389#
#q_1 = 2p_0 q_0 = 2*14+1 = 28#
If we stopped here then we would have:
#sqrt(193) ~~ 389/28 = 13.89bar(285714)#
Next iteration:
#p_2 = p_1^2 = n q_1^2 = 389^2 + 193*28^2 = 151321+151312 = 302633#
#q_2 = 2p_1 q_1 = 2*389*28 = 21784#
So:
#sqrt(193) ~~ 302633/21784 ~~ 13.892444#
Actually:
#sqrt(193) ~~ 13.8924439894498#
but as you can see this method converges quite rapidly.