What is the square root of 190?
It can be approximated as:
The square root of
If we factor
#190 = 2 * 95 = 2 * 5 * 19#
We can use a Newton Raphson type method to find successively better rational approximations to the irrational number
Let our first approximation be
We can use the following formula to get a better approximation:
#a_(i+1) = (a_i^2 + n)/(2a_i)#
See: How do you find the square root 28? for a slightly easier way of doing this. For simplicity here, I'll use the classic formula above.
#a_1 = (a_0^2 + n)/(2a_0) = (14^2+190)/(2*14) = 386/28 = 193/14 ~~ 13.7857#
#a_2 = 74489/5404 ~~ 13.78404885#
#a_3 = 11097222161/805077112 ~~ 13.784048752090222#