How do you find the square root of 193?

1 Answer
Aug 30, 2016

Answer:

#sqrt(193) ~~ 13.8924439894498# is an irrational number.

We can find approximations to it using a Newton Raphson method.

Explanation:

#193# is a prime number, so its square root does not have any simpler form. It is an irrational number a little less than #14# (since #14^2 = 196#). That is, it is not expressible in the form #p/q# for any integers #p, q#.

We can find approximations to it using a kind of Newton Raphson method.

Given a number #n# and an initial approximation #a_0# to #sqrt(n)#, derive progressively more accurate approximations by using the formula:

#a_(i+1) = (a_i^2 + n)/(2a_i)#

I like to reformulate this slightly using integers #p_i# and #q_i# where #a_i = p_i/q_i#. Then use these formulae to iterate:

#p_(i+1) = p_i^2+n q_i^2#

#q_(i+1) = 2p_i q_i#

If the resulting #p_(i+1)# and #q_(i+1)# have a common factor, then divide both by that factor before the next iteration.

Let #n=193#, #p_0 = 14# and #q_0 = 1#

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.