# 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.