Question

How do you find the square root of 193?

Answer 1

`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.