Question

How do you find square root of 138?

Answer 1

`sqrt(138) ~~ 39019777/3321584 ~~ 11.74734012447073`

Explanation:

The prime factorisation of `138` is:

`138 = 2 * 3 * 23`

Since this contains no squared terms, the square root cannot be simplified and not being a perfect square, it is irrational.

Note that:

`11^2 = 121 < 138 < 144 = 12^2`

So `sqrt(138)` is somewhere between `11` and `12`, closer to `12`.

Let us approximate it as `11 3/4 = 47/4`.

This is actually a very efficient approximation, since:

`47^2 = 2209 = 2208 + 1 = 4^2 * 138 + 1`

A much more formal way to find such an efficient initial approximation is to be found at https://socratic.org/s/aPLdnFSE

Next, consider the quadratic with zeros `47+4sqrt(138)` and `47-4sqrt(138)`...

`(x-47-4sqrt(138))(x-47+4sqrt(138)) = x^2-94x+1`

From this quadratic we can define an integer sequence recursively, using the rules:

`{ (a_0 = 0), (a_1 = 1), (a_(n+2) = 94a_(n+1)-a_n) :}`

The first few terms of this sequence are:

`0, 1, 94, 8835, 830396, 78048389`

The ratio between consecutive terms converges very rapidly towards `47+4sqrt(138)`

So we can approximate:

`sqrt(138) ~~ 1/4(78048389/830396-47)`

`color(white)(sqrt(138)) = 1/4(39019777/830396)`

`color(white)(sqrt(138)) = 39019777/3321584`

`color(white)(sqrt(138)) ~~ 11.74734012447073`