Question

What is the square root of 90?

Answer 1

`sqrt(90) = 3sqrt(10) ~~ 1039681/109592 ~~ 9.48683298051`

Explanation:

`sqrt(90) = sqrt(3^2*10) = 3sqrt(10)` is an irrational number somewhere between `sqrt(81)=9` and `sqrt(100) = 10`.

In fact, since `90 = 9 * 10` is of the form `n(n+1)` it has a regular continued fraction expansion of the form `[n;bar(2,2n)]`:

`sqrt(90) = [9;bar(2,18)] = 9+1/(2+1/(18+1/(2+1/(18+1/(2+1/(18+...))))))`

One fun way to find rational approximations is using an integer sequence defined by a linear recurrence.

Consider the quadratic equation with zeros `19+2sqrt(90)` and `19-2sqrt(90)`:

`0 = (x-19-2sqrt(90))(x-19+2sqrt(90))`

`color(white)(0) =(x-19)^2-(2sqrt(90))^2`

`color(white)(0) =x^2-38x+361-360`

`color(white)(0) =x^2-38x+1`

So:

`x^2 = 38x-1`

Use this to derive a sequence:

`{ (a_0 = 0), (a_1 = 1), (a_(n+2) = 38a_(n+1)-a_n) :}`

The first few terms of this sequence are:

`0, 1, 38, 1443, 54796, 2080805,...`

The ratio between successive terms will tend to `19+2sqrt(90)`

Hence:

`sqrt(90) ~~ 1/2(2080805/54796-19) = 1/2(1039681/54796) = 1039681/109592 ~~ 9.48683298051`