What is the square root of 90?

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Answer 1 · 2018-03-02T16:19:32

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

$sqrt(90) = sqrt(3^2*10) = 3sqrt(10)$ 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:

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