What is the square root of 90?

1 Answer
Mar 2, 2018

Answer:

#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#