How do you find the square root of 26/89?
1 Answer
Explanation:
Note that
So
We can find rational approximations to it.
Generalised continued fraction method
First here's a little theory...
Suppose:
#sqrt(n) = a+b/(2a+b/(2a+b/(2a+...)))#
Then:
#a+b/(a+sqrt(n)) = a+b/(a+color(blue)(a+b/(2a+b/(2a+b/(2a+...))))) = sqrt(n)#
Multiplying both ends by
#a^2+color(red)(cancel(color(black)(asqrt(n))))+b = color(red)(cancel(color(black)(asqrt(n)))) + n#
Subtracting
#b = n-a^2#
So if we want a generalised continued fraction to help us approximate a square root
Application
Let
Then:
#b = n-a^2 = 26/89 - 1/4 = (104-89)/356 = 15/356#
So:
#sqrt(26/89) = 1/2+(15/356)/(1+(15/356)/(1+(15/356)/(1+...)))#
We can truncate this to give rational approximations.
For example:
#sqrt(26/89) ~~ 1/2+(15/356)/(1+(15/356)/(1+15/356)) = 74273/137416 ~~ 0.54050#
Having found this, we can see that putting
It gives:
#b = 26/89 - (27/50)^2 = 119/222500#
So:
#sqrt(26/89) = 27/50+(119/222500)/(27/25+(119/222500)/(27/25+(119/222500)/(27/25+...)))#
Just a couple of steps of this give us:
#sqrt(26/89) ~~ 27/50+(119/222500)/(27/25) = 129881/240300 ~~ 0.540495#
which is correct to
This continued fraction expansion will give us approximately