How do you find the square root of 26/89?

1 Answer
Feb 13, 2017

Answer:

#sqrt(26/89) = 1/2+(15/356)/(1+(15/356)/(1+(15/356)/(1+...))) ~~ 0.5405#

Explanation:

Note that #26 = 2*13# and #89# (which is prime) have no common factors and no square factors.

So #sqrt(26/89)# is an irrational number with no simpler form.

We can find rational approximations to it.

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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+sqrt(n))# we get:

#a^2+color(red)(cancel(color(black)(asqrt(n))))+b = color(red)(cancel(color(black)(asqrt(n)))) + n#

Subtracting #a^2+asqrt(n)# from both sides we find:

#b = n-a^2#

So if we want a generalised continued fraction to help us approximate a square root #sqrt(n)# then pick #a# such that #a^2 < n# and derive #b = n-a^2#. If #a^2# is close to #n# then #b# will be smaller and the fraction will converge faster.

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Application

Let #n = 26/89# and #a=1/2#

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 #a=54/100 = 27/50# might be a better first approximation.

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 #6# decimal places.

This continued fraction expansion will give us approximately #3# more decimal places for each additional step we include.