Question

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

Answer 1

`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.