How do you find the square root of 23?

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Answer 1 · 2016-08-31T21:15:56

$sqrt(23) ~~ 1151/240 = 4.7958bar(3)$

$23$ is a prime number, so it is not possible to simplify its square root, which is an irrational number a little less than $5 = sqrt(25)$

As such it is not expressible in the form $p/q$ for integers $p, q$ .

We can find rational approximations as follows:

$23 = 5^2-2$

is in the form $n^2-2$

The square root of a number of the form $n^2-2$ can be expressed as a continued fraction of standard form:

$sqrt(n^2-2) = [(n-1); bar(1, (n-2), 1, (2n-2))]$

In our example $n=5$ and we find:

$sqrt(23) = [4; bar(1,3,1,8)] = 4+1/(1+1/(3+1/(1+1/(8+1/(1+1/(3+1/(1+...)))))))$

To use this to derive a good approximation for $sqrt(23)$ terminate it early, just before one of the $8$ 's. For example:

$sqrt(23) ~~ [4;1,3,1,8,1,3,1] = 4+1/(1+1/(3+1/(1+1/(8+1/(1+1/(3+1/1)))))) = 1151/240 = 4.7958bar(3)$

With a calculator, we find:

$sqrt(23) ~~ 4.79583152$

So our approximation is not bad.