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.