Question

What is the Square root of 21?

Answer 1

`21 = 3*7` has no square factors, so is not possible to simplify `sqrt(21)`

`sqrt(21) ~~ 4.583` is an irrational number whose square is `21`

Explanation:

`sqrt(21)` is not a rational number, so it cannot be expressed as `p/q` for some integers `p, q` and its decimal expansion does not repeat.

`sqrt(21) ~~ 4.58257569495584000658`

It is expressible as a repeating continued fraction:

`sqrt(21) = [4;bar(1,1,2,1,1,8)] = 4 + 1/(1+1/(1+1/(2+...)))`

To see how to calculate this see http://socratic.org/questions/given-an-integer-n-is-there-an-efficient-way-to-find-integers-p-q-such-that-abs-#176764

We can get a good approximation for `sqrt(21)` by truncating the continued fraction.

`sqrt(21) ~~ [4;1,1,2,1,1] = 4+1/(1+1/(1+1/(2+1/(1+1/1)))) = 55/12 = 4.58dot(3)`

This is a good approximation because `55^2 = 21*12^2 + 1`