Question

How do you find the square root of 15?

Answer 1

`sqrt(15)` is not simplifiable.

We can find rational approximations `31/8`, `244/63`

Explanation:

`15=3xx5` has no square factors, so `sqrt(15)` cannot be simplified.

It is not expressible as a rational number. It is an irrational number a little less than `4`.

Since `15 = 4^2-1` is of the form `n^2-1`, `sqrt(15)` has a fairly simple continued fraction expansion:

`sqrt(15) = [3;bar(1,6)] = 3+1/(1+1/(6+1/(1+1/(6+1/(1+1/(6+1/(1+...)))))))`

We can truncate this continued fraction expansion early to get rational approximations to `sqrt(15)`.

For example:

`sqrt(15) ~~ [3;1,6,1] = 3+1/(1+1/(6+1/1)) = 3+1/(1+1/7) = 3+7/8 = 31/8 = 3.875`

`sqrt(15) ~~ [3;1,6,1,6,1] = 3+1/(1+1/(6+1/(1+1/(6+1/1)))) = 3+1/(1+1/(6+1/(1+1/7)))`

`= 3+1/(1+1/(6+7/8)) = 3+1/(1+8/55) = 3+55/63 = 244/63 = 3.bar(873015)`

Actually:

`sqrt(15) ~~ 3.87298334620741688517`