Question

How do you find the exact square root of 41?

Answer 1

`sqrt(41)` is an irrational number. It cannot be expressed as one integer divided by another or by a terminating or repeating decimal.

`sqrt(41) ~~ 6.40312423743284868648`

Explanation:

`sqrt(41)` can be represented as a repeating continued fraction:

`sqrt(41) = [6;bar(2,2,12)] = 6+1/(2+1/(2+1/(12+1/(2+1/(2+...)))))`

We can use this expression to get rational approximations for `sqrt(41)`

For example,

`sqrt(41) ~~ [6;2,2] = 6+1/(2+1/2) = 6+2/5 = 6.4`

Alternatively, use a Newton Raphson type approach:

To find approximations for `sqrt(n)`, start with a reasonable approximation `a_0` and iterate using the formula:

`a_(i+1) = (a_i^2+n)/(2a_i)`

So if we started with `n=41`, `a_0=6`

`a_1 = (a_0^2+n)/(2a_0) = (6^2+41)/12 = 77/12 = 6.41dot(6)`

`a_2 = (a_1^2+n)/(2a_1) = ((77/12)^2+41)/(2*(77/12))`

`= (77^2 + 41*12^2)/(2*77*12) = (5929+41*144)/1848`

`= (5929+5904)/1848 = 11833/1848 ~~ 6.40314`