Question

What is the square root of 543?

Answer 1

`sqrt(543) ~~ 23.30236`

Explanation:

The prime factorisation of `543` is:

`543 = 3 * 181`

Since it has no square factors larger than `1`, the square root of `543` cannot be simplified.

It is an irrational number between `23 = sqrt(529)` and `24 = sqrt 576`.

Linearly interpolating, we can approximate:

`sqrt(543) ~~ 23+(543-529)/(576-529) = 23 14/47 ~~ 23.3`

For more accuracy, let `p_0/q_0 = 233/10` and iterate using the formulas:

`{(p_(i+1) = p_i^2+543 q_i^2),(q_(i+1) = 2p_iq_i) :}`

So:

`{ (p_1 = p_0^2+543 q_0^2 = 233^2+543 * 10^2 = 54289+54300 = 108589), (q_1 = 2 p_0 q_0 = 2 * 233 * 10 = 4660) :}`

Just this one iteration is sufficient to get `7` (nearly `8`) significant digits:

`sqrt(543) ~~ p_1/q_1 = 108589/4660 ~~ 23.30236`

If we want more accuracy, just iterate again.

Footnote

The exact repeating continued fraction for `sqrt(543)` is:

`543 = [23;bar(3,3,3,1,14,1,3,3,3,46)]`

from which it is possible to find the solution of Pell's equation:

`669337^2 = 543 * 28724^2 + 1`

which makes `sqrt(543) ~~ 669337/28724` a very efficient approximation.