Question

How do you find the simplest radical form of 433?

Answer 1

Hmmm...

Explanation:

If you mean the simplest form of `sqrt(433)` then it is `sqrt(433)` since `433` is a prime number.

If you mean the simplest expression involving a radical with value `433`, then you might choose `433 sqrt(1)` or possibly `sqrt(187489)` (since `433*433 = 187489`).

Approximations

Since `433` is a prime number, its square root cannot be simplified. In addition, it is not close to a square number or half way between two square numbers. So it is a little fiddly to get a rapidly convergent continued fraction expressing it.

Note that:

`20^2 = 400 < 433 < 441 = 21^2`

So a reasonable approximation is somewhere between `20` and `21`. Linearly interpolating, we find it is about `20+33/41 ~~ 20.8 = 104/5`

Then:

`433 - (104/5)^2 = 9/25`

Now:

`sqrt(a^2+b) = a+b/(2a+b/(2a+b/(2a+...)))`

So:

`sqrt(433) = 104/5+(9/25)/(208/5+(9/25)/(208/5+(9/25)/(208/5+...)))`

`color(white)(sqrt(104)) = 1/5(104+9/(208+9/(208+9/(208+...))))`

Hence, if we define a sequence recursively by:

`{ (a_0 = 0), (a_1 = 1), (a_(n+2) = 208a_(n+1)+9a_n) :}`

then the ratio between successive terms rapidly converges to `5sqrt(433)+104`

The first few terms are:

`0, 1, 208, 43273, 9002656`

So:

`5sqrt(433) ~~ 9002656/43273-104 = 4502264/43273`

So:

`sqrt(433) ~~ 4502264/(5 * 43273) = 4502264/216365 ~~ 20.808652046`