Re: Accuracy for irrationals (was: what am i doing wrong here)
> From: "Hayden McKay" <hmck1 at dodo.com.au>
> Subject: Re: what am i doing wrong here.
>
>
> I found another way to get the any root of a number to 100% accuracy.
> (below)
>
> -- atom = root(atom digit, integer factor)
> -- Calculate the root of a digit.
> -- Returns the desired root.
> global function root(atom a,integer n)
> a = power(a,(1/n))
> return a
> end function
You'll never get 100% accuracy for any number which is not a fraction with
some product of powers of 2 as the denominator, because the hardware is wired
that way.
However, there are some routes to go to get many irrationals with ANY desired
accuracy:
- fractions have a periodic pattern, so you can do some computations using
that pattern. However, it is generally better to represent them exactly as
{numerator,denominator} and perform calculations on the pir (doesn't work for
roots).
- algebraic numbers of degree 2, which means numbers like a+/-sqrt(b), where a
and b are fractions, have periodic continued fractions. Computing with these
is definitely trickier, but you may get any absolute accuracy you need. This
is quite handy for square roots.
- algebraic numbers of any degree, which means numbers that are roots of some
polynomial with integer coefficients, can be reached through continued
fractions or power series expansions. As long as you have an algorithm to
compute any desired term in these, any accuracy can be reached, and there's no
periodicity any longer, plus some other problems as you have to choose among
several roots most of the time.
- transcendent numbers, ie real numbers which are not algebraic, like PI, can
sometimes be reached through power series expansions. Same remark as above,
but there's no longer a finite sequence of integers on which you can perform
some computations.
By the way, do you need arbitrary precision computing? It costs both CPU time
and RAM.
CChris
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