Re: Correcting complex math

On Dec 20, 9:51 am, Colin Bartlett <[e-mail]> wrote:
> If this can be made to work, it might be useful.
> Two thoughts that occur to me are:
>
> 1. The code you show would probably need to use Rational(?),
>    otherwise 1/n (or 1.0/n) may not be what one thinks it is.

From a mathematical/engineering perspective it is
understood that a root n is a positive integer > 1.
In other words, what number 'r' multiplied by itself
n times equals 'x'.

A coding implementation can do the requisite checks on
all the inputs and raise needed exceptions/warnings.

Of course from an arithmetic perspective (a)^(1/n) can
take any value of n (integer, float, complex) but if
n not a positive integer (or interizable) number > 1
then the discussion here may not apply to those cases.

I think the syntax of (a)^(n^-1) vs (a)^(1/n) makes the
point more mathematical clearer that what you are doing
is finding the nth root of 'a' vs raising 'a' to a
fractional power.

> 2. What would one do with a number (Float or Rational)
>    which is arbitrarily close to 1/n but which is not 1/n?

If a number can be expressed as rational n = b/c then 'c' would be the
root, and 'b' would be the power and
(a)^b^(c^-1), otherwise see comment above.


> I'd also like to take the opportunity of this thread
> to suggest removing (!) something from Complex,
> unless anyone can make a good case for retaining it:
>
> require "complex"
> n = -2                #=> -2
> p = 1                 #=> 1
> cn = Complex( n, 0 )  #=> Complex(-2, 0)
> cp = Complex( p, 0 )  #=> Complex(1, 0)
> n > p                 #=> false
> cn > p                #=> true
> cn > cp               #=> true
>
> The Complex numbers are not an "ordered field",
> at least not in any normally useful sense.
>
> At the moment if either x or y is Complex
> then  x <=> y  is calculated as  x.abs <=> y.abs;
>
> # File complex.rb, line 312
>   def <=> (other)
>     self.abs <=> other.abs
>   end
>
> I think that can be misleading (see the examples above),
> and I would prefer it if "<=>" and "<", etc,
> were *not* defined for Complex,
> and if one wants to compare absolute values
> then one should do so explicitly.

I would agree that <, >, are ambiguous. It's not just
a matter of magnitude size but also angular 'size',

Remember, complex variables are really vectors with
magnitude and direction (angle).

Take the example of a ferris wheel ride. Where you
get on it is angular distance zero from your start.
Once the ride starts, it will eventually be rotating at
some constant speed, with changing angular direction.

However, the max angular distance from the start with be
when you are at the top of the ferris wheel, which is an
angular distance of PI (180) degrees from your start. All
other locations have CCP angular distance from the start.

Thus, any complex number is defined with polar rep of
|mag(a)| /_ angle(a)

Therefore, to find all the n roots of real values, I can
change my original code to make it more efficient and
flexible by making it just find the magnitude and angles
of the root locations.

def Complex.root(a,n,k)
  angle = (2*k+1)*PI/n    # for 'a' real negative
  angle += PI/n if a > 0  # for 'a' real positive
  mag = a.abs**n**-1
  # The kth root of (a)**n**-1
  mag*Complex(cos(angle),sin(angle))
end

The key is to find the appropriate angle rotation for
'a' positive or negative. Now by finding that angle
and mag values, all the numeric classes can be written
to find all the roots of real values with easy syntax.

In Integer/Float you could now do:

a.root(n) # default to +/- real root or first CCP
a.root(n,k) # kth ccw root
a.roots(n) # array of all (ccw) roots

To make it intuitive (and follow Matz's POLS) let k
roots go from 1st to kth, k = 1..n, which means that
the routine that uses k just needs to subtract 1 from
it before computing the angle.

These features can be integrated within all the other
numeric classes/modules as appropriate.