From: jzakiya <jzakiya@...>
Date: 2009-12-19T10:05:07+09:00
Subject: Re: Math errors

On Dec 18, 5:24 pm, jzakiya <jzak...@mail.com> wrote:
> On Dec 18, 2:13 pm, "William James" <> wrote:
>
>
>
> > Fleck Jean-Julien wrote:
> > > > That is WRONG, you cannot do that.
>
> > > Well, I never said that you should do that, I just explained how Ruby
> > > interpreted it...
>
> > > > That only works for odd roots of negative numbers.
> > > > The even root of negative numbers are imaginary.
>
> > > > -27**3**-1 => -3 **correct
> > > > -27**2**-1 => -5.19615242270663 **WRONG, its 5.196152i
>
> > > Sure. That's quite a hint why ** does not accept a negative number
> > > with a non integer exponent. To take into account all the special
> > > cases, you should first see if your exponent is a rational and in that
> > > case, see if the denominator is odd (after all due simplifications of
> > > course). In this case (and only this case), you could try to decipher
> > > a root for this negative number.
>
> > > Cheers,
>
> > def root base, n
> >   exp = 1.0/n
> >   return base ** exp if base >= 0 or n.even?
> >   -( base.abs ** exp )
> > end
>
> > --
>
> Remember
>
> i = (-1)^(1/2)
>
> i^1 =  i
> i^2 = -1
> i^3 = -i
> i^4 =  1
> Then it repeats, for example: i^5 = i*(i^4) = i
>
> For negative real value roots:
>
> x = (-a)^(1/n) where n is odd integer => x = -[a^(1/n)]
>
> But for negative real value roots where n is even:
>
> x = (-a)^(1/n) where n is even gives
>
> x = |a^(1/n)|*(-1)^(1/n)
> x = |a^(1/n)|*(i^2)^(1/n)
> x = |a^(1/n)|*(i)^(2/n)
> from e^(i*x) = cos(x) + i*sin(x) where x = PI/2
> x = |a^{1/n)|*e^(PI*i/2)^(2/n)
> x = |a^(1/n)|*e^(PI*i/n)
> x = |a^(1/n)|*(cos(PI/n) + i*sin(PI/n)) for n even
>
> (-256)^(1/2) = |256^(1/2)|*(cos(PI/2) + i*sin(PI/2))
>              = (16)(0 + i) = 16i
>
> (-256)^(/4) = |256^(1/4)|*(cos(PI/4) + i*sin(PI/4))
>             = (4)*(0.707 + 0.707*i)
>             = 2.828 + i*2.828
>             = 2.828*(1+i)
>
> Check in irb
>
> > require 'complex'
> > include Math
>
> x = Complex(-256,0)
>
> x**(1/2.0)
> => (9.79685083057902e-16+16.0i)
>
> X**(1/4.0)
> => (2.82842712474619+2.82842712474619i)

BTW there is an error (sort of) in 'complex' too

>require 'complex'
>include Math

> x = Complex(-27,0)
=> (-27+0i)

>y = x**(1/3.0) # or x**3**-1
=> (1.5+2.59807621135332i)  # should be (-3+0i)

>y**3
=> (-27.0+1.24344978758018e-14i)

> Complex(-3,0)**3
=> -27

Whenever you take the root n of a number you actually
get n values. If the value is positive you get n copies
of the same positive real value.

When you take the root of a negative real value you
get n roots too, for n even and odd.

For even odd, you get one real root and n/2 Complex Conjugate Pairs
(CCP).

Thus, for n=3 for (-27)^(1/3) the real root is x1=-3
and x2 is y above and x3 is the CCP of y.

For n=5, you get one real root and 2 pairs of CCPs, etc.

For n even, you get n/2 CCPs only.
So, for n=2 there is one pair of CCP roots.
For n=4 you get 2 different CCP roots, etc,
Thus for n even there are no real roots.

So, I think it's more intuitive (for most people)
to expect Complex(-27,0)**(1/n-odd) to return the real
root x1 only (i.e. (-3)*(-3)*(-3) = -27), so have it
act as Complex(-27,0).real (for n odd) be the default.

I guess complex variables aren't called complex for nothing. :-)