From: jzakiya <jzakiya@...>
Date: 2009-12-19T07:25:09+09:00
Subject: Re: Math errors

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)