Math errors

ruby-talk
1

(-3)**3 => -27

(-27)**(1/3.0) give NaN instead of -3

All odd roots (1/3,1/5,1/7 etc) of negative numbers should give
negative root values, as above, but ruby (in irb) gives NaN (not a
number), even when I require 'complex' .

Is this considered an error in Ruby?

2

Ruby 1.9 has Math.cbrt.

% ruby -e 'p Math.cbrt(-27)'
-3.0

···

2009/12/18 jzakiya <jzakiya@mail.com>:

(-3)**3 => -27

(-27)**(1/3.0) give NaN instead of -3

--
Tanaka Akira

3

jzakiya wrote:

(-3)**3 => -27

(-27)**(1/3.0) give NaN instead of -3

All odd roots (1/3,1/5,1/7 etc) of negative numbers should give
negative root values, as above, but ruby (in irb) gives NaN (not a
number), even when I require 'complex' .

Is this considered an error in Ruby?

irb(main):001:0> 1/3.0
=> 0.333333333333333

···

--

4

jzakiya wrote:

(-3)**3 => -27

(-27)**(1/3.0) give NaN instead of -3

All odd roots (1/3,1/5,1/7 etc) of negative numbers should give
negative root values, as above, but ruby (in irb) gives NaN (not a
number), even when I require 'complex' .

Is this considered an error in Ruby?

irb(main):006:0> (-27)**(1/3.0)
=> NaN
irb(main):007:0> -27 ** (1/3.0)
=> -3.0

irb(main):012:0> RUBY_VERSION
=> "1.8.7"

···

--

5

Removing the brackets across -27.0 does the trick, though would need to find the logic behind the same
(-27)**(1/3.0) gives NaN

-27**(1/3.0) gives -3

Regards,
Raghav

···

-----Original Message-----
From: jzakiya [mailto:jzakiya@mail.com]
Sent: Friday, December 18, 2009 12:15 PM
To: ruby-talk ML
Subject: Math errors

(-3)**3 => -27

(-27)**(1/3.0) give NaN instead of -3

All odd roots (1/3,1/5,1/7 etc) of negative numbers should give
negative root values, as above, but ruby (in irb) gives NaN (not a
number), even when I require 'complex' .

Is this considered an error in Ruby?

6

IN 1.9.2 I got:

~ $> ruby -ve 'p v=Math.cbrt(-27); p v**3'
ruby 1.9.2dev (2009-12-11 trunk 26067) [x86_64-darwin10.2.0]
-3.0
-27.0

~ $> ruby -ve 'p v=(-27)**(1/3.0); p v**3'
(1.5000000000000004+2.598076211353316i)
(-27.000000000000007+1.2434497875801753e-14i)

~ $> ruby -ve 'p v=(-27)**(Rational(1,3)); p v**3'
(1.5+2.5980762113533156i)
(-26.99999999999999+3.552713678800501e-15i)

Got easy results, but with all different values ...
So, the first is right, of course.
The second is a good approximation, for floats ...
And the last is almost good, while I would expect to be exact (I think it
should use Math.cbrt in this case)

I would probably interested to use BigDecimal, but it's impossible I think
to use it for Complex. (And actually it accept only Fixnum exponent)

Another error in the exception:
TypeError: wrong argument type BigDecimal (expected Fixnum)

But it accept Float, really :wink:

···

2009/12/18 Tanaka Akira <akr@fsij.org>

2009/12/18 jzakiya <jzakiya@mail.com>:
> (-3)**3 => -27
>
> (-27)**(1/3.0) give NaN instead of -3

Ruby 1.9 has Math.cbrt.

% ruby -e 'p Math.cbrt(-27)'
-3.0
--
Tanaka Akira

7

Hi,

···

In message "Re: Math errors" on Fri, 18 Dec 2009 17:30:15 +0900, "W. James" <w_a_x_man@yahoo.com> writes:

irb(main):007:0> -27 ** (1/3.0)
=> -3.0

It's -(27 ** (1/3.0)).

              matz.

8

jzakiya wrote:

(-3)**3 => -27

(-27)**(1/3.0) give NaN instead of -3

All odd roots (1/3,1/5,1/7 etc) of negative numbers should give
negative root values, as above, but ruby (in irb) gives NaN (not a
number), even when I require 'complex' .

Is this considered an error in Ruby?

I guess, you are right. THIS IS A 'MATH ERROR' in Ruby.

The ** method is not able to handle negative numbers raised to Floating points (that are not Integers)

9

This behavior was shown in ruby ree 1.8.6, 1.8.6, 1.8.7, 1.9.1

-27**1/3.0 => -9.0
-27 ** 1/3.0 => -9.0
-27 ** (1/3.0) => -3.0
-27**(1/3.0) => -3.0
(-27)**(1/3.0) => NaN
(-27) ** (1/3.0) => NaN

OK, the first two expression are evaluated as (-27**1)/3 => -9
The second two are correct (what I expected).
But the last two, WHY??

···

On Dec 18, 4:52 am, "Dasson, Raghav" <raghav.das...@sap.com> wrote:

jzakiya wrote:
> (-3)**3 => -27

> (-27)**(1/3.0) give NaN instead of -3

> All odd roots (1/3,1/5,1/7 etc) of negative numbers should give
> negative root values, as above, but ruby (in irb) gives NaN (not a
> number), even when I require 'complex' .

> Is this considered an error in Ruby?

I guess, you are right. THIS IS A 'MATH ERROR' in Ruby.

The ** method is not able to handle negative numbers raised to Floating points (that are not Integers)

10

Yukihiro Matsumoto wrote:

Hi,

>irb(main):007:0> -27 ** (1/3.0)
>=> -3.0

It's -(27 ** (1/3.0)).

              matz.

Thanks for the explanation.

···

In message "Re: Math errors" > on Fri, 18 Dec 2009 17:30:15 +0900, "W. James" > <w_a_x_man@yahoo.com> writes:

--

11

And just to verify:

(27)**(1/3.0) => 3.0

So Ruby is messing up parsing (-27)**(1/3.0)

···

On Dec 18, 10:51 am, jzakiya <jzak...@mail.com> wrote:

On Dec 18, 4:52 am, "Dasson, Raghav" <raghav.das...@sap.com> wrote:

> jzakiya wrote:
> > (-3)**3 => -27

> > (-27)**(1/3.0) give NaN instead of -3

> > All odd roots (1/3,1/5,1/7 etc) of negative numbers should give
> > negative root values, as above, but ruby (in irb) gives NaN (not a
> > number), even when I require 'complex' .

> > Is this considered an error in Ruby?

> I guess, you are right. THIS IS A 'MATH ERROR' in Ruby.

> The ** method is not able to handle negative numbers raised to Floating points (that are not Integers)

This behavior was shown in ruby ree 1.8.6, 1.8.6, 1.8.7, 1.9.1

-27**1/3.0 => -9.0
-27 ** 1/3.0 => -9.0
-27 ** (1/3.0) => -3.0
-27**(1/3.0) => -3.0
(-27)**(1/3.0) => NaN
(-27) ** (1/3.0) => NaN

OK, the first two expression are evaluated as (-27**1)/3 => -9
The second two are correct (what I expected).
But the last two, WHY??

12

Hello,

This behavior was shown in ruby ree 1.8.6, 1.8.6, 1.8.7, 1.9.1

-27**1/3.0 => -9.0
-27 ** 1/3.0 => -9.0
-27 ** (1/3.0) => -3.0
-27**(1/3.0) => -3.0
(-27)**(1/3.0) => NaN
(-27) ** (1/3.0) => NaN

OK, the first two expression are evaluated as (-27**1)/3 => -9
The second two are correct (what I expected).
But the last two, WHY??

As Matz said, the second two are evaluated as - (27**(1/3.0)), that is
you take the cubic root of +27 which is 3 and then distribute the
minus.
That way, you take the cubic root of a positive number which does not
raise any problem.

Cheers,

···

--
JJ Fleck
PCSI1 Lycée Kléber

13

That is WRONG, you cannot do that.
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

···

On Dec 18, 11:05 am, Fleck Jean-Julien <jeanjulien.fl...@gmail.com> wrote:

Hello,

> This behavior was shown in ruby ree 1.8.6, 1.8.6, 1.8.7, 1.9.1

> -27**1/3.0 => -9.0
> -27 ** 1/3.0 => -9.0
> -27 ** (1/3.0) => -3.0
> -27**(1/3.0) => -3.0
> (-27)**(1/3.0) => NaN
> (-27) ** (1/3.0) => NaN

> OK, the first two expression are evaluated as (-27**1)/3 => -9
> The second two are correct (what I expected).
> But the last two, WHY??

As Matz said, the second two are evaluated as - (27**(1/3.0)), that is
you take the cubic root of +27 which is 3 and then distribute the
minus.
That way, you take the cubic root of a positive number which does not
raise any problem.

Cheers,

--
JJ Fleck
PCSI1 Lycée Kléber

14

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,

···

--
JJ Fleck
PCSI1 Lycée Kléber

15

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

···

--

16

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)

···

On Dec 18, 2:13 pm, "William James" <> wrote:

17

> 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. :slight_smile:

···

On Dec 18, 5:24 pm, jzakiya <jzak...@mail.com> wrote:

On Dec 18, 2:13 pm, "William James" <> wrote:

18

Oooh, I just noticed:
Complex(-27**(1/3.0),0) => (-3.0+0i).

So that acts I expected/want.

So maybe it would be nice to be able to do:

Complex(-a**(1/n),0).roots or
Complex(a**(1/n)).roots where a is already complex
and have this return all the roots in an array so you
can see/pick which one(s) you want.

···

On Dec 18, 8:02 pm, jzakiya <jzak...@mail.com> wrote:

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. :slight_smile:

19

So if you have these methods:

Complex(a).root(n).roots # all the roots of a
Complex(a).root(n).real # the real root(s) if exists

Then for any value a (real or complex) you will get
what you want under the different conditions.

···

On Dec 18, 8:26 pm, jzakiya <jzak...@mail.com> wrote:

On Dec 18, 8:02 pm, jzakiya <jzak...@mail.com> wrote:

> 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. :slight_smile:

Oooh, I just noticed:
Complex(-27**(1/3.0),0) => (-3.0+0i).

So that acts I expected/want.

So maybe it would be nice to be able to do:

Complex(-a**(1/n),0).roots or
Complex(a**(1/n)).roots where a is already complex
and have this return all the roots in an array so you
can see/pick which one(s) you want.