Math errors

[jzakiya]

(-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?

[Tanaka Akira]

2009/12/18 jzakiya <jza...@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

[Benoit Daloze]

[Note: parts of this message were removed to make it a legal post.]

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 ;)

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

[W. James]

jzakiya wrote:

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

--

[W. James]

jzakiya wrote:

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"

--

[Yukihiro Matsumoto]

Hi,

In message "Re: Math errors"

on Fri, 18 Dec 2009 17:30:15 +0900, "W. James" <w_a_...@yahoo.com> writes:

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

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

matz.

[Dasson, Raghav]

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

[Dasson, Raghav]

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)

[jzakiya]

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??

[jzakiya]

And just to verify:

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

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

[Fleck Jean-Julien]

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

[jzakiya]

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

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

[Fleck Jean-Julien]

> 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.

[Unknown]

Yukihiro Matsumoto wrote:

Thanks for the explanation.

--

[Unknown]

Fleck Jean-Julien wrote:

def root base, n
exp = 1.0/n
return base ** exp if base >= 0 or n.even?
-( base.abs ** exp )
end

--

[jzakiya]

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)

[jzakiya]

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. :-)

[jzakiya]

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.

[jzakiya]

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.