Velocity Reviews > time complexity

# time complexity

Dik T. Winter
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Posts: n/a

 10-24-2007
In article <(E-Mail Removed)> http://www.velocityreviews.com/forums/(E-Mail Removed) (Richard Harter) writes:
> On Wed, 24 Oct 2007 00:58:16 GMT, "Dik T. Winter"
> <(E-Mail Removed)> wrote:

....
> > > > > And the computer on which you can express that value is?
> > > >
> > > >Strange, I thought that in a previous article I gave that value.
> > >
> > > Well, no, you gave an expression that would yield the value if
> > > evaluated with infinite precision. Not at all the same thing as
> > > the real value.

> >
> >In mathematics it is.

>
> Well, no, it is not.
>
> >Each real value has many representations, and
> >not necessarily each representation is some finite numerical representation.
> >So when asked for the solution of x**2 = 2, the mathematical result is
> >+sqrt(2) or -sqrt(2), and they are the real values.

>
> No, they aren't the real values - they are representations of the
> real values.

So 'sqrt(2)' expresses the real number, and there are lots of computers
where I can express it that way. 'sqrt(2)' is not different from '123'
in that aspect, both give rules on how to calculate the actual number
if there is any need.
--
dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131
home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/

Ben Bacarisse
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Posts: n/a

 10-24-2007
CBFalconer <(E-Mail Removed)> writes:

> Richard Harter wrote:
>> <(E-Mail Removed)> wrote:
>>

> ... snip ...
>>
>>> Each real value has many representations, and not necessarily
>>> each representation is some finite numerical representation. So
>>> when asked for the solution of x**2 = 2, the mathematical result
>>> is +sqrt(2) or -sqrt(2), and they are the real values.

>>
>> No, they aren't the real values - they are representations of the
>> real values.

>
> I agree with Dik. The 'representations' are attempts to express
> those values with some combination of digits and numerical base,
> which can never be exact, since the values are transcendental.

That's a leap! Neither of the number you quote are transcendental
although the number that is at the heart of this thread may well be.

--
Ben.

CBFalconer
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Posts: n/a

 10-24-2007
Ben Bacarisse wrote:
> CBFalconer <(E-Mail Removed)> writes:
>> Richard Harter wrote:
>>> <(E-Mail Removed)> wrote:
>>>

>> ... snip ...
>>>
>>>> Each real value has many representations, and not necessarily
>>>> each representation is some finite numerical representation. So
>>>> when asked for the solution of x**2 = 2, the mathematical result
>>>> is +sqrt(2) or -sqrt(2), and they are the real values.
>>>
>>> No, they aren't the real values - they are representations of the
>>> real values.

>>
>> I agree with Dik. The 'representations' are attempts to express
>> those values with some combination of digits and numerical base,
>> which can never be exact, since the values are transcendental.

>
> That's a leap! Neither of the number you quote are transcendental
> although the number that is at the heart of this thread may well be.

Are you claiming sqrt(2) is not transcendental? It is a long time
since I took math classes, and I have been operating under the
(possibly delusional) assumption that a number that cannot be
expressed exactly is transcendental. According to that any
non-rational value is transcendental. Maybe it has to depend on
the series expansion of the value.

--
Chuck F (cbfalconer at maineline dot net)
Available for consulting/temporary embedded and systems.
<http://cbfalconer.home.att.net>

--
Posted via a free Usenet account from http://www.teranews.com

Richard Harter
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Posts: n/a

 10-24-2007
On Wed, 24 Oct 2007 10:53:07 GMT, "Dik T. Winter"
<(E-Mail Removed)> wrote:

>In article <(E-Mail Removed)> (E-Mail Removed) (Richard Harter) writes:
> > On Wed, 24 Oct 2007 00:58:16 GMT, "Dik T. Winter"
> > <(E-Mail Removed)> wrote:

>...
> > > > > > And the computer on which you can express that value is?
> > > > >
> > > > >Strange, I thought that in a previous article I gave that value.
> > > >
> > > > Well, no, you gave an expression that would yield the value if
> > > > evaluated with infinite precision. Not at all the same thing as
> > > > the real value.
> > >
> > >In mathematics it is.

> >
> > Well, no, it is not.
> >
> > >Each real value has many representations, and
> > >not necessarily each representation is some finite numerical representation.
> > >So when asked for the solution of x**2 = 2, the mathematical result is
> > >+sqrt(2) or -sqrt(2), and they are the real values.

> >
> > No, they aren't the real values - they are representations of the
> > real values.

>
>So 'sqrt(2)' expresses the real number, and there are lots of computers
>where I can express it that way. 'sqrt(2)' is not different from '123'
>in that aspect, both give rules on how to calculate the actual number
>if there is any need.

Just so. Or more precisely, to calculate the "actual number" to
any desired precision as desired. Let us walk around that large
pool of quicksand labelled "what is a number" and focus on usage.

In practice a real number is any point on the real line; a
canonical representation is in the form of an infinite sequence
of digits in any convenient base. The original point I thought
you were making is that real numbers cannot in general be
represented on computers; that's why we use things like floating
point. I expect that we are both on the same page here. Perhaps
we should make a bow to the gods of topicality and leave it at
that.

Richard Harter, (E-Mail Removed)
http://home.tiac.net/~cri, http://www.varinoma.com
In the fields of Hell where the grass grows high
Are the graves of dreams allowed to die

pete
Guest
Posts: n/a

 10-25-2007
CBFalconer wrote:

> Are you claiming sqrt(2) is not transcendental?

The square root of two is irrational.
Transcendental numbers are also not the roots of any equation.
pi is transcendental.

--
pete

Richard Heathfield
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Posts: n/a

 10-25-2007
CBFalconer said:

<snip>

> Are you claiming sqrt(2) is not transcendental?

Yes.

A transcendental number is one that is not a root of any integer polynomial
equation.

sqrt(2) is a root of the equation: x^2 - 2 = 0

Therefore, sqrt(2) is not transcendental, QED.

> It is a long time
> since I took math classes, and I have been operating under the
> (possibly delusional) assumption that a number that cannot be
> expressed exactly is transcendental.

The word for which you are reaching is "irrational". It is indeed the case
that sqrt(2) is irrational.

--
Richard Heathfield <http://www.cpax.org.uk>
Email: -http://www. +rjh@
"Usenet is a strange place" - dmr 29 July 1999

Richard Heathfield
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Posts: n/a

 10-25-2007
pete said:

> CBFalconer wrote:
>
>> Are you claiming sqrt(2) is not transcendental?

>
> The square root of two is irrational.
> Transcendental numbers are also not the roots of any equation.

Close, but no banana.

> pi is transcendental.

It is the root of the equation x - pi = 0. Therefore, according to your
definition of "transcendental", pi is *not* transcendental. Nevertheless,
we know that it /is/. Therefore, your definition must be wrong.

--
Richard Heathfield <http://www.cpax.org.uk>
Email: -http://www. +rjh@
"Usenet is a strange place" - dmr 29 July 1999

user923005
Guest
Posts: n/a

 10-25-2007
On Oct 24, 5:03 pm, pete <(E-Mail Removed)> wrote:
> CBFalconer wrote:
> > Are you claiming sqrt(2) is not transcendental?

>
> The square root of two is irrational.
> Transcendental numbers are also not the roots of any equation.
> pi is transcendental.

A transcendental number is not the root of any integer polynomial.
There are equations which have pi as root(s).

CBFalconer
Guest
Posts: n/a

 10-25-2007
Richard Heathfield wrote:
> CBFalconer said:
>
> <snip>
>
>> Are you claiming sqrt(2) is not transcendental?

>
> Yes.
>
> A transcendental number is one that is not a root of any integer
> polynomial equation.
>
> sqrt(2) is a root of the equation: x^2 - 2 = 0
>
> Therefore, sqrt(2) is not transcendental, QED.
>
>> It is a long time since I took math classes, and I have been
>> operating under the (possibly delusional) assumption that a
>> number that cannot be expressed exactly is transcendental.

>
> The word for which you are reaching is "irrational". It is indeed
> the case that sqrt(2) is irrational.

No, I wasn't reaching for 'irrational'. I was misusing
polynomial equations.

--
Chuck F (cbfalconer at maineline dot net)
Available for consulting/temporary embedded and systems.
<http://cbfalconer.home.att.net>

--
Posted via a free Usenet account from http://www.teranews.com

Dik T. Winter
Guest
Posts: n/a

 10-25-2007
In article <(E-Mail Removed)> (E-Mail Removed) (Richard Harter) writes:
> On Wed, 24 Oct 2007 10:53:07 GMT, "Dik T. Winter"
> <(E-Mail Removed)> wrote:

....
> >So 'sqrt(2)' expresses the real number, and there are lots of computers
> >where I can express it that way. 'sqrt(2)' is not different from '123'
> >in that aspect, both give rules on how to calculate the actual number
> >if there is any need.

....
> In practice a real number is any point on the real line; a
> canonical representation is in the form of an infinite sequence
> of digits in any convenient base.

Some practice. Much of what I have done would have given wrong results
if (for instance) sqrt(2) had been given as a finite approximation.
--
dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131
home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/