time complexity

[Dik T. Winter]

Re: time complexity

In article <471e0dbd.11632 84015@news.sbtc .netcri@tiac.net (Richard Harter) writes:

On Tue, 23 Oct 2007 10:32:01 GMT, "Dik T. Winter"
<Dik.Winter@cwi .nlwrote:

....

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. 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.
--
dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131
home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/

[Richard Harter]

Re: time complexity

On Wed, 24 Oct 2007 00:58:16 GMT, "Dik T. Winter"
<Dik.Winter@cwi .nlwrote:

>In article <471e0dbd.11632 84015@news.sbtc .netcri@tiac.net (Richard Harter) writes:

On Tue, 23 Oct 2007 10:32:01 GMT, "Dik T. Winter"
<Dik.Winter@cwi .nlwrote:

>...

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.



Richard Harter, cri@tiac.net
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

[CBFalconer]

Re: time complexity

Richard Harter wrote:

<Dik.Winter@cwi .nlwrote:
>

.... 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 'representation s' are attempts to express
those values with some combination of digits and numerical base,
which can never be exact, since the values are transcendental.

--
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]

Re: time complexity

In article <471ebc7e.12080 20828@news.sbtc .netcri@tiac.net (Richard Harter) writes:

On Wed, 24 Oct 2007 00:58:16 GMT, "Dik T. Winter"
<Dik.Winter@cwi .nlwrote:

....

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]

Re: time complexity

CBFalconer <cbfalconer@yah oo.comwrites:

Richard Harter wrote:

><Dik.Winter@cw i.nlwrote:
>>

... 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 'representation s' 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.

[Richard Harter]

Re: time complexity

On Wed, 24 Oct 2007 10:53:07 GMT, "Dik T. Winter"
<Dik.Winter@cwi .nlwrote:

>In article <471ebc7e.12080 20828@news.sbtc .netcri@tiac.net (Richard Harter) writes:

On Wed, 24 Oct 2007 00:58:16 GMT, "Dik T. Winter"
<Dik.Winter@cwi .nlwrote:

>...

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, cri@tiac.net
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

[CBFalconer]

Re: time complexity

Ben Bacarisse wrote:

CBFalconer <cbfalconer@yah oo.comwrites:

>Richard Harter wrote:

>><Dik.Winter@c wi.nlwrote:
>>>

>... 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 'representation s' 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

[pete]

[ot]Re: time complexity

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]

Re: time complexity

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@
Google users: <http://www.cpax.org.uk/prg/writings/googly.php>
"Usenet is a strange place" - dmr 29 July 1999

[Richard Heathfield]

Re: [ot]Re: time complexity

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@
Google users: <http://www.cpax.org.uk/prg/writings/googly.php>
"Usenet is a strange place" - dmr 29 July 1999

[user923005]

Re: time complexity

On Oct 24, 5:03 pm, pete <pfil...@mindsp ring.comwrote:

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]

Re: time complexity

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
'transcendental '. I had entirely forgotton about roots of integer
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]

Re: time complexity

In article <471f6cc0.12531 42937@news.sbtc .netcri@tiac.net (Richard Harter) writes:

On Wed, 24 Oct 2007 10:53:07 GMT, "Dik T. Winter"
<Dik.Winter@cwi .nlwrote:

....

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/

[Ben Bacarisse]

Re: time complexity

CBFalconer <cbfalconer@yah oo.comwrites:

Ben Bacarisse wrote:

>CBFalconer <cbfalconer@yah oo.comwrites:

<snip>

>>I agree with Dik. The 'representation s' 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?

Done to death now but yes. Though "irrational " is often used for
numbers like sqrt(2) the term in not very helpful since transcendental
numbers are also irrational. The modern term is "algebraic" giving
the hierarchy:

naturals + integers + rational + algebraics + transcendentals = reals

(with each class including those to the right).

--
Ben.

[Philip Potter]

Re: time complexity

Ben Bacarisse wrote:

Done to death now but yes. Though "irrational " is often used for
numbers like sqrt(2) the term in not very helpful since transcendental
numbers are also irrational. The modern term is "algebraic" giving
the hierarchy:
>
naturals + integers + rational + algebraics + transcendentals = reals
>
(with each class including those to the right).

Not the way I see it. Using < for the subset-of relation, we have:

Naturals < integers < rationals < algebraics

But it is certainly not the case that algebraics < trancendentals! In fact

algebraics I trancendentals = 0

where I is intersection and 0 is the empty set.

Phil