Velocity Reviews > Re: subexpressions (OT: math)

# Re: subexpressions (OT: math)

Steve Howell
Guest
Posts: n/a

 06-02-2007

--- Steven D'Aprano
<(E-Mail Removed)> wrote:

> On Sat, 02 Jun 2007 05:54:51 -0700, Steve Howell
> wrote:
>
> >>
> >> def f(x): y = x*x; return sin(y)+cos(y);
> >>

> >
> > Although I know valid trigonometry is not the

> point of
> > this exercise, I'm still trying to figure out why
> > anybody would ever take the square of an angle.
> > What's the square root of pi/4 radians?

>
> Approximately 0.886 radians. It corresponds to the
> angle of a point on the
> unit circle quite close to (sqrt(2/5), sqrt(3/5)),
> or if you prefer
> decimal approximations, (0.632, 0.775).
>
> Angles are real numbers (in the maths sense), so
> just as reasonable an angle as pi/4 radians. Both
> are irrational numbers
> (that is, can't be written exactly as the ratio of
> two integers).
>

Yes, I understand that, but what is the geometrical
meaning of the square root of an arc length? And what
would the units be? If you take the square root of an
area, the units change from acres to feet, or from
square meters to meters.

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Steven D'Aprano
Guest
Posts: n/a

 06-03-2007
On Sat, 02 Jun 2007 08:29:59 -0700, Steve Howell wrote:

>
> --- Steven D'Aprano
> <(E-Mail Removed)> wrote:
>
>> On Sat, 02 Jun 2007 05:54:51 -0700, Steve Howell
>> wrote:
>>
>> >>
>> >> def f(x): y = x*x; return sin(y)+cos(y);
>> >>
>> >
>> > Although I know valid trigonometry is not the

>> point of
>> > this exercise, I'm still trying to figure out why
>> > anybody would ever take the square of an angle.
>> > What's the square root of pi/4 radians?

>>
>> Approximately 0.886 radians. It corresponds to the
>> angle of a point on the
>> unit circle quite close to (sqrt(2/5), sqrt(3/5)),
>> or if you prefer
>> decimal approximations, (0.632, 0.775).
>>
>> Angles are real numbers (in the maths sense), so
>> just as reasonable an angle as pi/4 radians. Both
>> are irrational numbers
>> (that is, can't be written exactly as the ratio of
>> two integers).
>>

>
> Yes, I understand that, but what is the geometrical
> meaning of the square root of an arc length?

about the square root of an angle.

> And what would the units be?

Angles are a ratio of two lengths, and are therefore dimensionless units.
So the square root of an angle is just another angle, in the same units,
and it requires no special geometric interpretation: the square root of 25
degrees (just an angle) is 5 degrees (just another angle).

(Note: I see that the Unix program "units" does not agree with me. It
tries to use angles as dimensionless in some contexts, but taking roots is
not one of those cases.)

Arc lengths are dimensional lengths. While you can take the square root of
a length, it doesn't have any (obvious) geometrical or physical
interpretation. One might even say it is meaningless, e.g. you can always
say that the square root of (say) 9 feet is 3 feet**(1/2), but there is no
physical meaning to that -- it doesn't correspond to anything measurable.

> If you take the square root of an
> area, the units change from acres to feet, or from
> square meters to meters.

Yes, because areas have dimension L**2, so square-rooting them has the
obvious geometrical interpretation of asking "what length, when squared,
has this area?".

I've seen interpretations of fractional powers of length as scaling
factors for fractals. It's a nice interpretation, but not meaningful in
this context.

--
Steven.

Leonhard Vogt
Guest
Posts: n/a

 06-03-2007
>> Yes, I understand that, but what is the geometrical
>> meaning of the square root of an arc length?

>
> That's a different question to your original question, which was asking
> about the square root of an angle.
>
>> And what would the units be?

>
> Angles are a ratio of two lengths, and are therefore dimensionless units.
> So the square root of an angle is just another angle, in the same units,
> and it requires no special geometric interpretation: the square root of 25
> degrees (just an angle) is 5 degrees (just another angle).

But sqrt(25°) = sqrt(25/180*pi) = 5*sqrt(180/pi) != 5°

Leonhard

Steven D'Aprano
Guest
Posts: n/a

 06-03-2007
On Sun, 03 Jun 2007 09:02:11 +0200, Leonhard Vogt wrote:

>> Angles are a ratio of two lengths, and are therefore dimensionless units.
>> So the square root of an angle is just another angle, in the same units,
>> and it requires no special geometric interpretation: the square root of 25
>> degrees (just an angle) is 5 degrees (just another angle).

>
> But sqrt(25°) = sqrt(25/180*pi) = 5*sqrt(180/pi) != 5°

Hmmm... perhaps that's why the author of the "units" program doesn't
treat angles as dimensionless when taking square roots.

Given that, I withdraw my claim that the sqrt of an angle is just an
angle. I can't quite see why it shouldn't be, but the evidence is fairly
solid that it isn't.

--
Steven

Stebanoid@gmail.com
Guest
Posts: n/a

 06-03-2007
On 3 , 14:05, Steven D'Aprano <(E-Mail Removed)>
wrote:
> On Sun, 03 Jun 2007 09:02:11 +0200, Leonhard Vogt wrote:
> >> bla-bla

>
> Hmmm... perhaps that's why the author of the "units" program doesn't
> treat angles as dimensionless when taking square roots.
>
> Given that, I withdraw my claim that the sqrt of an angle is just an
> angle. I can't quite see why it shouldn't be, but the evidence is fairly
> solid that it isn't.
>
> --
> Steven

angle is dimensionless unit.
To understand it: sin() can't have dimensioned argument. It is can't
to be - sin(meters)

it is difficult to invent what is a "sqrt from a angle" but it can be.

Gary Herron
Guest
Posts: n/a

 06-03-2007
http://www.velocityreviews.com/forums/(E-Mail Removed) wrote:
> On 3 , 14:05, Steven D'Aprano <(E-Mail Removed)>
> wrote:
>
>> On Sun, 03 Jun 2007 09:02:11 +0200, Leonhard Vogt wrote:
>>
>>>> bla-bla
>>>>

>> Hmmm... perhaps that's why the author of the "units" program doesn't
>> treat angles as dimensionless when taking square roots.
>>
>> Given that, I withdraw my claim that the sqrt of an angle is just an
>> angle. I can't quite see why it shouldn't be, but the evidence is fairly
>> solid that it isn't.
>>
>> --
>> Steven
>>

>
> angle is dimensionless unit.
>

Of course not! Angles have units, commonly either degrees or radians.

However, sines and cosines, being ratios of two lengths, are unit-less.
> To understand it: sin() can't have dimensioned argument. It is can't
> to be - sin(meters)
>

> it is difficult to invent what is a "sqrt from a angle" but it can be.
>

I don't know of any name for the units of "sqrt of angle", but that
doesn't invalidate the claim that the value *is* a dimensioned
quantity. In lieu of a name, we'd have to label such a quantity as
"sqrt of degrees" or "sqrt of radians". After all, we do the same
thing for measures of area. We have some units of area like "acre", but
usually we label areas with units like "meters squared" or "square
meters". That's really no stranger than labeling a quantity as "sqrt
of degrees".

Gary Herron, PhD.
Department of Computer Science
DigiPen Institute of Technology

Stebanoid@gmail.com
Guest
Posts: n/a

 06-03-2007
On 3 , 21:43, Gary Herron <(E-Mail Removed)> wrote:
> (E-Mail Removed) wrote:
>
> > angle is dimensionless unit.

>
> Of course not! Angles have units, commonly either degrees or radians.
>
> However, sines and cosines, being ratios of two lengths, are unit-less.> To understand it: sin() can't have dimensioned argument. It is can't
> > to be - sin(meters)

>
> No it's sin(radians) or sin(degrees).> it is difficult to invent what is a "sqrt from a angle" but it can be.
>
> I don't know of any name for the units of "sqrt of angle", but that
> doesn't invalidate the claim that the value *is* a dimensioned
> quantity. In lieu of a name, we'd have to label such a quantity as
> "sqrt of degrees" or "sqrt of radians". After all, we do the same
> thing for measures of area. We have some units of area like "acre", but
> usually we label areas with units like "meters squared" or "square
> meters". That's really no stranger than labeling a quantity as "sqrt
> of degrees".
>
> Gary Herron, PhD.
> Department of Computer Science
> DigiPen Institute of Technology

angle is a ratio of two length and dimensionless.
http://en.wikipedia.org/wiki/Angle#U...ure_for_angles

only dimensionless values can be a argument of a sine and exponent!
Are you discordant?

Stebanoid@gmail.com
Guest
Posts: n/a

 06-03-2007
On 3 , 22:07, "(E-Mail Removed)" <(E-Mail Removed)> wrote:
>
> angle is a ratio of two length and dimensionless.http://en.wikipedia.org/wiki/Angle#U...ure_for_angles
>
> only dimensionless values can be a argument of a sine and exponent!
> Are you discordant?

if you are discordant read more :
sine is a dimensionless value.
if we expand sine in taylor series sin(x) = x - (x^3)/6 + (x^5)/120
etc.
you can see that sin can be dimensionless only if x is dimensionless
too.

I am a professional physicist and a know about what I talk

Cameron Laird
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Posts: n/a

 06-03-2007
In article <4662675b\$(E-Mail Removed)>,
Leonhard Vogt <(E-Mail Removed)> wrote:
>>> Yes, I understand that, but what is the geometrical
>>> meaning of the square root of an arc length?

>>
>> That's a different question to your original question, which was asking
>> about the square root of an angle.
>>
>>> And what would the units be?

>>
>> Angles are a ratio of two lengths, and are therefore dimensionless units.
>> So the square root of an angle is just another angle, in the same units,
>> and it requires no special geometric interpretation: the square root of 25
>> degrees (just an angle) is 5 degrees (just another angle).

>
>But sqrt(25°) = sqrt(25/180*pi) = 5*sqrt(180/pi) != 5°
>
>Leonhard

Yes it is; that is, if you're willing to countenance the square root
of an angle at all, then there should be no problem swallowing

sqrt(pi radians / 180) = 1 sqrt(degree)

so that

sqrt(25 degrees) = sqrt(25) * sqrt(pi radians / 180)

= 5 * sqrt(degree)

If it helps, we can call

zilth := sqrt(pi radians / 180)

Measured in square-roots of a degree, a zilth is numerically 1.

Wildemar Wildenburger
Guest
Posts: n/a

 06-03-2007
(E-Mail Removed) wrote:
> if you are discordant read more :
> sine is a dimensionless value.
> if we expand sine in taylor series sin(x) = x - (x^3)/6 + (x^5)/120
> etc.
> you can see that sin can be dimensionless only if x is dimensionless
> too.
>
> I am a professional physicist and a know about what I talk
>
>

No you don't. I'm a student of physics, and I know better:

First of all, what you have presented here is called the MacLaurin
series. It is however a special case of the Taylor series, so you are
correct. I just thought I'd let you know. (Sorry to sound like a bitch
here, i love smartassing )

Let me start by saying that *if* x had a dimension, none of the terms in
your expansion would have the same dimension. A well well-versed
physicist's head should, upon seeing such a thing, explode so as to warn
the other physicists that something is terribly off there. How (ye
gods!) do you add one metre to one square-metre? You don't, that's how!

OK, the *actual* form of the MacLaurin series for some function f(x) is

f(x) = f(0) + x/1! f'(0) + x^2/2! f''(0) + ...

So in each term of the sum you have a derivative of f, which in the
case of the sine function translates to sine and cosine functions at the
point 0. It's not like you're rid of the function just by doing a
polynomial expansion. The only way to *solve* this is to forbid x from
having a dimension. At least *I* see no other way. Do you?

/W
(Don't take this as a personal attack, please. I'm a good guy, I just
like mathematical nitpicking.)