Magnitudes of numbers or lengths are easily compared.
The number 180 is ten times the magnitude of the number 18.
180 meters is ten times as long as 18 meters. How do I make
the corresponding comparison between an angle of 180 degrees
and an angle of 18 degrees?

-- Jeff, in Minneapolis

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It is the same comparison: put 10 18-degree-angles adjacent and with the same vertex, and you'll have a 180-degree angle.

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yep.


listen to Spock. he is emotionless but wise

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Those that lack education have a hard time understanding its value. - Cross

I think Jeff Root is looking for an appropriate way to say it along these lines:

"180 is ten times the magnitude of the number 18."
"180 meters is ten times as long as 18 meters."
"180 degrees is ten times _______ 18 degrees"

It sounds like a really simple question, but I couldn't figure out how to phrase it myself, either. The best I could come up with is:

"180 degrees is ten times the rotation of 18 degrees"

I don't think that exactly works, though.

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you could use Magnitude or angle


how about


ToSeek is 10 times the man as ....

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"I will do my best to understand and explain the universe from big to small without invoking miracles, unrepeatable events, or divine intervention. In place of those things I will use observations, mathematics, and science."


-Cross
My travel blog

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Those that lack education have a hard time understanding its value. - Cross

Hmm. I was thinking that it was somehow fundamentally
different. I had in mind, but neglected to say, that the
maximum possible angle for the situation I'm looking at is
180 degrees. But even specifying that, I now don't see
that it makes a difference.

Ah, well. Thanks!

-- Jeff, in Minneapolis

__________________
http://www.FreeMars.org/jeff/

"I find astronomy very interesting, but I wouldn't if I thought we
were just going to sit here and look." -- "Van Rijn"

"The other planets? Well, they just happen to be there, but the
point of rockets is to explore them!" -- Kai Yeves

think fractions.

__________________
"I will do my best to understand and explain the universe from big to small without invoking miracles, unrepeatable events, or divine intervention. In place of those things I will use observations, mathematics, and science."


-Cross
My travel blog

Some of my Astrophotography


Those that lack education have a hard time understanding its value. - Cross

Suggestions: 180 degrees is ten times the..

• rotation
• opening
• flare
• angular separation

of 18 degrees.

But I think this is a language question, not math.

Circular trigonometry has angles defined on unit circles.
The best way to think of those angles is as pie-slice areas on a unit circle.
Do it this way and you get to re-use the idea for hyperbolic trigonometry.

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-al

But note that JR's intuition might really have been asking, 180 degrees is ten times the angular size of 18 degrees, but 1800 degrees is not ten times the angular size of 180 degrees! (Although it is ten times the rotation, so TBCs suggestion is likely the best). The point is, the magnitude of angles really do work differently than normal numbers, and rotation works different than translation, due to the 360 degree ambiguity. The distinction is even deeper: I get the question from my students, why does the Earth's orbit induce parallax but not the Earth's rotation? Or, why is the Copernican revolution always stated in terms of the Earth's orbit, and not the simple fact that it is easier to explain diurnal motion by having one thing rotate rather than everything else in the universe go around? Also, there is the interesting distinction between linear momentum, which relates to symmetry under translation, and angular momentum, which relates to symmetry under rotation. Angular momentum tends to be conserved more strongly than linear momentum, in practice, which if you think about it is again because of the 360 degree ambiguity, which makes it easier for effects to be similar from all directions (a finite interval) than from all locations (an infinite one). So there is actually a lot of profundity behind JR's simple math question!

They're both conserved, right? What do you mean by more strongly?

I meant in an open system, where nothing is completely conserved. But angular momentum conservation often works better, it seems to me. The obvious example being an object in orbit, but yeah, it's not any kind of fundamental difference, you could come up with open systems that conserve linear momentum but not angular momentum I'm sure. I'm just saying that rotations and translations can have profoundly different properties that connect to the difference between looking at the world from different angles rather than from different locations. Idle musing, probably, I just thought JR was getting at this with his intuition.