Two Pi

[gull]

Why do we use radians as the de facto unit of angular measurement? Why not diametians? Using diametians seems nice to me, because then a circle is only pi around, and 1/2 pi corresponds to halfway around the circle.

Alternatively, why do we use pi to compare the circumference of a circle to its diameter? Why don't we use a constant that compares the circuference to the radius? It would be around 6.283. Again, this seems nicer. The new pi would correspond to one full revolution.

Are there answers to these questions? I never wondered about the incongruity of using diameters and radii before, and now I'm very curious. I searched a number of sources, but I was unable to find any answers.

[Siegmund]

Well, you CAN use any unit of measurement you want... but radians are the most natural one (and the only one that doesn't require you to carry around extra constants when you use the trig functions.)

The "naturalness" comes from what an angle is: two rays (half-lines), starting from the same point and moving outward in two directions from it. For any circle constructed on that point, the two rays intersect it, marking off two radii and an arc between them.

[bigpooch]

Just to add some clarification:

e^z = 1 + z + z^2 + z^3 +...

With i=sqrt(-1),

e^(ix) = 1 + (ix) - (x^2)/2! - i(x^3)/3! + (x^4)/4! +...

and e^(ix) = cos(x)+i sin x

or cos(x)=1-(x^2)/2!+...
and sin(x)= x-(x^3)/3!+...

where x is in radians.

If x is measured in some other units, there would be extra
constants.

[PairTheBoard]

Trying to clarify bigpooch's clarification. You can identify complex numbers with points on the plane according to their magnitude (distance from the origin) and their phase (angle with the x-axis). The unit of measurement for the magnitude is forced. It's then natural to use the same unit of measurement for the length of the circular arc which defines the angle with the x-axis, or phase.

Letting R1,R2 and P1,P2 be the respective magnitudes and phases of two complex number, their complex product which is complicated and difficult to work with when the numbers are in the form x+iy, now reduces to the simple form,

[R1,P1]*[R2,P2] = [R1R2,P1+P2]

It's nice to have the unit for measuring the arc length agree with the unit for measuring the distance to the origin. While you might be able to do it differently, I don't think it would be natural. And as bigpooch points out it would complicate the kinds of calculations he demonstrated, which are also foundation calculations for complex numbers.

Complex numbers are not to be sneezed at. I think in some ways they are more foundational than the reals.

PairTheBoard

[Borodog]

[ QUOTE ]
Complex numbers are not to be sneezed at.

[/ QUOTE ]

What if they are imaginary sneezes?

[PairTheBoard]

[ QUOTE ]
[ QUOTE ]
Complex numbers are not to be sneezed at.

[/ QUOTE ]

What if they are imaginary sneezes?

[/ QUOTE ]

Compared to real sneezes? Like Atheists and Theists, they would be out of phase with each other.

PairTheBoard

[bluesbassman]

[ QUOTE ]
Complex numbers are not to be sneezed at. I think in some ways they are more foundational than the reals.

PairTheBoard

[/ QUOTE ]

Meh, the complex field is so... pedestrian. I prefer the more general quaternions (http://en.wikipedia.org/wiki/Quaternion). Sure, we lose commutativity, but we can easily represent 3-D rotations. The complex numbers are trapped in 2-D flatland.

On the other hand, those eccentric octonions (http://en.wikipedia.org/wiki/Octonian) scare me.

[bigpooch]

Octonions are a bit pathological! Associativity is a much
more important property than commutativity, so quaternions
are more "normal" to examine. What does this have to do
with pi, anyway?