Re: A tough question about elementary mathematics

The poster formerly known as Colleyville Alan wrote:
<drmwecker@xxxxxxxxx> wrote in message
news:1151983796.836312.119830@xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
I think an important point has been missed:

In calculus and in analysis in general, we wish to have functions of a
real variable. That is, we wish the inputs to be plain old real
numbers.

In truth, we don't even need "radians" for this, and I would argue (as
I do in my teaching) that radian measure actually is a needless
mystification.

To define sin(t), where t is a real number, one winds from (1,0) on the
unit circle in the appropriate direction, arrives at a point (x, y) on
the unit circle, and takes sin(t) = the y-coordinate we just obtained.

I was trying to make a similar point, but did not think through the matter
very well and ended up making a silly statement. But you have made the
point that I was unable to articulate, that a real number goes into the sine
function and a y-coordinate comes out. If it so happens that the real
number is 0 then the sine is 0 if the real number is equal to pi/2 the sine
is 1 and if the real number is equal to pi/4, the sine is srt(2)/2.

A better way of stating this is from Wikipedia. The following text gets at
the point I was trying to make: "Although the radian is a unit of measure,
anything measured in radians is dimensionless. This can be seen easily in
that the ratio of an arc's length to its radius is the angle of the arc,
measured in radians; yet the quotient of two distances is dimensionless."

Radians are *not* unit of measure. Angles measured in radians
*are* numbers. Pure numbers.

There is no "protopype radian" with which you compare angles, and
such that one measures angles with proportions with respect to the
prototype.

This is possible, of course, because circles are such that the
ratio of the circumference to the radius is constant. Therefore,
once you are able to measure lengths (ie, once you have introduced
a unit for length) you are able to measure angles without needing
an extra unit. Of course, that all this works out fine is a
non-triviality.
That is, theorems.

-- m

.

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