Correspondence between Arc Lengths and Real Numbers in Trigonometry

[paulmdrdo1]

i'm kind of confused with what i read from my book in trig which says

"there's a one to one correspondence between the length of all arcs in the Unit Circle and the set of real numbers."
my understanding to this is that there's a corresponding real number to the length of all arcs of the unit circle. like in the real number line.

but as i read further the books says "that there's a 1-1 correspondence between the set of real numbers and all angles $\theta$ in standard position."

my whole understanding to this is when we have a unit circle all lengths of arcs in the unit circle corresponds to a unique real number and all angles $\theta$ in the unit circle also correspond to a unique real number. it seems that the length of all arcs is also the measure of the angle $\theta$ subtended by an arc on the unit circle.

what will happen if have a circle different from the unit circle? will this statement still hold for any other circles?

please correct my wrong notion if there's any.

 

[MarkFL]

You are correct, for the unit circle, we have:

$$s=\theta$$

That is, the arc-length is the same as the subtended angle. This is how radians are defined in fact. As long as the circle has a non-zero radius, that is, the circle is not degenerate, then there is the correspondence between the circumference, and indeed any portion of this circumference and the continuum of the reals.

 

[paulmdrdo1]

does it mean if we have a circle having r radius where r>1 there would still be a 1-1 correspondence between the lengths of all arcs in that circle and the set of real numbers? am i correct?

by the way what do you mean by "degenerate"?

 

[MarkFL]

does it mean if we have a circle having r radius where r>1 there would still be a 1-1 correspondence between the lengths of all arcs in that circle and the set of real numbers? am i correct?

This is true for any circle whose radius is not zero.

by the way what do you mean by "degenerate"?

A degenerate circle is one having a radius of zero, it is in fact just a point. We can refer to this as a degenerate circle. There are other degenerate cases (lines) of conic sections as well.

But, as long as the radius is greater than zero, no matter how small, and the magnitude of the angle subtended by an arc on the circumference is greater than zero, no matter how small, then we may claim (by Cantor) that there is a one-to-one correspondence to the set of reals and the set of points on the arc.

 

[paulmdrdo1]

thanks mark! i think I'm on the threshold on complete understanding of this matter. but there's another thing i wanted to be clear. we say that in the unit circle "the lengths of all arcs is also the measure of the angle θ subtended by an arc on the unit circle." it means $s=\theta$. but if we have a bigger radius this statement will not be true. is my observation correct?

i''m kinda mixing the idea of 1-1 correspondence and the equality of arc lengths to angle measure $\theta$.

 

[MarkFL]

The more general formula is:

$$s=r\theta$$

But, as Cantor showed, any open interval in the real continuum has a one-to-one correspondence with the entire continuum. So, the value of the radius is irrelevant as far as the correspondence is concerned, as long as the radius is not zero.

 

[paulmdrdo1]

now i know! :)

 

[bergausstein]

any open interval in the real continuum has a one-to-one correspondence with the entire continuum.

what does this mean?