A confusion regarding radians

Question

I read in my textbook that radians and real numbers can be considered as one and the same

The usual example was given in my book.

A unit circle had been taken with $O$ as center and $A$ as any point on its circumference. $B$ was another point on the circumference such that $\angle AOB$ is a positive angle.

Let $\angle AOB = \theta$ and the length of $arc$ $AB$ be equal to $s$

So, using a theorem regarding radians, we have : number of radians in $\theta$ = $\dfrac {s}{r}$
Here, r is equal to $1$ $unit$.So, $\theta = s$

What was further given in my book was : so, real number $s$ is the radian measure of $\theta$, this means that every positive real number is the radian measure of some positive angle.

This means that $\theta = s^c$, right?

I get everything till here (I think so...)

Now, I just wanna make clear that this does not mean that $\pi^c = \pi$, right?

Thanks!

@Shaun An extremely popular Indian Textbook : RD Sharma for 11th grade Mathematics — Rajdeep Sindhu, Apr 23 '20 at 17:54
Long story short, what I wanna know is that the proof that I stated in the question does not prove that one radian is equal to one, right? — Rajdeep Sindhu, Apr 23 '20 at 17:56
They are the same numerically, but that does not mean they have the same units. — Andrew Chin, Apr 23 '20 at 18:12
@AndrewChin Thanks. I might have misunderstood the statement that radians and real numbers can be considered as one and the same. So, what is the point of this proof? What do we actually prove here? — Rajdeep Sindhu, Apr 23 '20 at 18:16
Some of the answers to MSE question 1858945 "what is sine of a real number" may be helpful for you. — Somos, Apr 23 '20 at 18:51
Is one gram equal to the real number one? I one meter equal to the real number one? — Somos, Apr 23 '20 at 20:10
Nope, the units are different. So, are we just trying to prove that every real number is the radian measure of some angle just like every real number is the degree measure of some angle? — Rajdeep Sindhu, Apr 24 '20 at 07:23
I think I got it. What we're trying to prove is that if $x^c$ is some angle then $x \in \Bbb R$, just like degree measures angles are real numbers. Let me know if I'm right. — Rajdeep Sindhu, Apr 24 '20 at 11:00

A confusion regarding radians

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