I know that the the perimeter of a circle is
$$2\pi r$$
The problem is that $\pi$ is un-finite number. ( its decimal representation never ends)
Im having trouble to understand :
If I "cut" the circle and make it as a line : - and i look at this line :
the line has a finite length ! ( its length is NOt infinite !)
but it cant be since - it has an un-finite number inside it ( $2\pi r$)..... ( the $\pi$)
how can a line length is not infinite - but it has an un-finite number inside it...
can you please explain ?
The number $\pi$ is perfectly finite. It is just as finite as $4$, and indeed it is less than $4$. Drawing the the square of side $2$ that just contains the circle with radius $1$ shows that. The decimal representation of $\pi$ is non-terminating. There are plenty of numbers with a non-terminating decimal expansion that are a good deal more familiar than $\pi$. One example is $\frac{1}{3}$.
The decimal expansion of $\frac{1}{3}$, however, is periodic. If you want a number somewhat less mysterious than $\pi$ with a non-periodic decimal expansion, look at $\sqrt{2}$. This number represents the length of the diagonal of a square of side $1$. I expect that you do not think of the the length of that diagonal, or of the number $\sqrt{2}$, as infinite.
The arithmetic of rational numbers, that is, numbers of the form $\frac{a}{b}$, where $a$ and $b$ are integers, is, through long years of practice, familiar to almost everyone. There are some technical hurdles in dealing with the arithmetic of irrational numbers, but these were overcome a long time ago.