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 ?

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    $\begingroup$ The decimal representation of $\frac{1}{9}$ never ends either. And the decimal representation of $\sqrt{2}$ never ends and never repeats in much the same way as $\pi$. Having infinite digits is not the same as not being a finite number. $\endgroup$ – Thomas Andrews Mar 2 '12 at 19:30
  • $\begingroup$ I am really wondering if there is a mathematical proof of the real length of PI. I mean may be after 2 million digits after the decimal point we get no more digits. Could that be? $\endgroup$ – NoChance Mar 2 '12 at 20:13
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    $\begingroup$ @EmmadKareem No, there is a mathetmatical proof that $\pi$ is irrational, and there is a mathematical proof that only rational numbers terminate or repeat. So $\pi$ necessarily never terminates or repeats. $\endgroup$ – Thomas Andrews Mar 2 '12 at 20:18
  • $\begingroup$ @ThomasAndrews, this is very interesting, thanks. $\endgroup$ – NoChance Mar 2 '12 at 20:23
  • $\begingroup$ @Emmad: What do you mean by "length"? There are well-defined (and well-researched!) concepts you should learn ... basically all of the vocab words (in blue) on the wikipedia page for repeating decimal, such as "rational number", "decimal representation", and "periodic" $\endgroup$ – The Chaz 2.0 Mar 2 '12 at 20:23

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.

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    $\begingroup$ Im having trouble to understand : How can a number with infinite decimal numbers - can be finite in length ???? every x.yyy will have x.yyy+z.....x.yyy+z+m.....x.yyy+z+m+p ........ I can get it into my mind.... $\endgroup$ – Royi Namir Mar 2 '12 at 19:39
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    $\begingroup$ @Royi Namir: The size of a number is not related to the question of whether its decimal expansion is terminating or non-tterminating, periodic or not periodic. For example, certainly $\sqrt{2}<2$. But $2$ has a terminating decimal expansion, while the decimal expansion of $\sqrt{2}$ is not even periodic. $\endgroup$ – André Nicolas Mar 2 '12 at 19:47
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    $\begingroup$ There are many issues here: you are using number, numbers, finite, and infinite in strange and/or contradictory ways. Also, your z m and p are - at best - 10x smaller than the letter before it... $\endgroup$ – The Chaz 2.0 Mar 2 '12 at 19:49
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    $\begingroup$ @RoyiNamir When you get more advanced, you'll find out there is a notion called a "limit" which becomes crucial in understanding the idea of this. Also, consider Zeno's paradox: If you want to travel to a point $2$ feet away, you first have to travel to $1$ foot away from the point, then $\frac{1}{2}$ feet away, then $\frac{1}{4}$ feet away, etc. You have to go through infinitely many steps to arrive at a finite distance. See: en.wikipedia.org/wiki/… $\endgroup$ – Thomas Andrews Mar 2 '12 at 20:25
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    $\begingroup$ Note something else you should realize: The fact that Zeno's paradox is 2400 years old and we still teach it to students means that this is a problem that ages of young thinkers encounter, ponder, and struggle with. The nature of the infinite is in constant conflict with our early intuitions. @RoyiNamir $\endgroup$ – Thomas Andrews Mar 2 '12 at 20:33

Think about this, $\pi<4$. Now suppose you have a circle with radius $r=1$. Plug into your equation,

perimeter$=2\times pi\times r<2*4*1=8$ and 8 is a finite number.

You can also bound your perimeter from below, since $3<\pi$, then

perimeter$=2\times pi\times r<2*3*1=6$.

So for this particular circle of radius 1, your perimeter is between 6 and 8.

you might not be able to express the exact value of the perimeter in fractions, but that does not mean that the perimeter is infinite. For example, $1/5$ is the representation of the infinite number $.2\bar{0}$ and this value is definitely finite.


Let $r=1/2\pi$, then the circle has perimeter 1=1.0000000... And the radius which is a finite straight line has a non-repeating infinite decimal expansion.


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