# a Circle perimeter as expression of $\pi$ Conflict?

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...

• 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. Mar 2, 2012 at 19:30
• 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? Mar 2, 2012 at 20:13
• @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. Mar 2, 2012 at 20:18
• @ThomasAndrews, this is very interesting, thanks. Mar 2, 2012 at 20:23
• @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" Mar 2, 2012 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.

• 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.... Mar 2, 2012 at 19:39
• @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. Mar 2, 2012 at 19:47
• 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... Mar 2, 2012 at 19:49
• @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/… Mar 2, 2012 at 20:25
• 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 Mar 2, 2012 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.