Question about Pi?

Question

If it is true (now) that Pi has infinite decimals, then does that mean you can never know EXACTLY what the circumference of a circle is? (2(Pi)r).
Yet you can draw a circle that has an exact length? Or take a string of known exact measurable length and make a circle out of it.

What am I missing?

I understand that we can never measure anything exactly, but how can a circle get larger in exceedingly smaller increments but never allow precise measurement when it starts at a point and ends at the same point? It must be finite in that case. It can't go on indefintely. Logic tells me that's impossible.
It's either that, or Pi is an approximation to make up for a lack of our measuring capabilties.

Answer 1

Imagine a circle with the radius of the observable universe. Now imagine try to find the diameter of that circle by multiplying the radius by 2pi. If you have 50 decimal places of pi, you will get the diameter with an accuracy of much less than the diameter of a proton! There is no concievable physical reason to estimate pi to more places than that. The only reasons past that are purely mathematical.

Answer 2

Yes you can draw a circle that has an exact circumference. The radius or diameter would not be exact though, as you can only approximate it using pi.

In response to your additional details...
Yes, pi is approximate. We will never be able to calculate pi exactly because there is no exact number for pi. If you calculate it to 100 digits past the decimal point, you have an approximation. If you calculate it to 1000 digits, it's still approximate. It's not a lack of measuring capabilities. The number pi is actually generated using calculus equations, not just by measuring a circle as best we can, which was how people did it before calculus was invented a few hundred years ago. When you measure anything, you can always get a better ruler and measure it more exactly. Maybe you think something is 5 inches long, but is it actually between 5" and 5-1/8"? Or is it closer to 5-1/16"? Or just a hair longer than that? So the reason that the circumference of a circle can't be figured exactly actually has something to do with the continuity of length. The reason pi goes on indefinitely has more to do with the fact that there really is no good way to draw a relationship between a straight line and a circle.

Answer 3

Yes to the first part. C = 2πr and you are saying that you can never know exactly what the circumference of the circle is. But it has little to do with π and rather a great deal to do with r. How closely can you measure r?

If r is nominally 1 meter, then you can't measure it to much better than a few nanometers because of quantum uncertainties in the exact position of the electrons in the last layer of atoms on the end of whatever medium you use to draw the line. This means that you can't possibly know r to within better than 8 or 9 significant digits.

As to what you're missing..... An understanding of what is meant by an 'irrational' number. Consider two rational points on the number line. Call them a/b and c/d (with c/d > a/b). It should be pretty obvious that you can *always* find another rational point between them, no matter how close together they are and that that point will be at

a/b + (bc-ad)/2db = (ad + bc)/2db.

This is simply the point in the 'middle' between a/b and c/d and it's a/d plus one half the difference c/d - a/b. It's also, obviously, a rational number since both the numerator and denominator are integers.

Think about that for a moment...... No matter *how* close together two rational numbers are, you can always use the above 'method' to find another rational number that lies between them. No matter how infinitely small an interval gets, you can always 'cut it in half' to get a smaller interval (in the same way that, no matter how infinitely large an integer gets, you can always add 1 to it to get one that's bigger)

So how about irrationals? An irrational number is one that *can't* be written as the quotient of 2 integers. How can that be when, as we showed above, that the rationals are 'infinely close' together?

Consider √2. Say it can be written as a rational fraction (say p/q). Without loss of generality we may assume that p/q has been reduced to its lowest terms (that is, p and q have no factors in common). Then

√2 = p/q ==> 2 = p²/q² ==> p² = 2q²

Now this says that p² is even and so p must be even (the squares of even numbers are always even and teh squares of odd numbers are always odd) so let p = 2s and then p² = 4s² so that

4s² = 2q² ==> 2s² = q² and so q must also be even. But that means that p and q both have to have factors of 2 which contradicts the original assumption, so it must be that the original assumption was wrong.

Irrationals are a fundamentally different 'kind' of number that you can't write using the rationals (except as the sum of an infinite series of rational numbers)

Now.... There is a fairly well-known theorem from Analysis that says 'between any two rational numbers there is at *least* one irrational number' (I won't prove it because it's a kind of long and messy proof)

Think about that. No matter how close together we let two rational points become (and we know that we can always cut the 'distance' between them in half) there will always be at *least* one irrational number between them.

If you think about all of that long enough, it will all start to make sense. Math is no more 'intuitive' than quantum physics. Both of them will take you to places that simply have no equivalent in the 'normal' world. To understand them, you really have to work at it and think about it.

But it's worth it.


Doug

Answer 4

Well, we say that a circle of radius 1 has circumference of 2pi, and that is an exact way to say that, but we can't express that number exactly as a decimal.

If I were to take a string of 10 inches, and make a perfect circle from it, I know that it has circumference of 10, but the diameter would be 10/pi, which couldn't be expressed exactly as a decimal.

This is a really neat question, by the way!

Answer 5

Consider the diameter of a circle. Take that length (a line segment), and bent it into on arc. But it is bent into an arc in such a way that when it is put over the circumference of its' respective circle, it exactly overlaps it's circumference, it does not bend in towards the center of the circle or away from the centre of the circle.
Now, how many of these arc-ed line diameters do you need to complete one full circumference of it's own circle?
You need 3∙14159..... of them.
The ratio of any diameter of it's respective circumference or
the ratio of it's radius one of it's respective semi-circumference will always be
1 : 3∙14159.... . This number 3∙14159.... is known as pie (π).
I'm sure it would be possible to calculate out the area and circumference of a circle without using π and it's likely you would not have a non-repeating ever lasting decimal.
So the circumference and area of a circle are exact values but π is an never repeating ever lasting decimal.

Answer 6

I'm not positive that this is right, but I think Pi is just a ratio. So technically if you took the length of the string you are making the circle with and divided it by 2r, you would get Pi. If you know the circumfrence of several circles, and their radii you would always come up with the number Pi. Then logically, if you only knew the radius of a circle, you could multiply it by the number Pi, or something close enough to it, to get the circumfrence.

Answer 7

Yes, you can measure the exact circumference and the exact radius. But, when you "calculate" the circumference from the radius, the result is never exact because you do not know the "exact" value of pi.

Answer 8

It is not true that we can not know EXACTLY what the circumference of a circle is. We know EXACTLY it some thousand years ago.
In fact we can not draw any circle with exact length. We also can not measure exactly any value.
So you should separate what you know and what you can!

Answer 9

one way to think of ot might be this- certainly if you took a thumbtack and tied a piece of string to it, and tied a pencil to the other end, you could draw a "perfect" circle (and this is how compasses work).....BUT- a circle has 360 degrees, "right"?

if you took a circle big enough and drew 360 radians 1 degree apart you could still divide the distance between radians in half, and in half again, and in half again...

does this mean that a circle has "infinite" degrees?- no, but it certainly speaks to the inexactitude of geometry w/ respect to circular objects, hence the apparent infinity of pi.

p.s. do a google on "feynman's number"

Answer 10

can you measure the length of the string down to a million decimal points?