Accuracy in Measurement

Date: 02/08/2002 at 15:38:15
From: Rosemary
Subject: Pi

I understand that pi is the ratio of a circle's circumference to its 
diameter. Since pi is irrational, that implies that at the least, 
either the circumference or the diameter must be irrational. I don't 
understand how that is possible. 

If I had a piece of string one inch long and formed it into a circle, 
couldn't I theoretically measure the diameter of that circle? How 
could that measurement be irrational? Just because I can't measure 
it accurately, it doesn't mean that the true length of it is some 
never-ending decimal. 

So that's my question, if you understand it. Thanks.

Date: 02/08/2002 at 20:50:14
From: Doctor Jeremiah
Subject: Re: Pi

Hi Rosemary,

Your observation is very smart.

One of either the circumference or diameter is not rational. If you 
have a piece of string exactly 1 inch long and you make it into a 
perfect circle, the diameter of that circle will be an irrational 
value. You could try to measure it, but no matter how accurately you 
did measure it, it would still not be quite accurate enough. So you 
can never know exactly what the diameter is just by measuring.

And if you took that string and made a diameter with it, the 
circumference would be irrational and you would never be able to 
measure the circumference accurately enough. Whatever you measured 
would be close but not exact.

It's the same problem with a right angled triangle with short sides 
exactly 1 inch long:

               +
             + |
           +   | 1
         +     |
       +       |
     +---------+
           1

The long side is an irrational number. You can't find out what it is 
by measuring. You can get close, but it's still not exact.

The only way to calculate the exact length of these things is with 
algebra.

If you want to talk about this more, please write back.

- Doctor Jeremiah, The Math Forum
  http://mathforum.org/dr.math/   

Date: 02/13/2002 at 15:24:23
From: Rosemary
Subject: Pi

Thanks so much for your response. I guess it confirms what I was 
thinking, but I still don't really understand how it can be. I would 
like to continue to talk about this subject, but I really don't know 
what other questions to ask about it. Mathematically, it makes sense, 
but rationally, it doesn't. If there were some reason that the 
measurement could never be accurate enough, then perhaps I would 
understand better. However, I don't see how one length of something 
can be measured as an exact number and another cannot. There are 
hypoteneuses of right triangles that are rational numbers, so if they 
can be measured, I don't see why some others cannot, except for the 
fact that some formula says that they can't. 

In my mind, I know that these lengths are indeed finite, so that 
decimal must end...

Still confused,
Rosemary

Date: 02/13/2002 at 23:39:11
From: Doctor Jeremiah
Subject: Re: Pi

Hi Rosemary,

Nothing can be measured perfectly accurately. Some things can be 
measured "accurately enough," but that still isn't perfectly accurate.

For example, say you have a piece of string exactly 1 inch long (if 
you are a centimeter person then substitute centimeter for inch). Now, 
we want to measure this piece of string. Our ruler has inches on it 
and when we hold it up it looks close to 1, but is it exactly 1 or is 
it 1.00000000000000001 ?  

Well, to measure 1.00000000000000001 we need a ruler with markings
that are very close together. But let's say we can measure using a 
microscope and a very detailed ruler, and we find it isn't 
1.00000000000000001 .  Now the question is whether it is exactly 1 or 
if it is actually 1.000000000000000000000000000000001 - and how are
you going to measure that?

See?  No matter how accurately you measure, the actual value might not 
be exactly the value the ruler shows you. That is because rulers
don't have an infinite amount of accuracy. No ruler does.

So it doesn't matter if something is exactly 1 or if it is exactly Pi, 
you can't measure either one of them with enough exactness to be sure.
The number 1 and Pi both have an infinite number of decimal points 
(it's just that they are all zero in the one case), and there is no 
way to measure to an infinite accuracy.

Any measurement you make is an approximation to the real value. It 
might be a very good approximation and really, really close, but it 
will never be exact.

That's why algebra was invented. If you can't measure exactly how big 
something is, then you must calculate the exact size some other way.
And that's how we know Pi is irrational; not because we measured it 
and found out it had an infinite number of decimal points, but because 
algebra says that it must be like that. If we could measure something 
perfectly, we would find that it is the exact number 1 or the exact 
number Pi, but in reality we can't measure things that accurately.

What do you think?

- Doctor Jeremiah, The Math Forum
  http://mathforum.org/dr.math/   

Date: 02/13/2002 at 23:42:30
From: Doctor Peterson
Subject: Re: Pi

I'd like to contribute some thoughts. Let's forget the circle for the 
moment and look at your hypotenuse of a right triangle, or 
specifically the diagonal of a square.

First, a length is not a number - not until you have chosen a unit to 
measure it with. That is, we measure lengths as RATIOS of segments, so 
that the length of a side of our square in inches is the ratio of its 
length to that of a one-inch segment. So a given segment might have a 
rational length or an irrational length, depending on what unit you 
use to measure it with.

Take that square, for example. If we use the length of a side as our 
unit, then it is a unit square, and of course its sides have length 
one. But the diagonal is the square root of 2, which is irrational.

If instead we chose to use the diagonal as the unit, then the diagonal 
would have a rational length, and the sides would be irrational.

So line segments in themselves are not rational or irrational. Rather, 
two line segments may be "incommensurable," meaning that the ratio of 
their lengths is irrational. This idea goes back to the ancient 
Greeks, who at first assumed that in any figure, all the segments 
would have whole-number lengths (they only knew about whole numbers, 
or at least only trusted those numbers) if you chose a small enough 
unit to use. When it was discovered that the diagonal of a square was 
incommensurable with the side - meaning that they could not be 
measured as whole numbers of the same unit, because their ratio was 
not a fraction - it ruined a lot of their perfectly good proofs, and 
they had to start over.

Now let's get back to reality. The fact is, the numbers we talk about 
in math are not something you can ever measure. You can't get enough 
digits of a decimal to tell whether it is rational or not; and if you 
did measure it accurately enough you would find it is composed of 
atoms and doesn't have a definite end anyway. It's only in the ideal 
world of Euclidean geometry that we can take some segment as our unit 
and measure everything else exactly enough to know whether it is 
rational or not. Irrational numbers are irrelevant to the real world.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/   

Date: 02/14/2002 at 20:32:09
From: Rosemary
Subject: Pi

Ooooooooooh, I get it!
Thanks!