Accuracy in MeasurementDate: 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! |
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