Mathematics As AnalogyDate: 12/16/2003 at 17:35:01 From: Mark Subject: Theoretical Question about Circumference and Diameter Everyone knows that to date, the number representing pi is infinite, so, for the sake of convenience, we often round it off to a decimal place appropriate for whatever purpose we are calculating it. But can somebody explain why it's infinite, or so hard to calculate to the end? I mean, if pi is the actual relation of the circle's diameter to its circumference, how can this ratio be infinite when the diameter has a finite length, and if you were to disconnect the circle and measure it in a straight line, it too has a finite length. Is there a formula to explain this? Date: 12/17/2003 at 12:57:12 From: Doctor Ian Subject: Re: Theoretical Question about Circumference and Diameter Hi Mark, A lot of people get confused by thinking that pi is something that is determined by actually _measuring_ physical circles. It's not. Pi is the ratio of circumference to diameter for any _mathematical_ circle. But there are no mathematical circles in the real world, so we can't ever find pi by measuring something: Why Pi? http://mathforum.org/library/drmath/view/61017.html Does this help? - Doctor Ian, The Math Forum http://mathforum.org/dr.math/ Date: 12/17/2003 at 14:08:21 From: Mark Subject: Theoretical Question about Circumference and Diameter Thank you for the answer. That clears up a lot of questions, but it makes me wonder about something else. I hope you don't mind if I get a little philosophical here. Am I to assume then, that if pi as an irrational number highlights the distinction between the ideal circle and the real world circle, we can also view other geometric shapes in the same light? For example, parallel lines must never cross to be considered parallel, but I imagine then, this only occurs in the ideal because limitations of our measuring instruments would preclude two lines from being placed beside each other perfectly straight, and these imperfections would become apparent the longer the lines ran. Second, I saw that it can be mathematically demonstrated that pi is irrational, but I was wondering if we can still use the idea of physically measuring a perfect circle to illustrate the irrationality of pi in more layman's terms. That is, if we can imagine the idealized circle which doesn't exist in reality, does that mean the perfect curvature of this circle confuses the actual meeting points of a hypothetically perfect diameter line with the perimeter of the circle? I could understand how pi has no end if we thought of it in this way. Does this make any sense? Date: 12/17/2003 at 22:16:20 From: Doctor Ian Subject: Re: Theoretical Question about Circumference and diameter Hi Mark, >Am I to assume then, that if pi as an irrational number highlights >the distinction between the ideal circle and the real world circle, >we can also view other geometric shapes in the same light? Yes. It's not emphasized enough that the way math works is that we create idealized concepts, and then look for situations where we can set up _analogies_ between these concepts and things we observe in the world. For example, we might have a situation in the world where we have a collection of objects with unique identities (e.g., humans, or dates, as opposed to hydrogen atoms). Numbers also have unique identities, so we can use a number to represent each object. Going further, if the objects fill some space of possible objects, as with dates, then we assign consecutive numbers to consecutive objects. If they don't, as with people or accounts, then we can be more arbitrary in assigning numbers (e.g., social security or credit-card numbers). But when we do this, we give up the ability to calculate things like "distances" between pairs of objects. But as with any analogies, if you push them too hard, they break. >For example, >parallel lines must never cross to be considered parallel, In a flat space, anyway. In a spherical space, they can cross (e.g., the meridians of longitude on a globe intersect at the poles--each one is a "line" within the surface of the globe): Why is Pi a Constant? http://mathforum.org/library/drmath/view/57828.html >but I >imagine then, this only occurs in the ideal because limitations of >our measuring instruments would preclude two lines from being placed >beside each other perfectly straight, and these imperfections would >become apparent the longer the lines ran. Even before you get to that point, you have to deal with the fact that mathematical lines are infinite in extent! Normally, we model things with line segments instead of lines, and not crossing isn't sufficient to determine whether two line segments are parallel or not. But the main thing is that we can't make observations of physical objects, and then use those observations to draw conclusions about mathematical concepts. Suggestions, yes! But not conclusions. And this is a subtle but important point. Looking at objects in the world, we get ideas for mathematical idealizations of those objects; but then we define the idealizations within axiomatic systems that aren't in any way based on the world. And these axioms are the only valid sources of conclusions about the mathematical concepts they define. People often think that the point of math is to "describe the world", but this is a misunderstanding: What is Mathematics? http://mathforum.org/library/drmath/view/52350.html >Second, I saw that it can be mathematically demonstrated that pi is >irrational, but I was wondering if we can still use the idea of >physically measuring a perfect circle to illustrate the irrationality >of pi in more layman's terms: Sure, although I would drop the word "physically". One way to do that is by constructing an infinite series of approximations, each of which roughly corresponds to a "measurement". For example, suppose we construct a circle, and then construct the largest square that will fit inside it. The perimeter of the square is an approximation to the circumference of the circle, although not a very good one. :^D And thus the ratio of a diagonal of the square to the perimeter of the square is an approximation to the value of pi. If the diameter of the circle is 1, that's also the diagonal of the square; and each side of the square will have length sqrt(2)/2. So the perimeter would be 4 times sqrt(2)/2, or 2 times sqrt(2). Given a diameter of 1, this is also an approximation of pi: 2*sqrt(2) = 2.82 It's a little on the small side, which is what we'd expect, since the circumference is completely outside the perimeter except at four points of contact. Now, suppose instead of a square, we use a hexagon. (We always want to use a polygon with an even number of angles, so that a diagonal of the polygon is the same as the diameter of the circle.) We can do the same kind of calculation, and we'll arrive at a larger ratio of perimeter to diameter. The more angles we use, the closer the approximation will be. But to get all the way to pi, we'd have to use an infinite number of sides! The only way to get a value whose decimal expansion terminates or repeats would be to use a finite number of sides. But then we'd be computing something other than pi, wouldn't we? >If we can imagine the idealized circle which doesn't exist in >reality, does that mean the perfect curvature of this circle confuses >the actual meeting points of a hypothetically perfect diameter line >with the perimeter of the circle? I could understand how pi has no >end if we thought of it in this way. Does this make any sense? I'm not quite sure what you're getting at here. Can you say more about it? - Doctor Ian, The Math Forum http://mathforum.org/dr.math/ Date: 12/18/2003 at 01:13:24 From: Mark Subject: Theoretical Question about Circumference and Diameter This is very abstract for me so I'll do my best to explain: I understand that in a perfect circle (a circle which does not exist in reality) pi is irrational or infinite. We can only demonstrate this mathematically. We could never demonstrate this with a real world circle using physical measuring devices. But let's imagine someone measuring the diameter of this perfect circle with a perfectly precise measuring device that could give infinite measurement if required. (This also assumes the line segment representing the diameter of the circle is perfectly straight and passes through the real centre of the circle.) Since pi is infinite, we could never be able to visualize this perfect line touching the true edge at either poles of the circle. And this is my question: If we were trying to visualize the infiniteness of pi based on the whole technique of using diameter to find circumference, is the infiniteness of pi explained by the fact that we could never get the diameter to intersect exactly with the circle because of its perfectly symmetric curvature? I mean, if you could imagine drawing a perfectly straight line through the centre of a perfect square, it would not be hard to demonstrate that this line has a definite beginning and end, and runs exactly perpendicular to the two sides of the square, because they are flat. So I'm sure you could work out a formula to demonstrate the ratio of this line to the perimeter of this perfect square and show that this ratio is a finite number. Unless there is an error in my reasoning, it seems like the reason pi is irrational is that the perfectly circular shape of the circle makes it impossible to reach a conclusive measurement of its true diameter, and therefore the relation thereto to its circumference. And I suppose since it can be proven that pi is infinite, the hypothetical diameter of a perfect circle must also be infinite. It's amazing to think that an infinite line could exist within the closed space of a perfect circle. I bet the secret of the universe is contained within this contradiction. This may sound confusing, but I assure you I'm not crazy. Date: 12/18/2003 at 09:58:41 From: Doctor Ian Subject: Re: Theoretical Question about Circumference and diameter Hi Mark, >I understand that in a perfect circle (a circle which does not exist >in reality) pi is irrational or infinite. We can only demonstrate >this mathematically. We could never demonstrate this with a real >world circle using physical measuring devices. Right. >But let's imagine someone measuring the diameter of this perfect >circle with a perfectly precise measuring device Are you talking about a _physical_ measuring device? If so, how can you use a physical measuring device to measure a non-physical object? To measure a mathematical object, we can only use other mathematical objects. That was the point of using polygons to make successively closer approximations to the circumference of a circle. >that could give >infinite measurement if required. (This also assumes the line segment >representing the diameter of the circle is perfectly straight and >passes through the real centre of the circle.) The only thing that can behave this way is an equation, or a geometric construction, which amounts to the same thing. >Since pi is infinite, we could never be able to visualize this >perfect line touching the true edge at either poles of the circle. > >And this is my question: > >If we were trying to visualize the infiniteness of pi based on the >whole technique of using diameter to find circumference, is the >infiniteness of pi explained by the fact that we could never get the >diameter to intersect exactly with the circle because of its >perfectly symmetric curvature? No, because we're not really using diameter to find circumference. We're using various methods to find the circumference and the diameter separately, and then finding pi by taking the ratio of circumference to diameter. Only after we're confident that the ratio of circumference to diameter is constant for all circles can we go back and start using that ratio to find circumference in terms of diameter, or vice versa. Note that if we create a perfect mathematical circle, and create a perfect mathematical line passing through its center, the line _will_ intersect the circle at two points. Now, suppose we define the circle in such a way that its circumference is a rational number, like 1. The diameter will be diameter = circumference / pi a rational divided by an irrational... so as with pi, it won't be possible to specify the diameter with a terminating or repeating decimal expansion. On the other hand, suppose we define the circle in such a way that the diameter is a rational number, like 1. The circumference will be circumference = pi * diameter a rational times an irrational... so as with pi, it won't be possible to specify the circumference with a terminating or repeating decimal expansion. It has less to do with the perfect curvature of the circle, or the perfect straightness of the line, than with the properties of rational and irrational numbers. >I mean, if you could imagine drawing a perfectly straight line >through the centre of a perfect square, it would not be hard to >demonstrate that this line has a definite beginning and end, and runs >exactly perpendicular to the two sides of the square, because they >are flat. Actually, you'd have the same problem. To see why, imagine that we have a square, and that we define the largest circle that will fit inside it. This circle touches the square at four points, i.e., the midpoint of each side of the square. A vertical or horizontal line segment passing through the center of the square will also be a diameter of the circle. If one has a definite beginning and end, then so does the other. To make this more definite, let's say that the corners of the square are at (1,1), (-1,1), (-1,-1), and (1,-1). The circle is centered at the origin and has radius 1. The line segment from (0,-1) to (0,1) is the kind of segment you're describing; and it's also a diameter of the circle. The length is very definite, and very rational: 2. Similarly for the line segment from (-1,0) to (1,0). >So I'm sure you could work out a formula to demonstrate the ratio of >this line to the perimeter of this perfect square and show that this >ratio is a finite number. The length of this line segment would be the same as the length of a side of the square, right? So what if I define the square so that the length of each side is sqrt(2), or pi? Then it's an irrational number. I wonder if some of the problem that you're having with this comes from your use of the word "infinite" to describe what are really finite, irrational numbers. Because the decimal expansion of a number doesn't terminate or repeat, that doesn't mean that the number is infinite. The square root of two is a very finite number--it's the length of the hypotenuse of a right triangle whose legs have length 1. And we can specify it exactly by using square root or exponential notation: __ 1/2 \| 2 = 2 We just can't write out the decimal expansion using a finite number of digits. But that's an issue of notation, not size. An "infinite" number, on the other hand, would be something like the size of the set of integers. But we really use the term "transfinite" to talk about those numbers. >Unless there is an error in my reasoning, it seems like the reason pi >is irrational is that the perfectly circular shape of the circle >makes it impossible to reach a conclusive measurement of its true >diameter, and therefore the relation thereto to its circumference. Again, I'm not quite clear on what you mean by this. There are certainly interpretations of this statement that would make the statement correct, but I don't know if those are consistent with your understanding of what you're saying. I think the main difficulty is probably that you're trying to import the concept of direct measurement, which is a physical concept, into mathematics. We can't really measure things in mathematics, at least not the way we do in the world, by putting things next to each other and comparing the lengths. The most we can do is define the dimensions of some objects, and then use deduction to draw conclusions about other dimensions. >And I suppose since it can be proven that pi is infinite, the >hypothetical diameter of a perfect circle must also be infinite. Not at all. When we create a circle, we get to _define_ its diameter, or its circumference. We can specify that the circumference of a circle is 1 unit, or 5 units, or 12 times pi units, or whatever we want. Similarly, we can specify that the diameter is 1 unit, or 11 units, or sqrt(29) units. However, having specified either the circumference or the diameter, we know that the other is constrained by the _rule_ that the ratio of circumference to diameter has to be pi. That rule is what we use instead of a measuring tape. But we can't just say "Let a circle have diameter 1 and circumference 5", any more than we can say "Let a square have a side length of 1 and an area of 5". Or rather, we _can_ say it, but only if we're willing to be flexible about the shape of the space we're operating in! But in flat, Euclidean space, as soon as we specify that a particular shape is a "circle" or a "square", we're saying that it has a particular definition, and obeys particular rules. >It's >amazing to think that an infinite line could exist within the closed >space of a perfect circle. See above regarding this use of the word "infinite"... but in any case, is it more amazing than thinking that an infinite number of points can be contained in a finite line segment? >I bet the secret of the universe is >contained within this contradiction. I'll bet it's not. :^D The reason I'd make that bet is that what happens in mathematics has no necessary connection to what happens in the world. In mathematics, we make up definitions, and then play with them to see what conclusions we can derive from them. These definitions are not constrained by reality in any way. For example, in mathematics, we can define sets of infinite size, like the whole numbers: 0, 1, 2, 3, ... If the universe is finite (which it might be), then a set like this doesn't correspond to anything in the universe. But that doesn't mean we can't define the set, and play with it. Similarly, in mathematics, we can define continuous sets, like the real numbers between 0 and 1. Between any two numbers in this set, we can always find another number. If the structure of the universe is discrete (which it might be), then this kind of continuity doesn't correspond to anything in the universe. But that doesn't mean we can't define the concept, and play with it. And so on, and so on. As I said earlier, mathematics is really good for setting up analogies to real world objects and processes. For example, we might look at something like a container of water, and decide that for all practical purposes, the volume of the water in the container can be modeled using a real number. But if we start taking out half the water, then half the remaining water, and so on, at some point we get down to a single water molecule, right? But that never happens with the numbers. We can just keep dividing numbers in half forever, always getting more numbers. Does this mean that there is something wrong with the mathematics, or with our description of the world? Not at all. It just means that if you push _any_ analogy hard enough, it will eventually break. Where people get confused is in forgetting that when we use math to describe the world, we're _just_ making analogies. >This may sound confusing, but I assure you I'm not crazy. I don't think you're crazy. I think that like most people, you were taught to think that mathematics is somehow the "language of the universe". I was taught that, too. But it's not true. It's a language used by people who are trying to describe the universe. So I'd say that what you _are_ is ready to make a conceptual breakthrough that never occurs to most people about the nature of mathematics, and its relationship to the world. And when you make that breakthrough, I think you're going to find that mathematics is a lot more interesting, and a lot more fun, than you've ever imagined. - Doctor Ian, The Math Forum http://mathforum.org/dr.math/ Date: 12/18/2003 at 12:24:03 From: Mark Subject: Thank you Well, that definitely clears up my question and I've had a conceptual breakthrough. Thank you for the insight. I will never look at the physical world the same way. I now realize that at a certain level, math becomes more art than science. |
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