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How is π interpreted?

  1. Jul 1, 2015 #1
    The area of a circle is πr2. Here π is the constant that represents the ratio of the circumference to the diameter. But in deriving πr2, 2π is used as an angle, which is the upper bound of the integral ∫½ r2 dθ. So how the same π is used in 2 different meaning?
     
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  3. Jul 1, 2015 #2
    Area of a circle:

    A = πr2. π is the number that makes that equation true.

    That's the definition my calculus 1 professor gave the class.
     
  4. Jul 1, 2015 #3

    Mentallic

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    It's not two different meanings at all. If you instead used [itex]\tau=2\pi[/itex] (pronounced tau) which is in most ways a more appropriate circle constant, then the upper bound to the integral would be [itex]\tau[/itex], the circumference would be [itex]C=\tau r[/itex] and the area of the circle would be [itex]A=\frac{1}{2}\tau r^2[/itex].

    The definition of ##\pi## being the ratio of the circumference to the diameter was useful back in the day when diameters were used most often since they were easier to measure than the radius is the only reason ##\pi## is still being used today. It's hard to change from the old.

    Anyway, rant over, back to your question. The circle has a circumference ##C=2\pi r## which means that there are ##2\pi## radians in a full revolution of a circle. This is the reason your upper bound is ##2\pi##. Just because the constant coefficient of the area of a circle is ##\pi## has no bearing on your integral. Similarly if you calculate the volume of a sphere, you won't be using ##\frac{4}{3}\pi## as your upper bound.
     
  5. Jul 1, 2015 #4
    So π means it corresponds to 180 degree but not equal to 180 degree in absolute sense.
     
  6. Jul 1, 2015 #5

    Mentallic

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    ##\pi## is just a number like 1, -23 or ##\sqrt{2}##. If we want to express length, then we have to include units, such as 1cm, -23ft or ##\pi## light years. But when doing geometry and every number corresponding to a side length represents a certain unit length (and it usually doesn't matter whether it's cm, ft, or ly) then we either add the unit u to represent any unit length, or more conveniently, we ignore it altogether and just assume it's a unit length.

    Similarly for radians, we're expected to include units. The symbol for radians are either c or rad, so to represent 180o we write either ##\pi ^c## or ##\pi \text{ rad}##. But again just like before, when we're always dealing with angles, since radians are most commonly used then we can just exclude the radian symbols and it would be implied that they're radians. We don't want to mix this up with degrees though, so that's one reason why we always include the degree symbol (at least one of them has to always be included). No symbol hence implies radians.

    Radians are an angle measure, and degrees are too, so ##\pi^c \equiv 180^o##. They are the same thing just as 2.54cm = 1" (2.54 centimetres = 1 inch).
     
  7. Jul 1, 2015 #6
    Adel. You've asked a very subtle question that involves dimensionality as well as units, that is not in the usual habitate of typical mathematical concern. Good for you. You're thinking.


    It's a good idea not to confuse units and dimensionality of a quantity. For instance, radians per second has dimensions of [R/T] but units of [1/T]. (Most people don't make the distinction.)

    First, let us define ##\pi## as the ratio of the circumference of any circle divided by twice its own radius. It has dimension of length over length, or ##[L/L]## and units of length of length, ##[L/L]##.

    Now, which integral for the area you asking about? There is more than one.
     
    Last edited: Jul 1, 2015
  8. Jul 1, 2015 #7
    If I divide the circle into n triangles each has a small but finite base = r Δθ and sides r, then the area of this triangle is ½ r2 Δθ. The area of the circle i=∑½ r2 Δθ. When Δθ->dθ, ∑->∫. So the area of the circle =∫½ r2 dθ with the upper bound is 2π and the lower one is zero.. The area then= πr2.
     
  9. Jul 2, 2015 #8
    Forget degrees. That there are 360 degrees in a circle is an arbitrary designation. We could have just as easily said, "Let a circle be divided into 6024 equiangular arcs, and let each of these angles be called a "degree."
     
  10. Jul 2, 2015 #9
    This is very cool. But how do we know that ##C = 2\pi r##?
     
  11. Jul 2, 2015 #10
    Or is it that ##\pi## is the number that makes ##C = 2\pi r## true?
     
  12. Jul 2, 2015 #11

    micromass

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    Not really a good definition since it needs to be shown that the constant is independent of ##r##. Who knows, a priori it might be the case that the area of the unit circle is ##\pi##, but the area of a circle with radius ##2## is more like ##A = 3\cdot 2^2##. In fact, that this is not the case is quite deep and depends crucially on the parallel axiom: it is false on a sphere or on a hyperbolic plane. In those contexts, ##\pi## should be seen as related to the area of an infinitesimal circle.
     
  13. Jul 2, 2015 #12

    HallsofIvy

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    Actually it is more common to define [itex]\pi[/itex] as the circumference of a circle divided by the diameter of the circle. After, of course, showing that "all circles are similar"- that is, that circumference divided by diameter is the same for all circles.
     
  14. Jul 2, 2015 #13
    I think proving c=2πr may be inconsistent. Because at one point of the proof, π=c/2r will be again used as an axiom which makes using an axiom to prove it inconsistent by Godel incompleteness theorem.
     
    Last edited: Jul 2, 2015
  15. Jul 2, 2015 #14

    pwsnafu

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    That's not what Godel incompleteness theorems (because there's two) say. More to the point, Euclidean geometry is proven to be consistent and complete, and Godel's theorems don't apply.
     
  16. Jul 2, 2015 #15
    I mean π is equal to the ratio of the circumference to the diameter by definition. Nothing to be proved here.
     
  17. Jul 2, 2015 #16
    Do you mean Euclidean geometry can prove any of its 5 axioms?
     
  18. Jul 3, 2015 #17

    pwsnafu

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    It's important to understand in axiomatic set theory, "axiom" and "proof" are technical terms. The general mathematician's interpretation that "axioms cannot be proved" is inadequate. For a given theory (such as EG or PA or ZF), a subset of all sentences (technically called "well formed formulas") in the theory are designated as "axioms". A "formal proof" of a sentence is a finite sequence which ends with the sentence in question, and only contains axioms, assumptions or sentences which can inferred from previous sentences. This means, suppose "A" is an axiom, then one element sequence
    is infact a formal proof of A itself. In this narrow setting, yes, the axioms of Euclidean geometry have proofs. Another way to look at this is to define axioms as those sentences with trivial proofs.

    Godel's incompleteness theorems are technical results of Peano arithmetic, not Euclidean geometry. It turns out that arithmetic of the natural numbers has a structure that Godel's proof exploits, but that structure is not present in Euclidean geometry. Hence, Godel's theorems don't apply.

    Secondly, Godel's theorems are not talking about axioms. Goodstein's theorem is an example of a true-but-not-provable-within-PA theorem. You'll notice it's very different from the axioms of PA.
     
    Last edited: Jul 3, 2015
  19. Jul 3, 2015 #18
    So, is c=2πr a trivial proof that c=2πr?
     
  20. Jul 3, 2015 #19

    micromass

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    It is not usually taken as an axiom of Euclidean geometry.
     
  21. Jul 3, 2015 #20

    HallsofIvy

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    It is proved in Euclidean geometry that the circumference of any circle is proportional to its radius. [itex]\pi[/itex] is then defined as that constant of proportionality.
     
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