Question About the Pythagorean Theorem
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Question About the Pythagorean Theorem
I've always had a problem with the Pythagorean Theorem. Ever since I learned about it, I've had this nagging curiosity about why it works. It's always seemed so simple and elegant that I think there must be some fundamental reason why it's true. However, despite looking at a good portion of the 9001 proofs of the Pythagorean theorem, I have yet to find one that explains it satisfactorily. They always involve taking some geometrical shape(s) and shearing, rearranging, redrawing, etc. them in some way that admittedly proves that the Pythagorean Theorem is true despite leaving me no closer to understanding why it is true. Most of these proofs leave me feeling like the authors have found some clever way to point out applications of the Pythagorean Theorem but still have no real understanding of it. IE they can point out how some arrangement of shapes follows the Pythagorean Theorem but would be unable to take Euclid's axioms and make a logical stepbystep argument as to why they imply the Pythagorean Theorem. The only proof that seemed close to what I'm looking for I only vaguely remember; something about why x^2+y^2 was rotationally invariant. However I can't remember what it was.
Anyways, me and this guy I know were having an argument recently. I was taking about how I couldn't find a satisfactory proof (I guess I'm looking for something more like a derivation) and he pulled out some generic geometric proof by rearrangement. I told him that showing something is true isn't the same thing as showing why it's true and he told me that it is and I just don't understand the proof. So my question is: am I right that these proofs don't actually show why the theorem is true, or is he right that showing that something is true is equivalent to showing why it is true? Also, if I am right, is there a derivation of the Pythagorean Theorem that explains it at a more fundamental level than these geometric proofs?
Anyways, me and this guy I know were having an argument recently. I was taking about how I couldn't find a satisfactory proof (I guess I'm looking for something more like a derivation) and he pulled out some generic geometric proof by rearrangement. I told him that showing something is true isn't the same thing as showing why it's true and he told me that it is and I just don't understand the proof. So my question is: am I right that these proofs don't actually show why the theorem is true, or is he right that showing that something is true is equivalent to showing why it is true? Also, if I am right, is there a derivation of the Pythagorean Theorem that explains it at a more fundamental level than these geometric proofs?
Re: Question About the Pythagorean Theorem
You are both right in some sense.
There have been a lot of situations where I read a proof, and was fully convinced by it, but I did not (whether I realized this at the time or not) "deeply understand" it. A little while later (sometimes years), it clicked, suddenly it all made much more sense. In this sense, perhaps your friend is right. It's possible that in some time, one of the proofs that you've read will "click", and you'll realize that you "understand" it, and you didn't before.
On the other hand, it's certainly possible to have two different proofs of a proposition. One that is illuminating, and one that is not. The usual example is combinatorics. Consider this proposition:[math]\sum_{k=0}^n {n \choose k} = 2^n[/math]
Here is one proof, by induction. The base case in n=0. Indeed, [imath]{0\choose 0} = 1 = 2^0[/imath]. The inductive step goes like this:[math]\sum_{k=0}^{n+1} {n+1 \choose k} = \sum_{k=0}^{n+1} {n \choose k} + \sum_{k=0}^{n+1} {n \choose k1} = \sum_{k=0}^n {n \choose k} + \sum_{k=1}^{n+1} {n \choose k1} = 2^n+2^n = 2^{n+1}[/math]
The other proof: Let's count the number of subsets of {1,...,n}. Each element is either in the set or not. So, there are 2^{n} possibilities. But for each k there are [imath]n \choose k[/imath] subsets of size k. So, both sides are counting the same thing, so the numbers must be equal.
The first proof is straightforward, mechanical, and (in my eyes) unsatisfying. The second proof is "the right" proof of the proposition. It shows not only that it is true, but also why it is true. Here, your friend is wrong. I will grant that I don't "understand" the first proof completely (the symbols do all the thinking for me), and if I did, that proof would probably also show why it's true. Even if this is the case, the second proof simply does a better job of helping me understand why the statement is true.
This happens a lot in mathematics. Gauss went through, what, 10 proofs of quadratic reciprocity, and he was never satisfied with any of them. He kept looking for "the right" proof. Erdös imagined a book where God kept all the best proofs, and saying that a proof was "from the book" was a great compliment.
So, no, not all proofs are created equal. All correct proofs show that a proposition is true, but not all are good at imparting a sense of why it's true. If you are unsatisfied with all proofs that you have seen of the Pythagorean theorem, you are right to keep looking for another one. However, it would also be wise to keep some of the ones you have seen in the back of your mind. One of them might "click".
There have been a lot of situations where I read a proof, and was fully convinced by it, but I did not (whether I realized this at the time or not) "deeply understand" it. A little while later (sometimes years), it clicked, suddenly it all made much more sense. In this sense, perhaps your friend is right. It's possible that in some time, one of the proofs that you've read will "click", and you'll realize that you "understand" it, and you didn't before.
On the other hand, it's certainly possible to have two different proofs of a proposition. One that is illuminating, and one that is not. The usual example is combinatorics. Consider this proposition:[math]\sum_{k=0}^n {n \choose k} = 2^n[/math]
Here is one proof, by induction. The base case in n=0. Indeed, [imath]{0\choose 0} = 1 = 2^0[/imath]. The inductive step goes like this:[math]\sum_{k=0}^{n+1} {n+1 \choose k} = \sum_{k=0}^{n+1} {n \choose k} + \sum_{k=0}^{n+1} {n \choose k1} = \sum_{k=0}^n {n \choose k} + \sum_{k=1}^{n+1} {n \choose k1} = 2^n+2^n = 2^{n+1}[/math]
The other proof: Let's count the number of subsets of {1,...,n}. Each element is either in the set or not. So, there are 2^{n} possibilities. But for each k there are [imath]n \choose k[/imath] subsets of size k. So, both sides are counting the same thing, so the numbers must be equal.
The first proof is straightforward, mechanical, and (in my eyes) unsatisfying. The second proof is "the right" proof of the proposition. It shows not only that it is true, but also why it is true. Here, your friend is wrong. I will grant that I don't "understand" the first proof completely (the symbols do all the thinking for me), and if I did, that proof would probably also show why it's true. Even if this is the case, the second proof simply does a better job of helping me understand why the statement is true.
This happens a lot in mathematics. Gauss went through, what, 10 proofs of quadratic reciprocity, and he was never satisfied with any of them. He kept looking for "the right" proof. Erdös imagined a book where God kept all the best proofs, and saying that a proof was "from the book" was a great compliment.
So, no, not all proofs are created equal. All correct proofs show that a proposition is true, but not all are good at imparting a sense of why it's true. If you are unsatisfied with all proofs that you have seen of the Pythagorean theorem, you are right to keep looking for another one. However, it would also be wise to keep some of the ones you have seen in the back of your mind. One of them might "click".
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Re: Question About the Pythagorean Theorem
If you're looking for a proof that really explains why it's true, the best I can think of is this proof of the law of cosines:
Let u and v be vectors in R^{n}. Then [imath]u+v^2 = \sum_{i=0}^n{p_i(u+v)^2} = \sum_{i=0}^n{\left(p_i(u)^2 + 2p_i(u)p_i(v) + p_i(v)^2\right)} = u^2 + v^2 + 2(u \cdot v) = u^2 + v^2 + 2uvcos\theta[/imath], where [imath]\theta[/imath] is the angle between u and v.
Obviously, if u and v are orthogonal then this reduces to the Pythagorean theorem. This shows that the Pythagorean theorem is just a consequence of how we define the magnitude of a vector... Is that the kind of thing you were hoping for?
Let u and v be vectors in R^{n}. Then [imath]u+v^2 = \sum_{i=0}^n{p_i(u+v)^2} = \sum_{i=0}^n{\left(p_i(u)^2 + 2p_i(u)p_i(v) + p_i(v)^2\right)} = u^2 + v^2 + 2(u \cdot v) = u^2 + v^2 + 2uvcos\theta[/imath], where [imath]\theta[/imath] is the angle between u and v.
Obviously, if u and v are orthogonal then this reduces to the Pythagorean theorem. This shows that the Pythagorean theorem is just a consequence of how we define the magnitude of a vector... Is that the kind of thing you were hoping for?
Re: Question About the Pythagorean Theorem
for me, the theorem boils down to cos² x + sin² x = 1, which is proved
 dumbly with exponential (may not be satisfactory)
 by saying cos 2x = 2cos² x  1 = 1  2sin² x (so get a satisfactory geometrical proof of that version)
 dumbly with exponential (may not be satisfactory)
 by saying cos 2x = 2cos² x  1 = 1  2sin² x (so get a satisfactory geometrical proof of that version)
Re: Question About the Pythagorean Theorem
This reminded me of a page I read a while ago: Here it is.
A little way down there's a section called: Intuitive Look at The Pythagorean Theorem
Personnally I didn't find his proof more convincing than any other, but it looks like exactly what you're looking for.
A little way down there's a section called: Intuitive Look at The Pythagorean Theorem
Personnally I didn't find his proof more convincing than any other, but it looks like exactly what you're looking for.
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Re: Question About the Pythagorean Theorem
Wow that's an intuitive proof. Puts the whole thing in easilygraspable terms so that the result just falls out halfway through. I also like all similar examples of other uses PT can be put to at the end.
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Re: Question About the Pythagorean Theorem
Heh, I actually came up with that proof earlier, but forgot all about it. Nice.
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Re: Question About the Pythagorean Theorem
The rearranging is HOW it's true.
The Axioms are WHY it's true. If you want to know WHY something is true, you just need to look at axioms. If you don't assume the axioms, then you've got nothin'
The Axioms are WHY it's true. If you want to know WHY something is true, you just need to look at axioms. If you don't assume the axioms, then you've got nothin'
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Re: Question About the Pythagorean Theorem
My opinion is that your friend is right. "Why?" is a good question to ask when human motives are involved, but it's much less meaningful when it comes to explaining how the world works. In my eyes, asking why in situations like this is a pathetic fallacy.
Re: Question About the Pythagorean Theorem
But you would still agree that some proofs of a proposition are more "illuminating" than others, right?
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Re: Question About the Pythagorean Theorem
The best proofs are probably those that use the most elementary concepts. For example, the Pythagorean theorem is a cakewalk, if you assume similar triangles. However, you're losing the rawness that accompanies more elementary proofs.
I mean, do you want a short proof, with a lot of black boxes, or a longer proof, proven with a minimal tool set? Which best articulates the nature of the system anyway? I don't know if either is more "illuminating."
Either way, I'm partial to Pythagorean proofs that use trapezoids without similar triangles.
I mean, do you want a short proof, with a lot of black boxes, or a longer proof, proven with a minimal tool set? Which best articulates the nature of the system anyway? I don't know if either is more "illuminating."
Either way, I'm partial to Pythagorean proofs that use trapezoids without similar triangles.
Re: Question About the Pythagorean Theorem
Personally I prefer a reason or a "why" to the truth of a theorem too, rather than a just a "proof" which proves it to be true but gives no deeper insight as to how it comes to be true, I always feel something is missing if I don't have this . I think there is a word for mathematicians who are of the belief that a theorem is only truly proved via such an construction , funnily enough this word is "constructionist". I read that there are constructionists who for example believe that the diagonal proof shows nothing but that there is no onetoone mapping from the rationals to the reals, as opposed to it proving the conecpt of different "levels" of infinity
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Re: Question About the Pythagorean Theorem
antonfire wrote:But you would still agree that some proofs of a proposition are more "illuminating" than others, right?
Yes with a but. There are statements and explanations about any proposition that more illuminating than others. Whether these are also proofs or not really has nothing to do with it. For me, I want a good, convincing proof and enough explanation to have a full understanding of the proposition. But these are not always, or even often, the same, and forcing them to overlap I think is not worthwhile.
Whether it's a proof or not, I don't think an explanation should strive to answer "why" but rather "how", anyway. A much better diagnostic of understanding in my opinion than "why" (because it's so vague) is "would the proposition hold if things were different in a certain way?" Like is the Pythagorean Theorem true in nonEuclidean space.

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Re: Question About the Pythagorean Theorem
I think that maybe there isn't an "intuitive" proof for the Pythagorean Theorem. Maybe there never will be. And there certainly doesn't have to be; it was probably discovered by accident. Some guy might have been playing with right triangles, observed that this was true for all those that he tried, and then tried to prove that it would be true for any triangle.
Perhaps the P.T. is just up there with other questions like "Why do atoms tend towards their lowest energy state?", something that has been observed(/proven) to be true but is hard (or perhaps impossible) to really explain why it is in fact true.
Perhaps the P.T. is just up there with other questions like "Why do atoms tend towards their lowest energy state?", something that has been observed(/proven) to be true but is hard (or perhaps impossible) to really explain why it is in fact true.
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Re: Question About the Pythagorean Theorem
So, take the sqrt of each side:
sqrt(a*a + b*b) = c
That is the distance equation for flat euclidean space. You can change this and end up with spaces that are "shaped" differently 
c = ( a^{n} + b^{n} )^{1/n}
is a class of norms called the L_{n} norms. In that case:
c^{n} = a^{n} + b^{n}
for an arbitrary n!
In short, another way of looking at this kind of "why" question is find a related question for which the answer is different.
(Note that the L^{n} norms are not, in general, inner product spaces  so the question of "what does orthogonal mean" is a bit trickier!)
sqrt(a*a + b*b) = c
That is the distance equation for flat euclidean space. You can change this and end up with spaces that are "shaped" differently 
c = ( a^{n} + b^{n} )^{1/n}
is a class of norms called the L_{n} norms. In that case:
c^{n} = a^{n} + b^{n}
for an arbitrary n!
In short, another way of looking at this kind of "why" question is find a related question for which the answer is different.
(Note that the L^{n} norms are not, in general, inner product spaces  so the question of "what does orthogonal mean" is a bit trickier!)
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Re: Question About the Pythagorean Theorem
It seems to me that you don't want a proof per se of the PT, but rather the trail of thinking that Pythagoreas (or whoever first figured it out) had when he wrote down that beautiful equation. How did someone figure out that the square of the length of any right triangle's hypotenuse was equal to the square of lengths of the other two sides summed together? If that is what you're trying to answer, then I'm right along with you, but I don't let it nag me as much.
That's not true, I worry worry worry about stuff I don't fully understand all the time and I fear that when I die I still won't know enough physics or mathematics to be satisfied.
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Re: Question About the Pythagorean Theorem
My favorite "proof" is the following image (stolen from wikipedia). C^{2} = A^{2} + 2AB + B^{2}  4(.5 * AB)
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Re: Question About the Pythagorean Theorem
I found a nice algebraic proof on Wolfram's Mathworld. In complex numbers, real numbers and imaginary numbers are orthogonal. The addition of them gives a complex number which can be written in polar form giving it's length and angle: [imath]a + bi = c e^{\theta i}[/imath].
To this, we can take the complex conjugate of both sides: [imath]a  bi = c e^{\theta i}[/imath]. Now, multiplying these two equations, we get:
[math](a + bi) (a  bi) = c e^{\theta i} c e^{\theta i}[/math]
[math]a^2  (b^2) = c^2 e^{\theta i  \theta i}[/math]
[math]a^2 + b^2 = c^2 e^0[/math]
[math]a^2 + b^2 = c^2[/math]
I was happy to find this proof. I keep running into other domains where the fricken' 2norm runs around with an unexplained sense of importance. Now I can think about this proof and try and justify its iniquitousness in mathematics.
To this, we can take the complex conjugate of both sides: [imath]a  bi = c e^{\theta i}[/imath]. Now, multiplying these two equations, we get:
[math](a + bi) (a  bi) = c e^{\theta i} c e^{\theta i}[/math]
[math]a^2  (b^2) = c^2 e^{\theta i  \theta i}[/math]
[math]a^2 + b^2 = c^2 e^0[/math]
[math]a^2 + b^2 = c^2[/math]
I was happy to find this proof. I keep running into other domains where the fricken' 2norm runs around with an unexplained sense of importance. Now I can think about this proof and try and justify its iniquitousness in mathematics.
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Re: Question About the Pythagorean Theorem
Let <.,.> be an arbitrary inner product.
Let <a,b> = 0  ie, a and b are orthogonal.
If c = a+b, then
<c,c> = <a+b,a+b> = <a,a>+2<a,b>+<b,b> = <a,a>+<b,b> [Footnote 1]
Let . be the natural norm on the inner produce space. Then we have
c^{2} = a^{2} + b^{2}
whenever a and b are orthogonal and c=a+b
("orthogonal" is the generalization of "at right angles"  in short, the theorem holds in any space where "at right angles" or "orthogonal" makes sense. . .)
[Footnote 1]  this is only strictly correct with a realvalued norm. With a nonrealvalued norm, you end up with 2 Re<a,b> instead of 2 <a,b>. But that still ends up being 0, so it doesn't matter.
Let <a,b> = 0  ie, a and b are orthogonal.
If c = a+b, then
<c,c> = <a+b,a+b> = <a,a>+2<a,b>+<b,b> = <a,a>+<b,b> [Footnote 1]
Let . be the natural norm on the inner produce space. Then we have
c^{2} = a^{2} + b^{2}
whenever a and b are orthogonal and c=a+b
("orthogonal" is the generalization of "at right angles"  in short, the theorem holds in any space where "at right angles" or "orthogonal" makes sense. . .)
[Footnote 1]  this is only strictly correct with a realvalued norm. With a nonrealvalued norm, you end up with 2 Re<a,b> instead of 2 <a,b>. But that still ends up being 0, so it doesn't matter.
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Re: Question About the Pythagorean Theorem
Yakk wrote:Let . be the natural norm on the inner produce space. Then we have
c^{2} = a^{2} + b^{2}
whenever a and b are orthogonal and c=a+b
I've used that idea many times (in both C and general i.p. spaces), but never before seen that explicit connection to Pythagoras. That's a really nice proof.
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Re: Question About the Pythagorean Theorem
I can't help shake the feeling that some of the "proofs" are running around in circles  that the fact that they prove the Pythagorean Theorem is only because they're defined in a system that was designed such that it would hold. For example, the norm and complex number proofs both rely on a definition of "orthogonal" which presumably relates to the definition of the trigonometric functions, which in turn is based on right angled triangles. I'm not saying they're definitely wrong, but I can't see a line of thought that avoids that circular reasoning.
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Re: Question About the Pythagorean Theorem
The trig functions can be defined analytically, without reference to right triangles.

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Re: Question About the Pythagorean Theorem
Heh, I'm surprised no one has brought up the fact that the theorem actually has to be 'postulated' in the more general sense.
All the proofs you've seen involve drawing geometric shapes and shearing them and such because the Pythagorian theorem actually furnishes only a single particular way to measure distances on a manifold. All Riemannian manifolds are locally flat and consequently locally satisfy this rule, but in general Pythagorian's theorem is not something that can be derived via any analytical method. There is a bilinear, symmetric function that tells us how to measure distance on a manifold: the metric. Riemannian manifolds are locally equipped with a Euclidean metric, making the Pythagorian theorem hold true.
PseudoRiemannian and nonRiemannian manifolds don't have to satisfy the Pythagorian theorem. Spacetime, for instance, is a pseudoRiemannian manifold equipped locally with a Minowski metric, not a Euclidean one. If one side of a 'triangle' extends in the t direction and another in the x direction, the segment joining the two edges has a length (t^2)+(x^2). There is a distinct sign difference for the t coordinate than what you would expect by naively applying the P. theorem.
So there ya go. That might be why it's always seemed strange to you.
All the proofs you've seen involve drawing geometric shapes and shearing them and such because the Pythagorian theorem actually furnishes only a single particular way to measure distances on a manifold. All Riemannian manifolds are locally flat and consequently locally satisfy this rule, but in general Pythagorian's theorem is not something that can be derived via any analytical method. There is a bilinear, symmetric function that tells us how to measure distance on a manifold: the metric. Riemannian manifolds are locally equipped with a Euclidean metric, making the Pythagorian theorem hold true.
PseudoRiemannian and nonRiemannian manifolds don't have to satisfy the Pythagorian theorem. Spacetime, for instance, is a pseudoRiemannian manifold equipped locally with a Minowski metric, not a Euclidean one. If one side of a 'triangle' extends in the t direction and another in the x direction, the segment joining the two edges has a length (t^2)+(x^2). There is a distinct sign difference for the t coordinate than what you would expect by naively applying the P. theorem.
So there ya go. That might be why it's always seemed strange to you.
Re: Question About the Pythagorean Theorem
Never had the 'diagrams are not proofs' rule beaten into you, huh? Actually that image does remind me of the geometric proof that I can't quite remember and I'm fairly sure that the nonrigorous sketch proof that one would think of based on that is the most illuminating.jmorgan3 wrote:My favorite proof is the following image
He means to say, the Pythagorean only really holds on flat surfaces: it's a theorem in Euclidean geometry which is where distances works how you expect them to as opposed to nonEuclidean geometries, where they don't.hitokiriilh wrote:something
Re: Question About the Pythagorean Theorem
jmorgan3 wrote:My favorite proof is the following image (stolen from wikipedia). C^{2} = A^{2} + 2AB + B^{2}  4(.5 * AB)
Holy snaps, that's awesome.
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Re: Question About the Pythagorean Theorem
the tree wrote:Never had the 'diagrams are not proofs' rule beaten into you, huh? Actually that image does remind me of the geometric proof that I can't quite remember and I'm fairly sure that the nonrigorous sketch proof that one would think of based on that is the most illuminating.jmorgan3 wrote:My favorite proof is the following image
I haven't heard that rule, but I'm not a math major. The diagram does help me see the 'why' of the Pythagorean theorem, more so than any rigorous proof. Could you explain why it doesn't prove PT?
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Re: Question About the Pythagorean Theorem
It's a valid proof, but a diagram can easily hide some premises that seem "obvious" from the diagram, but still need to be proven.
In this case, for instance, it hasn't shown that, if you take a square with sidelengths of C, and add those 4 righttriangles, that you get a bigger square. It's true, you do, but it still has to be proven in order for the full proof to be rigorous.
The "diagrams are not proofs" rule is basically because, while in this case the premises implied by the diagram are true, in other cases they can be false (and just look true because of a misleading diagram). See: that erroneous proof that all angles are equal... the diagram shows an angle going one way, and the proof assumes it goes that way, but it actually goes the other way.
In this case, for instance, it hasn't shown that, if you take a square with sidelengths of C, and add those 4 righttriangles, that you get a bigger square. It's true, you do, but it still has to be proven in order for the full proof to be rigorous.
The "diagrams are not proofs" rule is basically because, while in this case the premises implied by the diagram are true, in other cases they can be false (and just look true because of a misleading diagram). See: that erroneous proof that all angles are equal... the diagram shows an angle going one way, and the proof assumes it goes that way, but it actually goes the other way.
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Re: Question About the Pythagorean Theorem
That's very interesting. I hadn't realized all the assumptions that go into making even a simple diagram. I've added quotes around "proof" in my first post to acknowledge its proper status.
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EDIT: Only .05 of a second from posting on 31415. Darn.
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Re: Question About the Pythagorean Theorem
the link below will 100% satisfy you.
http://www.scribd.com/doc/5371209/pythagorastheorem
http://www.scribd.com/doc/5371209/pythagorastheorem

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Re: Question About the Pythagorean Theorem
Macbi wrote:This reminded me of a page I read a while ago: Here it is.
A little way down there's a section called: Intuitive Look at The Pythagorean Theorem
Personnally I didn't find his proof more convincing than any other, but it looks like exactly what you're looking for.
That's an "Aha!" proof if I ever saw one. And (to me at least) it answers the WHY question. Why does c² = a² + b² hold for a right triangle?
Spoiler:
The formal (nongraphic) proof:
Spoiler:
They'll learn to like it someday.
 Ludwig van Beethoven
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Re: Question About the Pythagorean Theorem
lalitwarkde wrote:the link below will 100% satisfy you.
http://www.scribd.com/doc/5371209/pythagorastheorem
I have many a problem with this "proof", not the least of which is the fact that he seems to pull out the [imath]\sqrt{2}[/imath] factor for the diagonal of a square out of thin air.
 gmalivuk
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Re: Question About the Pythagorean Theorem
Yeah. Seems like anyone satisfied for *why* sqrt(2) plays a part is already going to be pretty satisfied by why the PT is true.
Re: Question About the Pythagorean Theorem
lalitwarkde wrote:the link below will 100% satisfy you.
http://www.scribd.com/doc/5371209/pythagorastheorem
That link is terrible.
 NathanielJ
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Re: Question About the Pythagorean Theorem
lalitwarkde wrote:the link below will 100% satisfy you.
http://www.scribd.com/doc/5371209/pythagorastheorem
Am I the only one who thinks that link was posted as a joke...?
Re: Question About the Pythagorean Theorem
Else, if any of the given proofs does not satisfy you I recommend looking at http://www.cuttheknot.org/pythagoras/index.shtml. There are, according to the site, 79 proofs of the Pythagorean Theorem.
Re: Question About the Pythagorean Theorem
TacTics wrote:I found a nice algebraic proof on Wolfram's Mathworld. In complex numbers, real numbers and imaginary numbers are orthogonal. The addition of them gives a complex number which can be written in polar form giving it's length and angle: [imath]a + bi = c e^{\theta i}[/imath].
To this, we can take the complex conjugate of both sides: [imath]a  bi = c e^{\theta i}[/imath]. Now, multiplying these two equations, we get:
[math](a + bi) (a  bi) = c e^{\theta i} c e^{\theta i}[/math]
[math]a^2  (b^2) = c^2 e^{\theta i  \theta i}[/math]
[math]a^2 + b^2 = c^2 e^0[/math]
[math]a^2 + b^2 = c^2[/math]
I was happy to find this proof. I keep running into other domains where the fricken' 2norm runs around with an unexplained sense of importance. Now I can think about this proof and try and justify its iniquitousness in mathematics.
A formal proof should be in terms of the axioms of the logical system. So in this case a formal proof should be in terms of the ZermeloFraenkel axioms of set theory and the field axioms of the real numbers.
Actually I like the proof given above in terms of exp(i theta). Many students learn the definition that exp(i theta)= cos(theta)+isin(theta) and learned that cos^2(theta)+isin^2(theta)=1 by using the pythagorean theorem. But we can take an alternate definition of this function so that the proof doesn't depend on the Pythogrean Theorem (of course the proof of the Pythagorean Theorem shouldn't depend on the Pythagoreath Theorem!).
We can define the function by it's power series exp(i theta)=sum from n=0 to infinity( (i*theta)^n/n!) and define cos(theta)=Real(exp(i theta)) and sin(theta)=Imag(exp(i theta). Of course to really use the above proof you need to show that 1) The power series converges. 2)exp(z1)exp(z2)=exp(z1+z2). 3) For any complex number z, there exists a complex number w such that z=exp(i w). To prove all of this in terms of the axioms mentioned above you would have to take a course in real analysis. These proofs are given in the first pages of Real and Complex Analysis by Walter Rudin. So it really does take alot of work to prove it in terms of the basic axioms (at least a few pages!). Good luck!
Re: Question About the Pythagorean Theorem
The "point" of the Pythagorean theorem is that the definition of distance is invariant under rotation. From the modern perspective, rotation is actually the more fundamental concept, and distance (and the Pythagorean theorem) arises naturally from it, rather than the other way around. Any proof using properties of the complex numbers is using the fact that the structure of the complex numbers encodes in a fundamental way the properties of 2dimensional rotation.
There are several ways to really drive home this point. One is the following "physical" proof of the Pythagorean theorem. Consider a fish tank filled with water in the shape of a triangular prism, so that there are two identical right triangles with sides a, b, c, a', b', c' and opposite vertices A, B, C, A', B', C' such that P and P' are connected by parallel edges for P = A, B, C. Attach the edge BB' to a pole and allow it to pivot freely. This system is in static equilibrium, so the total torque of the fish tank about the pole must be zero. But the torque must be proportional to c^2  a^2  b^2. (To see this, note that the water exerts pressure on each rectangular face proportional to the corresponding side length of the triangle. It helps to draw the diagram and stare at it carefully.)
The point of this proof is that the definition of torque is invariant under rotation.
If you want to go as far as using power series, you should do it with a clear sense of why the sine and cosine have particular power series representations. These power series ultimately come from the fact that a particle whose acceleration is perpendicular to and proportional to its velocity travels in a circle. When thinking about axiomatic mathematics it is never a good idea to lose sight of your physical and geometric intuition.
There are several ways to really drive home this point. One is the following "physical" proof of the Pythagorean theorem. Consider a fish tank filled with water in the shape of a triangular prism, so that there are two identical right triangles with sides a, b, c, a', b', c' and opposite vertices A, B, C, A', B', C' such that P and P' are connected by parallel edges for P = A, B, C. Attach the edge BB' to a pole and allow it to pivot freely. This system is in static equilibrium, so the total torque of the fish tank about the pole must be zero. But the torque must be proportional to c^2  a^2  b^2. (To see this, note that the water exerts pressure on each rectangular face proportional to the corresponding side length of the triangle. It helps to draw the diagram and stare at it carefully.)
The point of this proof is that the definition of torque is invariant under rotation.
If you want to go as far as using power series, you should do it with a clear sense of why the sine and cosine have particular power series representations. These power series ultimately come from the fact that a particle whose acceleration is perpendicular to and proportional to its velocity travels in a circle. When thinking about axiomatic mathematics it is never a good idea to lose sight of your physical and geometric intuition.
 doogly
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Re: Question About the Pythagorean Theorem
phlip wrote:It's a valid proof, but a diagram can easily hide some premises that seem "obvious" from the diagram, but still need to be proven.
In this case, for instance, it hasn't shown that, if you take a square with sidelengths of C, and add those 4 righttriangles, that you get a bigger square. It's true, you do, but it still has to be proven in order for the full proof to be rigorous.
The "diagrams are not proofs" rule is basically because, while in this case the premises implied by the diagram are true, in other cases they can be false (and just look true because of a misleading diagram). See: that erroneous proof that all angles are equal... the diagram shows an angle going one way, and the proof assumes it goes that way, but it actually goes the other way.
That's an example of an incorrectly drawn diagram not being a valid proof. But an incorrectly written proof is also invalid!
"Diagrams are not proofs" is a terrible rule. "I have a harder time proofreading diagrams than lines of text" is a totally valid matter of taste that seems to be popular but is by no means a matter of fact.
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Noc: A larval Doogly. They grow the tail and stinger upon reaching adulthood.
Keep waggling your butt brows Brothers.
Or; Is that your eye butthairs?
Re: Question About the Pythagorean Theorem
I guess your question has more to do with philosophy than with mathematics. I remember discussing this issue at length in college during a class called Fundamentals of Mathematics. In essence, you are wondering if a proof satisfies your understanding of the truth that it ultimately demonstrates. Of course, showing that something is true doesn't mean that you understand why it is true. In a way the beauty of mathematics is that we can find truth in things that to some extent are so difficult to understand that if it weren't for the abstraction underlying it, it might not be possible to come to know these truths. Sometimes, the understanding comes after the proof, sometimes it's the other way around. The feeling one has when one cannot reconcile two contradictory beliefs is known as cognitive dissonance. Often, it is this feeling that drives mathematical derivation. Sometimes it is plain intuition.
In short, I think what you meant is that understanding why something is true is not the same as showing that it is true, or knowing why it is true. The main reason both you and your friend were right is that you were using two different meanings for "why". You were searching for understanding while he was searching for logical argument. Since the proofs you read don't match the way you understand the Pythagorean theorem, you feel at odds with the proofs you have found so far.
At least that's what I gather.
In short, I think what you meant is that understanding why something is true is not the same as showing that it is true, or knowing why it is true. The main reason both you and your friend were right is that you were using two different meanings for "why". You were searching for understanding while he was searching for logical argument. Since the proofs you read don't match the way you understand the Pythagorean theorem, you feel at odds with the proofs you have found so far.
At least that's what I gather.
 Yakk
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Re: Question About the Pythagorean Theorem
doogly wrote:That's an example of an incorrectly drawn diagram not being a valid proof. But an incorrectly written proof is also invalid!
"Diagrams are not proofs" is a terrible rule. "I have a harder time proofreading diagrams than lines of text" is a totally valid matter of taste that seems to be popular but is by no means a matter of fact.
A value of a proof over the proposition it proves is that it is easier to proofread the proof.
The proposition and the axioms already tell you if it is a true statement, assuming it isn't indeterminate.
What the proof provides is an easiertoverifythangenerate argument that the statement is indeed true.
The problem with geometrical proofs is that they are difficult to verify. Now, many good geometrical presentations of proofs are amazingly succinct ways to express a way to symbolically prove the statement without having to write out the symbols, and with sufficient capability to do in the reader this is valid.
Back to the original posters issue with "how, but not why": many proofs are generated one way, then displayed the other way. When they present a proof, they present you with a polished exterior to "what was going on in order to generate the proof". In the case of ancient mathematics, multiple people may have done this, rebuilding a portion of the proof in order to make it more slick.
This often generates a beast that is, as I noted, easy to verify  but it doesn't help you (other than exposing you to tricks) understand what was going through the heads of the people who actually generated it, which can help you with generating proofs yourself. And being able to regurgitate a proof of a theorem is not the same as being able to build the theorem's proof on demand: the second is closer to the problem of "why" than the first.
One of the painful things about our time is that those who feel certainty are stupid, and those with any imagination and understanding are filled with doubt and indecision  BR
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