Csi whodunnit math answers the Pythagorean theorem


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  1. 11.12.2004, 10:15#1

    Pythagorean theorem

    Hi, so my problem:
    For a specialist thesis in math, we should choose a proof for the Pythagorean theorem and describe it, i.e. create a proof (of course not our own). Now I have informed myself and found a lot of evidence, but I don't know how to explain it, can someone post me an explanation without the sentence being evidence for himself?

    Pythagorean theorem:
    a² + b² = c²

    And what about the reverse of the sentence?

  2. 12/11/2004, 3:12 PM Top # 2

    37 years old
    626 articles since 12/2003
    so I'll give it a try

    for a triangle with a right angle:

    a -> Kathethe
    b -> opposite cat
    c -> hypotenuse

    the squares with the lengths of the cathetus (ie a² + b²) together have exactly the area of ​​the square with the lengths of the hypotenuse (pronounced c²).

    the reverse ?!

    maybe

    a² = c²-b²
    b² = c²-a²

    ?!

    "without that the sentence is evidence of itself?"

    I don't understand ... just insert numbers ?!

  3. December 14th, 2004, 5:05 pm Top # 3

    41 years old
    from Ulm
    332 posts since 11/2002
    Quote by Angel1
    Hi, so my problem:
    For a specialist thesis in math, we should choose a proof for the Pythagorean theorem and describe it, i.e. create a proof (of course not our own).
    The following pages may help you:


    Quote by Angel1
    Now I have informed myself and found a lot of evidence, but I don't know how to explain it,
    I think at least one of these procedures, which is presented and explained on the pages mentioned (I'm sure you've found something like that in this direction), can be expanded a bit so that it can be conveyed to classmates in an understandable way (and that's what it's about yes, probably, right?).

    Quote by Angel1
    can someone post me an explanation without that the sentence is evidence of itself?
    When I read your post this morning, I initially thought of possibly proving the sentence vectorially (e.g. via the scalar product). But somehow I stumbled upon exactly the problem, namely that you had to use some things that actually come from the Pythagorean Theorem. For example the Euclidean norm (length) of a vector. Then there is the question of what is defined and what is inferred from what (or set up as a sentence).

    I therefore find these geometric proofs nicer, because there you don't run the risk of using the theorem yourself again in the proof (as if you do it vectorially, for example) (and then it would no longer be a proof).

    Quote by Angel1
    And what about the reverse of the sentence?
    What do you mean exactly by "inversion"? That from a² + b²! = C² it follows that a triangle is not right-angled ("! =" Stands for "not equal to")? I would only conclude that logically if the proposition itself is proven.

    According to the motto ("A -> B" means: "From A follows B" or "If, then").
    • Pythagorean theorem says: Right-angled triangle is present -> a² + b² = c² is always fulfilled.
    • Pythagorean theorem is proven.
    • Logical reasoning: If the following applies: A -> B, then also: (not B) -> (not A).


    I have no idea whether this is enough for you or whether you can get ahead with it. HTH!

    Greeting,
    Steffen

  4. 12/14/2004, 5:29 PM Top # 4

    41 years old
    from Ulm
    332 posts since 11/2002
    Quote by Warrior
    so I'll give it a try

    for a triangle with a right angle:

    a -> Kathethe
    b -> opposite cat
    c -> hypotenuse

    the square with the length of the cathetus (ie a² + b²) together has exactly the area of ​​the square with the length of the hypotenuse (pronounced c²).
    You have reproduced the Pythagorean theorem nicely in words, but you have not proven it in general. ;-)

    Quote by Warrior
    "without that the sentence is evidence of itself?"

    I do not understand...
    In the proof of an assertion or a sentence, the content of the sentence itself must not be used. In the proof you can only use things that are defined (axioms) or sentences and assertions that have already been proven. In a proof, you trace a statement (such as the Pythagorean theorem, for example) back to simple, obvious things or to other, already proven theorems.

    An example of "false evidence" where exactly this is not taken into account. Suppose I want to prove that 3 = 5 (which is bullshit).

    Then I could just go and write:
    [center] 3 = 5 (equation 1) [/ center]

    From this I can in turn conclude:
    [center] 5 = 3 (equation 2) [/ center]

    In the end, I just turned the "equation Eq. 1" around. Yes, that's allowed. If I go over and add the two "equations" (which are actually obviously nonsense) (which you are also allowed to do if you do it on both sides), then it says:

    [center] 5 + 3 = 3 +5 [/ center]

    And that simplifies to:
    [center] 8 = 8 [/ center]

    And that this is true ("8 = 8") is confirmed by every child. So you can infer and say that "3 = 5". ;-)

    Well, where was the mistake: I used the (wrong) statement "3 = 5" directly in the proof itself: I started with it in the proof.

    But if I am to prove an assertion (such as the Pythagorean theorem), I cannot assume that it is correct, I have to show it. Precisely for this reason I am not allowed to use the possibly false assertion (as long as it is not proven, it may be false) in the proof. Because, as the trivial example with "3 = 5" shows, it can happen that something wrong turns out to be something right (here: "8 = 8").

    Suppose the Pythagorean theorem is wrong (which of course it is not, but the point is that the person who proves it doesn't know it in advance) and I want to prove it and make the mistake, the theorem in the proof itself to use. Then it can happen that the proof looks logically correct afterwards and one thinks that the Pythagorean theorem is correct.

    Therefore, you always have to be very careful (and with complex proofs you can lose track of things) that you do not use the statement to be proven in its proof. In my opinion, the example with "3 = 5" shows what you can do wrong here.

    Quote by Warrior
    just insert numbers ?!
    That is not proof either. It could be that the world is so crazy that it works for 10,000 numbers, but for 10,001. Don't pay any more.

    You want to show it in general, and using concrete values ​​doesn't work. That would be an example, but no proof of the correctness of a statement that claims to work for all numbers. You are only showing that the sentence applies to this specific example (and that is not the point of the matter).

    The numerical example can still be helpful: If you are presented with a statement that claims to apply to all numbers and you only find one counterexample (e.g. a pair of numbers for which the statement does not apply), then the statement wrong. This means that the numerical example is sufficient to make a (false) statement fall, but is by no means sufficient to prove the general validity of a specific statement.

    By the way, there are a number of statements in mathematics that have not yet been proven (ie "guesswork"). You suspect that they hold (because experience tells you, for example), but so far you have not been able to prove them formally.
    You try to use mainframes or clusters to test whether you might find a contradiction somewhere. But of course the mainframe does not prove anything, because even if it calculates a few hundred years and tries out more and more numbers, this does not say anything about the general validity of the assumption. However, if the calculator only finds a combination of numbers for which the guess fails, you can confidently abort the calculation process and reject the guess.

    I hope it's reasonably clear what I mean here. ;-)

    Greetings,
    Steffen

  5. 12/14/2004, 6:13 PM Top # 5
    I have finished work. Thanks for your tips!

  6. 12/24/2004, 9:47 PM Top # 6
    A specialist job in one day? WoW, I sat on my weeks

  7. 12/24/2004, 10:46 PM Top # 7
    vip: oxy
    37 years old
    out of the callsign after the COLUMN
    1,948 articles since 05/2001
    [edit] wrong ... [/ edit]

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