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Math Help - Structure and Method (for the human)

  1. #1
    No one in Particular VonNemo19's Avatar
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    Detroit, MI
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    Structure and Method (for the human)

    Hey guys. I'm looking for someone to do a proof. Well, actually several proofs, but we'll start with something easy (Not easy for me, though. Why? Maybe I'm stupid.) Here's the thing. Someday in the future, I would like to be super awesome at proof writing, and I've been studying the method of many proofs. Pythagoras proved his theorem geometrically with a smaller square insribed in a larger square, with the vertices of the smaller square lying in the four respective sides of the larger square. Simple, yet genious. Riemann, with his rectangles, summing them together, showed that the area under a curve could be approximated to any degree of accuracy. L'hopital, taught us that indeterminate forms were nothing to be afraid of. These people all have something in common. They were able to solve problems. Now I'm not pretending to know how these people thought, but I believe that they're approach to these daunting tasks were not only similar, but EASY.

    In my opinion, I think it goes like this:
    1) most problems start with a different problem. 'Man, if I could just solve this quadratic equation, that would be great!'

    2)Identifying the problem with the problem. 'Why can't I solve this thing?! I shouldn't have to sit here for 2 freakin weeks testing all of the factors of of these enormous coefficients!'

    3)The quest begins. 'O.K. There's got to be an easier way.'

    4)Stating the problem in general terms. 'Well, I'm gonna state this thing in general terms, because I want this to work for all quadratics. I Know that all second degree equations can be written like this: ax^2+bx+c=0'

    5)Racking the brain. 'I've been working on this thing for 4 months. Maybe I should just set myself on fire, wrap a bag around my head, take some pills and jump off of a bridge.'

    6) A coulpe of teases. ' Oh my God! I've got it! Oh. Wait. No I don't. Stupid! stupid! stupid!.'

    7)The eureka. 'Oh! What if I..... That's IT!!! The trinomial square! I'll build a simple, stupid, little trinomial square!'

    8) The final proof. 'OK, let's see here......

    ax^2+bx+c=0

    ax^2+bx=-c

    x^2+\frac{b}{a}x=-\frac{c}{a}

    Now... Ha Ha Ha. I gotcha ya little bastard! You don't stand a chance before the great and powerful trinomial square.'

    x^2+\frac{b}{a}x+(\frac{b}{2a})^2=-\frac{c}{a}+(\frac{b}{2a})^2

    And now for the good stuff.

    (x+\frac{b}{2a})^2=-\frac{c}{a}+\frac{b^2}{4a^2}

    (x+\frac{b}{2a})^2=\frac{b^2-4ac}{4a^2}

    x=\frac{-b\pm{\sqrt{b^2-4ac}}}{2a}

    8) The dance. Whoooooooooooo!!!!!!

    9)The Ego.

    And that's how it goes. I've given this a lot of thought. Think about Old Isaac Newton, and the apple. Sounds like the eureka to me.

    Any way, let's start with something simple.

    The sum of cubes. x^3+a^3=x^2-ax+a^2

    If you don't like this one. pick one. I just want to study your structure and method.

    I CHALLENGE YOU ALL!!! DO A PROOF IF YOU DARE!

    I'll submit one myself on this thread. Give me a minute though.
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by VonNemo19 View Post
    Hey guys. I'm looking for someone to do a proof. Well, actually several proofs, but we'll start with something easy (Not easy for me, though. Why? Maybe I'm stupid.) Here's the thing. Someday in the future, I would like to be super awesome at proof writing, and I've been studying the method of many proofs. Pythagoras proved his theorem geometrically with a smaller square insribed in a larger square, with the vertices of the smaller square lying in the four respective sides of the larger square. Simple, yet genious. Riemann, with his rectangles, summing them together, showed that the area under a curve could be approximated to any degree of accuracy. L'hopital, taught us that indeterminate forms were nothing to be afraid of. These people all have something in common. They were able to solve problems. Now I'm not pretending to know how these people thought, but I believe that they're approach to these daunting tasks were not only similar, but EASY.

    In my opinion, I think it goes like this:
    1) most problems start with a different problem. 'Man, if I could just solve this quadratic equation, that would be great!'

    2)Identifying the problem with the problem. 'Why can't I solve this thing?! I shouldn't have to sit here for 2 freakin weeks testing all of the factors of of these enormous coefficients!'

    3)The quest begins. 'O.K. There's got to be an easier way.'

    4)Stating the problem in general terms. 'Well, I'm gonna state this thing in general terms, because I want this to work for all quadratics. I Know that all second degree equations can be written like this: ax^2+bx+c=0'

    5)Racking the brain. 'I've been working on this thing for 4 months. Maybe I should just set myself on fire, wrap a bag around my head, take some pills and jump off of a bridge.'

    6) A coulpe of teases. ' Oh my God! I've got it! Oh. Wait. No I don't. Stupid! stupid! stupid!.'

    7)The eureka. 'Oh! What if I..... That's IT!!! The trinomial square! I'll build a simple, stupid, little trinomial square!'

    8) The final proof. 'OK, let's see here......

    ax^2+bx+c=0

    ax^2+bx=-c

    x^2+\frac{b}{a}x=-\frac{c}{a}

    Now... Ha Ha Ha. I gotcha ya little bastard! You don't stand a chance before the great and powerful trinomial square.'

    x^2+\frac{b}{a}x+(\frac{b}{2a})^2=-\frac{c}{a}+(\frac{b}{2a})^2

    And now for the good stuff.

    (x+\frac{b}{2a})^2=-\frac{c}{a}+\frac{b^2}{4a^2}

    (x+\frac{b}{2a})^2=\frac{b^2-4ac}{4a^2}

    x=\frac{-b\pm{\sqrt{b^2-4ac}}}{2a}

    8) The dance. Whoooooooooooo!!!!!!

    9)The Ego.

    And that's how it goes. I've given this a lot of thought. Think about Old Isaac Newton, and the apple. Sounds like the eureka to me.

    Any way, let's start with something simple.

    The sum of cubes. x^3+a^3=x^2-ax+a^2

    If you don't like this one. pick one. I just want to study your structure and method.

    I CHALLENGE YOU ALL!!! DO A PROOF IF YOU DARE!

    I'll submit one myself on this thread. Give me a minute though.
    Why not start with what has been done before:

    1. How to Solve It, George Polya

    2. How to Solve Mathematical Problems, Wayne Wicklegren

    3. Solving Mathematical Problems, Terence Tao

    etc.

    CB
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  3. #3
    No one in Particular VonNemo19's Avatar
    Joined
    Apr 2009
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    Detroit, MI
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    Thanks Cap'n! You know what's hap'nin'!

    These links you have provided have shown me many things. However, I must say, you yourself did not take up the challenge. Again, I say, do a proof. It doesn't have to be novel or brilliant. It can just be something from your past. An old friend, Or perhaps, and old enemy. I myself am taking up the law of cosines. Simple, yes, but a proof just the same. I'm doing it on paint right now. It will be posted before the night's end.

    Enduring gratitude, sir,

    VonNemo.
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  4. #4
    No one in Particular VonNemo19's Avatar
    Joined
    Apr 2009
    From
    Detroit, MI
    Posts
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    Here's my logic:

    a=a

    b=b

    c=c

    d=b-c\cos{A}

    \sqrt{c^2-c\cos{A}}=BD and \sqrt{a^2-(b-c\cos{A})^2}=BD, therefore,

    \sqrt{c^2-(c\cos{A})^2}=\sqrt{a^2-(b-c\cos{A})^2}

    squaring

    c^2-(c\cos{A})^2=a^2-(b-c\cos{A})^2

    c^2-(c\cos)^2{A}=a^2-[b^2-2bc\cos{A}+(c\cos{A})^2]

    c^2=a^2-b^2+2bc\cos{A}

    a^2=b^2+c^2-2bc\cos{A}

    And there's my proof. I did it about 2 years ago. WITHOUT EVER SEEING EVEN SO MUCH AS A HINT! I had to basically from scratch again because I forgot how. Now, is anyone ELSE gonna get some of this? If you put this in 2 column form youre awesome. I want to see some proofs!
    Last edited by VonNemo19; September 19th 2009 at 10:49 PM.
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