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Thread: visualizing math

  1. #1 visualizing math 
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    Hi there,

    I have a problem with math, if I can't visualize it I have problems understanding it...

    Taking the standard equation (also called implicit?) for a circle:



    I totally understand what this equation is saying. It's saying that the squared radius of the circle is equal to the length of the point at coordinates by Pythorean theorem. (assuming origin is )

    The general equation of the circle is:

    whenever

    I understand how the general form results from the expansion of the standard form.

    But my question is, would one attempt to visualize the general equation? Or simplify and attempt to visualize the standard form (which is easy to visualize).

    I used the above for an example. There is quite a bit of stuff I've come across in math that I can't even visualize. What are your thoughts on that?

    Thanks!

    EDIT: On another note, looking at the fundamental theorem of algebra, I can see how it arises from the properties of numbers, but how the hell would you visualize it? Or would one not attemp that? Hence it being called abstract algebra (please correct me if I'm wrong)?


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  3. #2 Re: visualizing math 
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    Quote Originally Posted by rgba
    Hi there,

    I have a problem with math, if I can't visualize it I have problems understanding it...

    Taking the standard equation (also called implicit?) for a circle:



    I totally understand what this equation is saying. It's saying that the squared radius of the circle is equal to the length of the point at coordinates by Pythorean theorem. (assuming origin is )

    The general equation of the circle is:

    whenever

    I understand how the general form results from the expansion of the standard form.

    But my question is, would one attempt to visualize the general equation? Or simplify and attempt to visualize the standard form (which is easy to visualize).

    I used the above for an example. There is quite a bit of stuff I've come across in math that I can't even visualize. What are your thoughts on that?

    Thanks!

    EDIT: On another note, looking at the fundamental theorem of algebra, I can see how it arises from the properties of numbers, but how the hell would you visualize it? Or would one not attemp that? Hence it being called abstract algebra (please correct me if I'm wrong)?
    Visualizing mathematics is a rather personal thing. Different people see things in different ways.

    I have a friend who was one visiting a famous mathematician in Paris. This mathematician opened the meeting by asking him "Are you a picture guy or a numbers guy ?" By this he meant do you think in pictures or in numbers ? He only talks with pictures guys because he does not think he can communicate well with numbers guys.

    How you visualize these things is something that you will have to work out for yourself.


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  4. #3 Re: visualizing math 
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    Quote Originally Posted by rgba
    Hi there,

    I have a problem with math, if I can't visualize it I have problems understanding it...

    Taking the standard equation (also called implicit?) for a circle:



    I totally understand what this equation is saying. It's saying that the squared radius of the circle is equal to the length of the point at coordinates by Pythorean theorem. (assuming origin is )

    The general equation of the circle is:

    whenever

    I understand how the general form results from the expansion of the standard form.

    But my question is, would one attempt to visualize the general equation? Or simplify and attempt to visualize the standard form (which is easy to visualize).

    I used the above for an example. There is quite a bit of stuff I've come across in math that I can't even visualize. What are your thoughts on that?

    Thanks!

    EDIT: On another note, looking at the fundamental theorem of algebra, I can see how it arises from the properties of numbers, but how the hell would you visualize it? Or would one not attemp that? Hence it being called abstract algebra (please correct me if I'm wrong)?
    Circle equation: can be rewritten as:


    Therefore you have a circle centered at and
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  5. #4 Re: visualizing math 
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    Quote Originally Posted by rgba
    Hi there,

    I have a problem with math, if I can't visualize it I have problems understanding it...

    Taking the standard equation (also called implicit?) for a circle:



    I totally understand what this equation is saying. It's saying that the squared radius of the circle is equal to the length of the point at coordinates by Pythorean theorem. (assuming origin is )

    The general equation of the circle is:

    whenever

    I understand how the general form results from the expansion of the standard form.

    But my question is, would one attempt to visualize the general equation? Or simplify and attempt to visualize the standard form (which is easy to visualize).

    I used the above for an example. There is quite a bit of stuff I've come across in math that I can't even visualize. What are your thoughts on that?

    Thanks!

    EDIT: On another note, looking at the fundamental theorem of algebra, I can see how it arises from the properties of numbers, but how the hell would you visualize it? Or would one not attemp that? Hence it being called abstract algebra (please correct me if I'm wrong)?


    I think you can do this with grids of lines.
    You hae the vertical lines distance 'x' apart and the horizontal lines distance 'y'
    apart.
    With x=y it is fairly simple to see.
    I notice this with bathroom tiles which are square, if you cut them in half diagonally
    if is faiirly easy to see that the 4 of those hallfs will make a square equal to the length of the diagonal.

    It is then not to difficult to expand that to tiles of any rectangular shape, which are two right angled triangles stuck togeather.


    Going on to circles or whatever when x=y the think of your grid where x=y, the square grid, the angle of the square on the long side is 45 degrees.

    So you could go all around the circle to get alll the possible angles and rectangle combinations, if you could automatically draw the appropiate rectangles and square everything should become clear.



    There is a visual 'proof' here
    http://www.cinderella.de/files/HTMLD...ythagoras.html

    Howeverit wold have been better if the yellowand brown squares were drawn
    on the outside of the final picture, and lines extended to the grids I mentioned above.
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  6. #5  
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    Visualising mathematics can in certain cases be redundant particularly when its geometrical sometimes its just "abstract nonesense" sorry i could not help.
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  7. #6  
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    Quote Originally Posted by jonbon
    Visualising mathematics can in certain cases be redundant particularly when its geometrical sometimes its just "abstract nonesense" sorry i could not help.
    I'm not sure what you meant by "abstract nonsense". That term is commonly applied to category theory.

    But category theory is often depiected in diagrammatic form with lots of arrows, and therefore is visualized in just that manner.
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  8. #7  
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    Quote Originally Posted by jonbon
    Visualising mathematics can in certain cases be redundant particularly when its geometrical sometimes its just "abstract nonesense" sorry i could not help.
    Maybe it is that maths which cant be visualised that is nonsense!
    Sometiimes sloppy grammer and punctuation can produce the same effect :wink:
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  9. #8  
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    DRrocket thats exactly what i was refering so you know what im talking about eg homology can be described as abstract nonsense and the arrows etc arent really
    geometrical theyre just us trying to come to terms with abstract nonsense.
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  10. #9  
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    Quote Originally Posted by DrRocket
    Visualizing mathematics is a rather personal thing. Different people see things in different ways.

    I have a friend who was one visiting a famous mathematician in Paris. This mathematician opened the meeting by asking him "Are you a picture guy or a numbers guy ?" By this he meant do you think in pictures or in numbers ? He only talks with pictures guys because he does not think he can communicate well with numbers guys.

    How you visualize these things is something that you will have to work out for yourself.
    That's interesting, thanks for that

    Esbo: thanks! thats cool

    I guess it's really a matter of how you see things, guess as you learn more stuff it becomes much more intuitive both visually and numerically.
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  11. #10  
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    I did all my high school and university courses before calculators, computers etc, were used on a regular basis. With a few glitches I always got 'A's but I felt like a fraud. I never 'really' knew what I was doing. I rarely could visualize the concepts. I passed because of hard work and not because any insight into what the 'larger' picture was.

    That has continued into one of my hobbies, playing the guitar. I have trouble 'hearing' the music just as I couldn't visualize math. I am a 'good' player but that's only because I've played for 40 years and, again, hard work. If I want to learn a new song or classical piece, I look up the tabs, chords, etc. and within a few hours have the song down...but there's no way I can 'hear' a piece (unless it has simple chord sequences) and start to play it. In contrast, a lot of guitar newbies can pluck out a song with no help after only a year or so of playing. What they can't do is play it 'well' but that is because of a lack of technique, rhythm etc.
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  12. #11  
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    Quote Originally Posted by rgba

    Esbo: thanks! thats cool
    You seem fairly sincere in trying to learn mathematics.

    Therefore I offer the strong advice to ignore Esbo. His next correct statement regarding mathematics will be the first one. He has demonstrated more misconceptions and outright fallacious ideas than I have every encountered in one individual in several decades in mathematics.
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  13. #12  
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    Quote Originally Posted by DrRocket
    Quote Originally Posted by rgba

    Esbo: thanks! thats cool
    You seem fairly sincere in trying to learn mathematics.

    Therefore I offer the strong advice to ignore Esbo. His next correct statement regarding mathematics will be the first one. He has demonstrated more misconceptions and outright fallacious ideas than I have every encountered in one individual in several decades in mathematics.
    Thanks DrRocket. I'll certainly heed your advice in the future.
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  14. #13  
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    Quote Originally Posted by DrRocket
    Quote Originally Posted by rgba

    Esbo: thanks! thats cool
    You seem fairly sincere in trying to learn mathematics.

    Therefore I offer the strong advice to ignore Esbo. His next correct statement regarding mathematics will be the first one. He has demonstrated more misconceptions and outright fallacious ideas than I have every encountered in one individual in several decades in mathematics.

    Rgba don't worry about Dr Rocket, he is just a bit upset that I made him look a fool, although he is quite capable of doing that by himself.


    Anyway there is another one here (not so good)
    http://www.cut-the-knot.org/pythagor...yPythPWW.shtml


    and more stuff on the link with loads of proofs.
    http://www.cut-the-knot.org/pythagoras/index.shtml
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  15. #14  
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    uh huh flame war starting???

    funny thing, i just asked how one would attempt to visualize maths... no need to turn things ugly...
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