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Nothing to do with teaching - need opinion of maths buffs

Discussion in 'Mathematics' started by manxli, Feb 26, 2012.

  1. Hi
    Nothing to do with teaching unless you've very advanced students. Following puzzle:
    If SQR(2) is an irrational number = 1.414........(ad infinitum), is it possible for a right angled triangle to even exist theoretically.
    Consider RA Triangle with two sides of 1 and 1 , then hypotenuse = SQR(2)
    In a theoretical world of perfect lines and angles, would this mean that any hypotenuse will ALWAYS be either too short or too long to fit a perfect figure? How do you specify the length of the hypotenuse to the drawer? When should he stop drawing the line?
    Remember, we're talking about a perfect diagram with lines of zero width and perfect angles.
    My solution? I'm going to open a bottle of beer. :eek:)
    Any thoughts?


     
  2. Maths_Mike

    Maths_Mike New commenter

    irrational doesnt mean not exact.
    and what about a 3,4,5 triangle?
     
  3. PaulDG

    PaulDG Occasional commenter

    As the distance between two points.

    When his line reaches the second point

    Measuring that line, is indeed impossible (on a theoretical level, though as practical as measuring any other line on a practical level).

    Constructing it is trivial in both theoretical and practical realms.
     
  4. Karvol

    Karvol Occasional commenter

    Don't you use the special rulers with measurements of sqrt 2, sqrt 3, sqrt 5, sqrt 7 and sqrt 10?
    They are standard issue in IB exams for constructions.
     
  5. "If SQR(2) is an irrational number = 1.414........(ad infinitum), is it
    possible for a right angled triangle to even exist theoretically."
    Yes. Of course. Any argument to the contrary is stupid. Root 2 'exists' in exactly the same way that 2 'exists'.

     
  6. Hi
    Thanks for the prompt responses. I started with a little fib I'm afraid. It does have something to do with teaching. I'm not a mathematician and actually teach Music and EFL but sometimes have to cover for sick colleagues in the Swiss special school where I work. I've tried this puzzle with a few older supposedly 'low achievers' and its amazing how animated and enthusiastic they become once they've grasped the suggestion.
    Isn't this the way to promote enquiring minds or does it only serve to confuse? Does anyone have other puzzles or peculiarities as lesson starters?
    Regards
     
  7. Anonymous

    Anonymous New commenter

    How long is the coast line around the UK mainland?
    That is an interesting question. Do you measure every little point etc? How accurate can you be?

     
  8. How much of the coast erodes between the time of starting to measure and completing? Oh no the tide is coming back in!
    Going back to the traingle, attempting to measure on the quantum scale of any real existing figure will affect the reality as in Heisenberg's uncertainty principle anyhow. i.e. The more accuracy demanded the more uncertain the position.
     
  9. Exactly the same way? Doesn't any proof of the existence of the square root of two rely on the axiom of completeness?
     
  10. Once you get down to very small scales, the whole concept of distance may not even exist. Think smaller than the scales for quantum mechanics, uncertainty principle and string theory: quantum foam is an example. Once lengths get to a certain theoretical accuracy, they become impossible to realise in the physical universe. No one has ever drawn a 3, 4, 5 triangle exactly, and nobody ever will.
     
  11. Don't try to show off on this forum - there are some big beasts lurking in the undergrowth.
     
  12. Oh I don't think we're showing off. Quite the reverse, I think we're showing what we don't know. But maybe you're right about the beasts!
    "The devil is in the detail" http://en.wikipedia.org/wiki/The_Devil_is_in_the_details
    Maybe Maths should be reclassified under humanities and the arts?

     
  13. the
    lim as n->0 of n/n is 1.
    taking the limit is not the same as
    what is the value of n/n when n = 0
     
  14. I don't know about diagrams and physical triangles but Professor Tim Gowers 'says' we define root 2 into existence, along with all the other irrationals to get the real numbers which...we like.
     
  15. Hmmm....
    Bit like, what is the difference between a mathematician, physicist, quantum physicist and an accountant?
    Mathemetician: 2 + 2 = 4
    Physicist: 2 + 2 = 3.8 -> 4.2
    Quantum Physicist: 2 + 2 = 4 (If no one is looking)
    Accountant: "What would you like 2+2 to be?"
    OK, if we can define a number to exist, where does it exist? In our heads or in the real world? Can any number 'exist'?
     
  16. Andrew Jeffrey

    Andrew Jeffrey New commenter

    Yes. They exist as generalisations. Failure to appreciate this is why some children get turned off maths very young as they are not helped to make the HUGE jump from 'two ice creams' to 'two'. This problem eventually develops into not understanding algebra. *gets back in box*
     
  17. Hi Andrew
    Yes, I think you've hit it on the head. Maths is a sort of language that learners need to understand, just as spoken/written language exists in our heads and not in the real world as such. Some linguists I know of have even looked at applying linguistic theory to the language of maths.
    http://www.scribd.com/doc/25352725/Mood-Analysis-SFL - This research paper shows an attempt to bridge the real to the abstract with young learners. I also seem to remember the now defunct teachers' TV going on about 'maths stories'.
    At the start of this thread, I mentioned looking after maths classes for sick colleagues and certainly noticed that many of the learners had no idea what use their maths exercises were and what the point was. It must be very easy to lose sight of that when one is under pressure to cover curriculum and deliver grades.
    Good food for thought.
     

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