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Pythagoras Theorem

  1. Nov 12, 2009 #1
    Hi,

    I have a really stupid question :smile:

    suppose the base and perpendicular are both length 1 (whatever units)
    then the hypotenuse's length comes out to be SquareRoot of 2

    but square root of 2 is 1.41421356............. (not a fixed number)

    so that means if i physically measure the hypotenuse upto the accuracy of 3 decimal places i would get 1.414

    but the actual length is bigger then that, its bigger by .00021356.... which is not much

    but the point is "Since the square root of 2 is not an exact number" with whatever arbitrary precision i choose to measure the hypotenuse its length will always be somewhat bigger than what i just measured

    Whats going on.. is there s disconnect between the real and mathematics world

    Thoughts please
     
  2. jcsd
  3. Nov 12, 2009 #2
    math is exact, humans aren't. i think that's all it is
     
  4. Nov 12, 2009 #3
    The Pythagoreans dealt with this problem 2000 years ago: the discovery that not all numbers are the ratio of two integers--particularly the case you mention. They referred to these as "unutterables."

    It's not just lower bounds, we can get upper bounds too. We consider the the Pellian equation Y^2-2X^2 = plus or minus 1. Take (1-sqr2)^n and get a series of terms (1-sqr2)^2 = 3-2sqr2, (1-sqr2)^3 = 7-5sqr2, etc.

    We then arrive at a series of lower and upper bounds on the square root of 2. 1/1<sqr2; 3/2>sqr2; 7/5<sqr2, 17/12 >sqr2; 41/29<sqr2....etc.

    Those are the best approximations possible considering the size of the integers.
     
    Last edited: Nov 12, 2009
  5. Nov 12, 2009 #4
    First, the square root of 2 IS a fixed number. It has an exact value. The problem is you are using an underpowered definition of what a "number" is. Numbers are more than their decimal places. Rational numbers, the kind you learned to add and multiply in elementary school, are represented by fractions of integers. You don't learn much about irrational numbers in school. Most people know that pi has a non-repeating decimal expansion, but beyond that, nothing.

    The details are complicated (and interesting), but aren't something you learn unless you're a mathematician (the class is usually called Real Analysis). But almost everyone can get by in life with rational numbers. Rational numbers can always get "close enough" to any irrational number. And close enough goes a long way in life.

    There's a ton of disconnect between math and the real world. Math is a formalization of our ideas about the world and our understanding of the world is pretty darn weak.
     
  6. Nov 12, 2009 #5

    Mentallic

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    But you know what. You wouldn't have been able to get the side lengths of 1 to the exact length either - or in another case, if you take some length and define it to be 1, you would still have a rounding error on the other length that you try to make equal 1. It's an error that we just account for.
    It's not the fact that the number is irrational that we can't measure it, it's just our precision that can't measure it perfectly accurately.

    If we could measure it perfectly, you'll find that as you magnify the ruler's reading, you'll keep following an infinite string of decimals, sqrt(2).
     
  7. Nov 12, 2009 #6
    You're describing truncating, not rounding.
     
  8. Nov 13, 2009 #7

    HallsofIvy

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    square root of 2 certainly IS a "fixed number". It just is not a terminating decimal nor is it a rational number.

    Absolutely not true. square root of 2 is as "exact" a number as 0 or 1 or 1/3. It is the measurement that is not exact. And no one expects measurement to be exact. That's why you have to talk about "accuracy".

    No, that's not true either. For example, if you measure to an accuracy of 7 decimal places you would get 1.4142136 and the correct result is somewhat less than that. But it is "bigger" or "smaller" by less than your accuracy.

    All measurement is approximate, just as you said when you talked about "accuracy". Nothing new about that. As any scientist knows, it is the measurement that is the problem.
     
  9. Nov 13, 2009 #8
    Thanks a lot for clarifying but how come square root of 2 is fixed?
     
  10. Nov 13, 2009 #9

    HallsofIvy

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    It is a specific number! It doesn't change! That's what 'fixed' means. What did you think it meant?
     
  11. Nov 13, 2009 #10
    Root(2) is a "Fixed Number"
    Just as we say that 1/3 is a fixed number.. we can represent it on the number line..(not using a ruler.. but a compass or divider)
    the same way we can represent root(2)..maybe we can.. don't know about root(2).
    Does anyone know if we can do so?
     
  12. Nov 13, 2009 #11
    OF COURSE YOU CAN! This was easily discovered in the days of Pythagoras. Simply draw a one unit X, and a one unit Y, and then CONNECT THE DIAGONIAL.
     
  13. Nov 13, 2009 #12
    Ha, I think I hear a face plant.
     
  14. Nov 14, 2009 #13
    Well, if that is too harsh a way to put it, I think we can just start with Xmonte, who started this problem by saying:

    Hi,

    I have a really stupid question

    suppose the base and perpendicular are both length 1 (whatever units)
    then the hypotenuse's length comes out to be SquareRoot of 2


    and HallsofIvy repeated the same statements..
     
  15. Dec 28, 2009 #14

    Thanks.
     
  16. Dec 28, 2009 #15

    HallsofIvy

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    Any number that is "Algebraic of order a power of 2" can be "constructed" using straight edge and compass (or compass alone assuming that marking 2 points "gives" the line through those two points). Any number that is not algebraic of order a power of 2 cannot.

    Of course, [itex]\sqrt{2}[/itex] satisfies [itex]x^2- 2= 0[/itex] and so is algebraic of order 2.
     
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