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Non integer square roots and pi = irrational?

  1. Dec 21, 2005 #1
    Since one can construct the length of a non-integer square root by drawing accurate triangles, and can draw a circle with a circumference of pi, then shouldn't one be able to plot corresponding non-integer square roots and pi on a number line? I know these numbers are supposedly irrational, but if their length can be determined at a certain scale, then how can they be irrational and why can't they be plotted on a nuymber line?
     
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  3. Dec 21, 2005 #2

    shmoe

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    What do you mean by non-integer square root (and "accurate triangles" for that matter? 1/4 is not an integer yet sqrt(1/4) is rational.

    pi isn't "supposedly irrational," it is irrational, you can find a proof easily enough with google

    I have no idea what you mean by "but if their length can be determined at a certain scale".

    What does plotting a number on a number line have to do with being irrational? What do you even mean by "plotting a number on the number line"? Do you mean straightedge/compass type of construction? If so how do you hope to construct a line of length pi? (hint-you can't, pi is transcendental) Maybe you mean something else by "construct"?
     
  4. Dec 22, 2005 #3

    Tide

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    You can certainly plot irrational numbers on the number line. E.g. construct two orthogonal number lines. Mark off 1 unit on each from their intersection. The length of the diagonal is irrational but you can rotate the diagonal about one of its ends onto the same number line. You've just plotted [itex]\sqrt 2[/itex].

    Of course, transcendental numbers pose a problem as schmoe pointed out. In principle, you can plot them by successive approximation using their decimal representations.
     
  5. Dec 22, 2005 #4

    matt grime

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    You can only construct some square roots this way, and they can be translated to the number line, I suppose, but how do you mathematically cut a circle and flatten it out?


    What do you mean by "plotted", and what makes you think rationality or otherwise has anything to do with it?

    Usually we mean something like: given a straight line and two marked points that we'll call 0 and 1 can you using a compass and straightedge (possibly fixed compass, possibly a ruler marked in the same units) to mark on certain other points of relative size.


    In that sense you cannot even mark on many rational numbers.

    I have no idea what determined at a certain scale even means.
     
  6. Dec 22, 2005 #5
    Any length can be plotted on a line. Lengths are not rational, irrational or transendental, numbers are. You also assume that just because a line can be plotted on graph paper that its length can be determined according to the scale of the graph paper. This is not true, only an approximation of the length can be determined at best, even using the highest possible magnification. But of course the length can be always determined to the nearest unit at the scale of the graph paper. Just as it can to the nearest unit of a ruler. Consider a rational number a/b. In theory, one could construct a number line of length ab, by first plotting the point 0 to the far left of a straight line and then by repeatedly plotting a new point one unit to the right of the rightmost point plotted thus far ab times by using a compass. Now let the scale of the graph paper be such that a units combined together is equal to one. In this way a point corresponding to a/b would have been exactly plotted on the number line. A irrational number is simply one that can't be plotted on a number line by this manner of construction because it can't be expressed as a fraction like a/b.
     
  7. Dec 22, 2005 #6

    mathwonk

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    ploting a given number legth on the real line is equivalent to finding a straight line with that length. so you can plot any hypotenuse of a ny triangle that way. e.g. sqrt(2).


    Numbers that can be plotted usig euclidean geometry are called constructible numbers, i.e. numbers that can be constructed from the original choicew of a unit length, using translation and circles. these constructions are described by quadratic equations, so only sqrt roots of square roots, etc can be s described.

    e.g. most cube roots cannot.


    to be irtionalk is another matter.

    so there is a hierarchy of more and more complicated lkengths: rational, constructible, algebraic, all reals including transcendental.

    only the first two can be described by euclidean geometric constructions.

    on the other hand, essentially any length, including pi can be approximated to within any desired accuracy by an infinite sequence of straight ine lengths. so given any desried degree of error, you could plot and construct a length that is within the desried error of pi, on the line.

    a nice sourde for this material, is michael artin's book, algebra, or Theodore Shifrin's book, a geometric approach to algebra.
     
  8. Dec 23, 2005 #7

    selfAdjoint

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    The OP's question seems to be, are all irrationals comparable, so that irrational pi being taken as a unit, irrational square root 5 comes out as, I don't know what he wants, maybe a linear combination with integer coefficients.

    And the answer of course is no, this doesn't happen. Irrationals are more complicated than that.

    The nearest correct theory to this idea seems to be Galois theory, where you study field extensions by different radicals and get theorems about how they relate.
     
  9. Dec 26, 2005 #8
    Thank you, everyone, for your help. Sorry I was unclear when posting my question, I will clarify what I meant. A non-integer square root is a number that is not an integer that when multiplied by itself equals an integer. All non-integer square roots of integers are irrational.

    My question was this: I was taught in school that one cannot plot the exact length of an irrational number on a number line becuse it continues on forever. This is makes sense, but in theory, isn't it possible to draw the exact length of an irrational number by drawing the two legs of a right triangle, for example 2 inches each, and then connecting the ends of the two legs with a hypotenuse of the square root of 8 (an irrational number)?
     
  10. Dec 26, 2005 #9

    matt grime

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    does it? what does 'plot' mean? what is the number line (i'm serious, what is it apart from some nice hand wavy picture for kids)? what has the decimal expansion got to do with it? is 1/3 plottable?



    i have several issues with this, starting with the notion of constructing anything in the real world that is some perfect example of a purely theoretical mathematical construct.

    everything in maths is true or false under certain hypthoses. starting from a ruler with the units 1,2,3 etc marked on your teacher was working under the hypthosis that it was possible only to mark onto a ruler rational division points, apparently because you're makring on their decimal expansion. there is no reason to suppose that one can even do this, or that this is a good assumption, but what she said was certainly true under that assumption. it also means one cannot plot 1/3, 1/7 or fractions of 1/p for any prime other than 2 or 5.

    what you're doing is extending the allowable constructions to include geometric ones, eg by allowing constructions with a compass as well. this means you can plot on the square root of 2, and in general appeal to galois thoery tells us all the constructible numbers in this sense. it is still by no means all numbers (does it allow even 1/3 to be plotted you should ask yourself). in any case under your new assumptions the old ones about decimal expansions are dropped, though presumably you're allowing yourself to plot all of the old rationals like 1/5 though that is no longer necessarily true.
     
  11. Dec 26, 2005 #10

    HallsofIvy

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    I suspect that you were not taught that but that is your hazy recollection of what you were taught (unless you had the misfortune to have had a school teacher who was teaching his/her hazy recollection!). It is true that if you try to write out an irrational number in decimal form, you can't do it but that has nothing to do with "plotting a length". Every real number corresponds to an exact point on the number line (that is, basically, the definition of "number line".). If by plot, you mean a Euclidean compass-straightedge construction, it can be shown that only those numbers that are "algebraic of order a power of two" (a number is "algebraic of order n" if it can be found as the root of a polynomial equation with integer coefficients of degree n but no such equation of lower degree) can be "constructed" in that way: that includes [itex]\sqrt{2}[/itex] (which is algebraic of order 2: it satisifies x2- 2= 0) but not [itex]^3\sqrt{2}[/itex] (which is algebraic of order 3: it satisfies x3-2= 0 but no equation of lower degree with integer coefficients), or [itex]\pi[/itex] which is not algebraic of any order- it is "transcendental".
     
    Last edited: Dec 26, 2005
  12. Dec 26, 2005 #11

    JasonRox

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    I wouldn't be surprised if some teachers teach their own "theories" to kids.
     
  13. Jan 1, 2006 #12
    Well since you can construct similar triangles and parallel lines with a straight edge and a compass, I think it is possible in theory to plot rational parts of any unit length and multiples thereof. For instance to plot 2/3 on unit graph paper you would draw a line with a straight edge through points <3,0> and <0,1> and construct a parallel line that passes through point <2,0>. The point that this parallel line intersects the axis between points <0,0> and <0,1> corresponds to 2/3.
     
    Last edited: Jan 1, 2006
  14. Jan 1, 2006 #13

    shmoe

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    Given lengths a and b, using a compass and straightedge, you can construct lengths a+b (also a-b), a*b, and a/b, so starting with a unit length you can certainly construct any rational multiple you like. The issue was what exactly the OP means by "construct" or "plot".
     
    Last edited: Jan 1, 2006
  15. Jan 18, 2006 #14
     
  16. Jan 18, 2006 #15

    Hurkyl

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    Why?

    Your argument seems to be: "I cannot imagine how this could be true. So, it must be false" -- that's a tremendously egotistic view, don't you think? :smile:


    P.S. while words like "terminates" or "repeats" are specific terminology about decimal numbers. They cannot be applied to numbers in general.
     
  17. Jan 18, 2006 #16

    HallsofIvy

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    You are declaring that by fiat? Once you have set up your coordinate system there exist a unique point corresponding to every real number and a unique real number for each point. That's because the real number system is "complete". The fact that the line is a connected set (with the usual metric) follows from the Least Upper Bound Property.
     
  18. Jan 18, 2006 #17
    By using common sense, anyone can see it is false. I express my views strongly, and for me to say "I think this, but it's probably wrong, and you probably know better" is weak. I have confidence in what I think to be true, and in an arguement, it is best not to act unsure of yourself.

    P.S. You accusing me of being egotistical does not answer the question. Are you implying I am stupid because I cannot imagine something that you think other people can (nobody can fathom infinity).

    P.P.S. I am applying them to decimal numbers. What do you mean they can't be applied to numbers in general?

    P.P.P.S. Can you imagine it to be true?
     
  19. Jan 18, 2006 #18
    I realize that you are probably much more knowledgeable about this than me and probably have received much more schooling than I have, but if you disregard about all the classes you've taken, books you've read, and all the theorems and whatnot, and think for yourself, then you will come to the same conclusion as me. Maybe the definition of a number line is a line that has exact corresponding points for all the real numbers. But just because that's what the internet says, or what some book says written by some guy with a PhD from CIT or whatever, doesn't mean it's true. I also realize that just because I say one thing doesn't mean it's true either. I am just encouraging you to use your own ability to think and reason, which is more important than what anybody else says. If you do this and still believe that there is an exact point on the number line for a number that continues ad infinitum, then I would question your sanity. I tried to be polite before this, but now this is angering me. :grumpy:
     
  20. Jan 18, 2006 #19

    JasonRox

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    You should come to the same conclusions as a mathematician.

    Mathematics is very rigorous and there is "no" way around it.
     
  21. Jan 18, 2006 #20
     
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