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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 by a moderator: 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
     
  22. Jan 18, 2006 #21

    Hurkyl

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    You must temper your confidence with reason. You are so caught up in trying to avoid any show of weakness that you are, no offense intended, making a fool of yourself.

    You have, so far, demonstrated many of the characteristics of people we call crackpots.
    (1) You are extremely insistent on your correctness.
    (2) You base your position on "common sense" -- and the assumption that everybody else's common sense must agree with yours, and those who disagree with you are simply refusing to listen to their common sense.
    (3) You base your position on your inability to imagine the alternative -- and the assumption that everybody else must also be unable to imagine the alternatives.


    The typical crackpot also tends to ignore any evidence that they may be incorrect -- you've one-upped them, though: you've acknowledged evidence you may be incorrect (you know full well how to construct a line segment of irrational length), and have come to the realization that there's a problem... yet you still maintain your position with absolute confidence.


    I hadn't noticed these tendancies in your earlier posts -- I suppose you're just responding to a perceived attack on your person?


    Of course I'm not saying you're stupid: everybody has their own capabilities.

    However, I am explicitly saying that you're being blinded by your egotism. You refuse to accept that everybody does have their own capabilities, and that someone else might be able to do something you cannot at the moment, such as being able to "fathom" infinity.

    In my estimation, based on watching many people assert this point, the main reason many people have trouble "fathoming" infinity is simply because they're convinced that it's "unfathomable" -- I could tell them about all sorts of things like cardinality, the extended reals, and non-archmedian fields. I could tell them about how using the adjective "infinite" is usually more appropriate than the noun "infinity", and so on. However, these people will invariably respond (roughly) that I cannot possibly be speaking about infinity, simply because I understand what I'm talking about.


    This is another example of the same problem. You hold this belief that your ability to think and reason is absolutely perfect -- anyone who disagrees with your conclusions must be mistaken or insane or something.


    You were responding to this quote of HoI:
    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".).

    Where he makes a point of separating the notions of the "decimal form" and a "real number".

    One of the hangups some people have, yourself included it seems, is separating these two notions. In their mind, a decimal number is the "only way" to write a real number. When they see other ways of denoting a real number, such as:

    [itex]\sqrt{8}[/itex] is the unique number x such that x² = 8 and x > 0

    or

    [itex]\sqrt{8}[/itex] is the length of the hypotenuse of a (Euclidean) right triangle whose sides have length two

    they think "These denotations are simply different ways of saying 2.828...".


    Another hangup people have, again yourself included it seems, is that the notation 2.828... "really means" that you're supposed to start with 2, and then you continue on to 2.8, and then to 2.82, and then to 2.828, and so forth. And since this algorithm never finishes, they think that the decimal notation cannot represent an actual number. Since they believe that this is the only way you're ever allowed to actually handle a real number, they have problems.

    Incidentally, they aren't too far from the mark -- but they refuse to budge at all, and thus never see the light. :frown: In the rigorous sense, a decimal number is simply a function that allows you to compute something called the "n-th digit", whatever that means. When I write something like:

    0.454545...

    this is shorthand for "the n-th digit of this number is 4 if n is negative and odd, 5 if n is negative and even, and 0 otherwise". (The places are numbered ...(3)(2)(1)(0).(-1)(-2)(-3)...)

    This is something we can manipulate in its entirety -- we're not doomed to forever add digits one at a time, never getting anywhere. I can, for instance, add it to 0.545454... and prove (in finite time!) that the result is equal to 0.999999..., which is known to equal 1.


    I was hoping it would make you aware that you are projecting yourself onto others -- that maybe you would come to realize on your own that not everybody will agree on what you maintain is "common sense".

    Another characteristic of logical debates is burden of proof. When we say to you that there does exist a point on the number line for every real number, you are (generally) justified in asking us to prove our assertion, and it would be unfair for us to ask you to either accept it or prove us wrong.

    Conversely, when you tell us that there cannot exist a point for each irrational number, we are justified in asking you to prove your assertion, and it is unfair for you to ask us to accept it or disprove you.

    To state it more succinctly:

    When someone makes an assertion, the burden is on that person to justify their assertion. There is no burden on everybody else to disprove that person.


    You're wrong.

    If some whacko on the internet defines "the number line" to consist of all barnyard animals and nothing else, then in that context, "the number line" really does mean the collection of all barnyard animals and nothing else.

    However, all mathematicians have essentially agreed on the default meaning of "number line". Furthermore, it is exactly this "number line" that is taught in schools.

    So, when no alternative is specified, the term "number line" refers precisely to what is written in that book from the PhD from CIT.



    And finally

    If you want to make this about strength, instead of mathematics, then I'll win: I wield the bigger stick. :wink:
     
    Last edited: Jan 18, 2006
  23. Jan 19, 2006 #22
    I don't think you are implying that Leonardo doesn't have the capacity to fathom reality. He seems quite intelligent and it is easy to see how some very deep mathematical concepts can be confusing to very capable individuals. I believe that math is a very concrete science, but it at best only provides only a rough blue print of the real world. Atoms have definite size. Points on a number line have no size. I think Leonardo is confusing reality with mathematics and is upset that he isn't getting his point across. A little more understanding on both sides is appropriate.
     
  24. Jan 19, 2006 #23

    mathwonk

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    this forum is certainly almost infinitely indulgent of crackpots. i do not know if that is a good or bad thing, but it gets tiresome.
     
  25. Jan 19, 2006 #24
    First of all, I would like to thank Ramsey for being so generous and reasonable.

    Secondly, I would like to make a few replies to Hurkyl regarding his assault.

    As a 14 year-old, I do not take any drugs or smoke or drink, but you seem to know an awful lot about how crackpots think! :wink: I am curious as to how you became so knowledgeable about such a relevant topic.

    You said that I've acknowledged evidence that I may be incorrect because I know full well how to construct a line with an irrational length. Yes, I was using this point to ask my question, for my question was about how that and the "impossibility" of finding the exact point on number line of an irrational number seem to contradict eachother.

    Yes, the reason I was angry last night was partly because a "perceived attack on my person" and that it seemed I might not be getting my point across very clearly. I will try to explain it again, but differently so that maybe it will make more sense.

    I am going to forget about the number line in this explanation, only because it seems to cause confusion and arguments over the technicalities of its properties and draws the attention away from the purpose of this thread.

    I hope we can all agree that one can draw a hypotenuse of irrational length. I say this disregarding human error and/or the width of the line, etc. Pretend it is exact, in theory. My confusion is about how this can be done when the decimal representation of the length of the line never terminates or repeats.

    Hurkyl: I am sorry for offending you if I did and I forgive you for offending me. In the future, instead of throwing insults back and forth at each other, arguing over who is stronger than who, and calling others "crackpots", let's act like mature, civilized, individuals, and have an intelligent discussion. :smile:
     
  26. Jan 19, 2006 #25
    First of all, I would like to thank Ramsey for being so generous and reasonable.
    Secondly, I would like to make a few replies to Hurkyl regarding his assault.
    As a 14 year-old, I do not take any drugs or smoke or drink, but you seem to know an awful lot about how crackpots think! :wink: I am curious as to how you became so knowledgeable about such a relevant topic.
    You said that I've acknowledged evidence that I may be incorrect because I know full well how to construct a line with an irrational length. Yes, I was using this point to ask my question, for my question was about how that and the "impossibility" of finding the exact point on number line of an irrational number seem to contradict eachother.
    Yes, the reason I was angry last night was partly because a "perceived attack on my person" and that it seemed I might not be getting my point across very clearly. I will try to explain it again, but differently so that maybe it will make more sense.
    I am going to forget about the number line in this explanation, only because it seems to cause confusion and arguments over the technicalities of its properties and draws the attention away from the purpose of this thread.
    I hope we can all agree that one can draw a hypotenuse of irrational length. I say this disregarding human error and/or the width of the line, etc. Pretend it is exact, in theory. My confusion is about how this can be done when the decimal representation of the length of the line never terminates or repeats.
    Hurkyl: I am sorry for offending you if I did and I forgive you for offending me. In the future, instead of throwing insults back and forth at each other, arguing over who is stronger than who, and calling others "crackpots", let's act like mature, civilized, individuals, and have an intelligent discussion. :smile:
     
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