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Gaps In The System of Rational Numbers

  1. Dec 10, 2014 #1
    I have a question which comes from Rudin's Principles of Mathematical Analysis; specifically, from the introduction.

    In example 1.1, the author clearly shows that no rational numbers satisfy the equation ##p^2 = 2##.
    So, I am trying to imagine myself in a scenario in which I am in a time before the discovery of irrational numbers, which provide us with solutions to equations as ##p^2 = 2##. Furthermore, I am one of these pre-irrational-number mathematician who does a proof to demonstrate that ##p^2= 2## has no rational solutions.

    Here is my question: What would make me think that there existed any solutions; and why would we think
    that one of the irrational numbers—namely, ##\sqrt{2}##—would be one such solution to ##p^2=2##? In other words, does ##p^2 = 2## "get" solutions because we want it to have solutions?

    Also, below this example the author attempts to demonstrate that the rational number system has gaps. But would this only be true if we wanted ##p^2 = 2## to have solutions, right? Two questions: (1) why does it matter if the rational numbers are incomplete, and why would we fill in the gaps with numbers and then refer to them as irrational numbers (Note: I am not questioning the name given to these sorts of numbers, but the numbers themselves); (2) by augmenting our number system by including these numbers which fill the gaps, we form the real number system, how do we know that ##\mathbb{R}## does not have any gaps?

    In short, why should we want ##p^2 = 2## to have solutions?
     
    Last edited: Dec 10, 2014
  2. jcsd
  3. Dec 10, 2014 #2
    Maybe because we can draw a triangle and, by Pythagoras's theorem, deduce that one side must have that length?

    I suppose it's a matter of taste. We could debate whether irrational numbers exist "out there" in the real world, but inventing (or discovering?) irrational numbers allows us to do more trigonometry etc. which might turn out to be useful.
     
  4. Dec 10, 2014 #3
    So, we constrain them to exist for pragmatic purposes? I suppose that that is reasonable!
     
  5. Dec 10, 2014 #4
  6. Dec 10, 2014 #5

    Erland

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    One answer is appealing to geometry, as madness mentioned. In a square with side length 1, the length of diagonal cannot be expressed as a rational number. But the diagonal is clearly a line segment, so it should have a length.

    We can also appeal to algebra. Mathematicians don't like unsolvable equations. If an equation, say ##x^2=2##, has no solution within a certain number system, we invent an extension of the number system such that that the equation has a solution in the extended system. One can view negative numbers, fractions, and imaginary (and complex) numbers in this way.

    As for the question about gaps in the real number system, there are no such gaps. The real number system is defined in such way that no such gaps can occur. There are some alternative definitions of the real numbers, but they give rise to isomorphic structures, and they all satisfy the following axiom, given by Dedekind:

    If A and B are nonempty disjoint sets of real numbers, such that A ∪ B = ℝ, and such that for all x ∈ A and y ∈ B, it must hold that x < y, then there is a an r ∈ R which is either the greatest number in A or the smallest number in B.

    This is not true with ℚ instead of ℝ, since ℚ has gaps, for example the gap produced by the irrational number ##\sqrt 2##.

    Rudin mentions this, if I remember correctly.
     
  7. Dec 10, 2014 #6
    That seems strange. So, it is not possible to come across another "equation" and find that the real numbers are insufficient, as we did with ##p^2 = 2## and the rational numbers? Didn't the pre-irrational-number people think that ##\mathbb{Q}## was complete?
     
  8. Dec 10, 2014 #7

    Mark44

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    The real numbers are not sufficient for this equation: x2 + 1 = 0. For this one you need the complex numbers.
     
  9. Dec 10, 2014 #8
    Oh, yes, complex numbers. Well, I suppose I could ask the very same question with complex numbers.
     
  10. Dec 10, 2014 #9
    Note that the extension to complex numbers wasn't exactly filling a "gap" like irrational numbers do, it was extending them in a different way. I could always add an extra element to the real number line if I like, but it wouldn't fill a "gap". A gap implies something about the ordering of numbers, i.e. if there is a gap between a and b then there should be some other (non-real) number that has a value in between any two real numbers.

    At a push, you might consider the hyperreal numbers (http://en.wikipedia.org/wiki/Hyperreal_number) as filling a gap (at infinity), but to me it's more like sticking another one on the end.

    There are also lots of extensions of complex numbers (http://en.wikipedia.org/wiki/Hypercomplex_number), but as far as I know they don't fill gaps in the usual complex numbers.
     
  11. Dec 10, 2014 #10

    lavinia

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    Starting with a compass and a straight edge, one can construct many lengths among them all of the rational numbers. One can also construct the diagonal of a square. It is natural if you believe that all numbers are rational to ask which rational number the length of this diagonal is. Using an elementary proof, the ancient Greeks discovered that it is not a rational.

    If not all numbers are rational, one might ask whether all numbers are constructible with a ruler and straight edge. In the 18'th century Gauss classified all possible constructible numbers and was able to show elementary examples such as the length of a side of a regular septagon that are not constructible. This meant that not only are all numbers not rational but not all of them are constructible either.

    If one thinks of ruler and compass constructions as constructing points in the plane, then one may think of these points as complex numbers. As such they all lie in iterated quadratic extensions of the rationals since ruler and compass constructions can only create square roots of known lengths.
    This of course is nowhere near all of the real or complex numbers and if one thinks about it the number of constructible numbers must be countably infinite since they are constructed iteratively starting with two points.

    One might try to generalize construction to taking roots of arbitrary polynomials (not just quadratic) starting with rational coefficients and iteratively including coefficients from the extension fields. Even though many new numbers arise, there are still only countably infinitely many of them. Numbers like pi are not obtained in this way. I do not know when it was first discovered that pi is transcendental but I imagine it would have stunned mathematicians much the way the discovery of the irrationality of the square root of 2 stunned the Pythagoreans. After all one can construct pi simply with only a compass.
     
    Last edited: Dec 11, 2014
  12. Dec 10, 2014 #11
    Okay, so I have another question.

    Below this example the author endeavors to show that there are gaps in the rational number system, which appears that it can only be done if you accept that ##p^2 = 2## must have some solution.

    To do this, the author considers all of the rational number solutions ##p^2 < 2##, and let ##A## contain all of these solutions; he also considers solutions to ##p^2 > 2##, and lets ##B## contain these solutions. Evidently, by showing that ##A## does not have a largest element, and that ##B## does not have a smallest element, we will have shown that there is a gap.

    He proceeds by supposing ##A## has a largest element, which he calls ##q##, so that ##p < q## ##\forall p \in A##. He then says that we can associate with each rational number ##p > 0## the number

    ##q = p - \frac{p^2 - 2}{p+2}##.

    Why would you do such a thing?

    ##q = p - \frac{(p+2)(p-2)}{p+2} = p - (p-2) = 2##

    Why would you define ##q## to be ##2##?

    Another more pressing question I have is, why would showing ##A## has no greatest element, and ##B## no smallest, show that there is a gap?
     
  13. Dec 10, 2014 #12

    lavinia

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    I don't know the history but it seems like ruler and compass constructions may be thought of as a program for constructing all of space, a failed program of course.
     
    Last edited: Dec 10, 2014
  14. Dec 10, 2014 #13

    Mark44

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    Where did the equation above come from? ##\frac{p^2 - 2}{p+2} \neq \frac{(p+2)(p-2)}{p+2}##
     
  15. Dec 10, 2014 #14

    Erland

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    If you mark the sets A and B on the number line, and if A has no greatest element and B no smallest, isn't it then geometrically obvious that there is a "hole" between the two sets, corresponding to a point on the number line?

    As for complex numbers, the field ℂ is algebraically complete: Every algebraic equation ##a_n z^n + a_{n-1} z^{n-1} + \dots + a_1 z + a_0=0##, with complex coefficients and ##a_n≠0##, has a complex root. This is the content of the Fundamental Theorem of Algebra.
    http://en.wikipedia.org/wiki/Fundamental_theorem_of_algebra
     
    Last edited: Dec 10, 2014
  16. Dec 10, 2014 #15
    Oh, whoops...Thank you for showing me my error Mark44. However, I am still confused as to why we would define ##q## as ##p - \frac{p^2 -2}{p+2}##. Certainly it it plausible to relate ##q## and ##p##, because we already have some relationship between them, that ##p < q##. But if we define ##q## in such a manner, as ##p## minus some quantity involving ##p##, won't this make ##q## less than ##p##?
     
  17. Dec 10, 2014 #16

    Erland

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    Is it ##p## or ##q## which is the supposed largest element in ##A##?

    Anyway, ##p - \frac{p^2 -2}{p+2}=p + \frac{2-p^2}{p+2}>p##, since ##p^2<2##.
     
  18. Dec 11, 2014 #17

    lavinia

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    - The idea of the proof is to show that there is no largest rational number less than the square root of 2. So given such a rational, one must find another that is bigger but still less than the square root of 2. Check that if ##p^2 < 2## then

    ## (p - \frac{p^2-2}{p+2} )^2 < 2##

    as well.

    - "Gap" just mans that there are numbers that are not rational. If you think of the number line then intuitively there must be holes and these holes separate the rationals into disjoint sets.

    As it turns out, there is no possible procedure for constructing all of the real numbers. One must define them in another way. All equivalence classes of Cauchy sequences of rationals is one way. Another is all Dedekind cuts. But there is no way to construct all of these cuts or equivalence classes. This is because the number of real numbers is a larger infinity than the infinity of the integers.
     
    Last edited: Dec 11, 2014
  19. Dec 11, 2014 #18

    mathwonk

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    As others have said, the geometric line preceded the real numbers. It was the desire to represent points on the line by numbers that led to the discovery of irrational numbers, and real numbers in general. This becomes clear if you start your mathematical education at the beginning, namely with Euclid. The need to define the ratio of two lengths, and to prove the principle of similarity for all triangles with the same angles, leads to the approximation of arbitrary lengths by rational lengths, in Book V or VI I believe in Euclid. See especially Book VI, Propositions 1 and 2.
     
  20. Dec 11, 2014 #19
    There are three kinds of gaps that I'm aware of. One of them is the gaps created by ruler and compass constructions, needed for the geometry of the ancient Greeks. You fill those in by throwing in square roots (including nested square roots). Another gap kind of gap is holes where there should be a solution to some algebraic equation. But what's most relevant for the real numbers is the gaps created by sequences that want to converge. Any sequence whose terms get arbitrarily close together ought to be converging to something (Cauchy sequences). What the real numbers really do is fill in those holes, so that any Cauchy sequence does converge to something. This is needed to get "obvious" facts like the intermediate value theorem to work. The intermediate value theorem is equivalent to completeness.

    The precursor to the real numbers was Eudoxus' theory of proportions, which is the subject of Book 5 of Euclid's Elements. Dedekind's Dedekind cuts construction of the real numbers was inspired by this ancient Greek theory. It was only later in the 19th century that the need for filling all these gaps to get calculus to work was really appreciated.

    The real numbers "exist", not in the real world, but mathematically, because we can construct them from the ground up, starting with things like the empty set and the ability to stick whatever sets we have together into another set, and the existence of an infinite set, and all that stuff. So, we don't just postulate the existence of the real numbers. We can construct them from simpler things, which we do have to postulate the existence of, but it's somewhat more satisfying to postulate something simple like some basic things about sets than to start with the real numbers.
     
  21. Dec 11, 2014 #20

    FactChecker

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    Not really. The way irrational numbers can be defined in modern math as limits of real numbers, there is no logic that allows anyone to deny they exist. It wouldn't be hard to define a series of rational numbers that come arbitrarily close to the solution of p2=2. The limit of that sequence would define the solution. Then you can prove that the solution is irrational.

    In the time of Pythagoras, they didn't believe in irrational numbers, but that was a different. Zero and negative numbers had not been invented and numbers were thought of as having mystical, religious properties. Geometry was all they really believed in. So the existence of an isosceles right triangle with a hypotenuse of sqrt(2) was a given. Pythagoras religiously believed that there was a rational number for that length. His religious beliefs did not allow irrational numbers. Some believe that his followers killed the man who proved that the square root of 2 was irrational. Here is an amusing (I think) account of it:
     
    Last edited: Dec 11, 2014
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