Another "stupid" question about the square root of a prime number

Discussion in 'General Math' started by Doug Wedel, Sep 1, 2008.

  1. Doug Wedel

    Doug Wedel Guest

    In a previous post it was shown by those obviously more learned than I that
    the square root of a prime number is an irrational number. And again I am
    curious, perhaps again revealing my ignorance, but this irrational number
    that is the square root of a prime, we could actually start to compute it,
    and I wonder, what if anything can we say about the actual stream of digits
    that would be produced by the computation of the square root of a prime
    number? Would they bear any formal similarity to the stream of digits
    produced by, say, generating pi?
     
    Doug Wedel, Sep 1, 2008
    #1
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  2. What do you mean by similarity? What is the difference between
    ordinary similarity and formal similarity? Why are you limiting
    yourself to the square root of prime numbers?

    All irrational numbers, whether transcendental (such as pi), the
    square root of a prime, the square root of any other non-perfect
    square integer, etc, share certain properties. When expressed in
    decimal (or other integer base): they never terminate and they never
    form an infinitely repeating pattern. What more are you looking for?
     
    Barry Schwarz, Sep 2, 2008
    #2
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  3. Doug Wedel

    Doug Wedel Guest


    Actually that was what I was looking for! However a point of confusion
    remains. Wikipedia informs me that all transcendentals are irrational and
    most irrationals are transcendentals.

    Which other irrationals besides the square roots of prime numbers are _not_
    transcendental?
     
    Doug Wedel, Sep 2, 2008
    #3
  4. Doug Wedel

    Rick Decker Guest

    Things like the solutions of x^7 - x + 2 = 0, for example. Look up
    "algebraic numbers". Basically, if you have any polynomial p(x) with
    integer coefficients and p(x) = 0 has no rational solutions,
    then the solutions will be irrational and not transcendental.


    Regards,

    Rick
     
    Rick Decker, Sep 2, 2008
    #4
  5. May I suggest that you try using google. I suggested in an earlier
    post that
    you seemed not to understand the meaning of transcendental. One might
    think
    that the suggestion would inspire someone truly interested in the
    subject to
    look up the answer using google. Do you plan on doing ANY study on
    this subject?
    Or do you just want answers handed to you?

    A transcendental number is a number that is NOT the root of any
    polynomial.
    Equally clearly, any irrational number that IS the root of a
    polynomial is NOT
    transcendental. Such numbers form a set commonly referred to as the
    algebraic
    numbers. The algebraic numbers union the transcendentals form the
    reals. They
    are disjoint sets. This is all elementary, pre-calculus mathematics.
     
    Pubkeybreaker, Sep 3, 2008
    #5
  6. Doug Wedel

    Doug Wedel Guest


    Perhaps you you don't notice when your students are studying and trying?
    After your original post I did indeed read up to see how I had misused the
    word
    "transcendental" and as my post itself clearly states, I researched
    irrational
    numbers in the Wikipedia as well.
    I hope it's not impertinent to ask: is there any way to estimate what
    proportion of
    irrationals are transcendental? The Wikipedia says "almost all." Are we
    talking 99 percent?
    Or 99.9999999999999999 percent?
     
    Doug Wedel, Sep 3, 2008
    #6
  7. Doug Wedel

    W. Dale Hall Guest

    Again, from Wikipedia:

    http://en.wikipedia.org/wiki/Transcendental_number


    However, transcendental numbers are not rare:
    indeed, almost all real and complex numbers are
    transcendental, since the algebraic numbers are
    countable, but the sets of real and complex
    numbers are uncountable. All transcendental
    numbers are irrational, since all rational
    numbers are algebraic. (The converse is not true:
    not all irrational numbers are transcendental.)

    From there, Google

    cardinality countable uncountable

    and take your pick of ~41000 hits.

    Dale
     
    W. Dale Hall, Sep 3, 2008
    #7
  8. Doug Wedel

    W. Dale Hall Guest

    ... stuff deleted ...

    Here, I ignored the fact that Mr. Wedel quoted Wikipedia
    and referred him to Wikipedia yet again.

    The relevant portion of my article should have been
    to note that he had no appreciation for how "almost"
    one gets with "almost all".

    The sets (transcendental, algebraic) of numbers are
    both infinite. However, the latter (algebraic) can
    be put in 1:1 correspondence with the natural numbers,
    while the former (transcendental) can be put in 1:1
    correspondence with the reals.

    In other words, if you're comparing "how many" or
    "what proportion" of transcendentals there are,
    it's like this:

    #transcendentals : #reals :: 1:1

    #transcendentals : #algebraics :: #reals : #naturals
     
    W. Dale Hall, Sep 3, 2008
    #8
  9. Doug Wedel

    Doug Wedel Guest

    This is lapidary. Thanks.
     
    Doug Wedel, Sep 4, 2008
    #9
  10. No. We are talking 100%. You need to learn some measure theory
    to understand this. An alternative to measure theory is realizing
    that
    the algebraic numbers are COUNTABLE, while the transcendentals are
    uncountable. Thus, the algebraic numbers form a subset of density
    0 in the reals. While both sets are infinite, one set (the
    transcendentals)
    is "vastly" larger.

    They can't be put into 1-1 correspondence. In a vague sense (and not
    very precise) one
    could say that for EACH algebraic number there are uncountably
    infinitely
    many transcendentals.

    Look at it another way. You know that there are infinitely many
    integers.
    You know that there are infinitely many integers that are perfect
    squares.
    Now ask: what fraction of the integers are perfect squares?

    The number of integers <= than N^2 is N^2. The number of squares is N.
    The fraction of squares to integers is then N/N^2. Now let N go to
    infinity in the
    limit.
     
    Pubkeybreaker, Sep 4, 2008
    #10
  11. Doug Wedel

    akitaishi Guest

    Hi ,

    I think there is a difference between transcendentals and irrationals.

    If I may boldly put, transcendental are more irrational than
    irrational numbers.
    I could discuss this further if anyone wishes to discuss more.

    Jason Akitaishi
     
    akitaishi, Oct 14, 2008
    #11
  12. You may not. You don't know enough mathematics to have an opinion
    on the subject.
    There is a topic in mathematics known as "measures of irrationality".
    I suggest that you look it up.

    Liouville's theorem shows that the transcendentals are far
    easier to approximate by rationals than the algebraics.
    You don't know enough about the subject to conduct a meaningful
    discussion.
     
    Pubkeybreaker, Oct 14, 2008
    #12
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