finding a rational number between a rational and irrational

Discussion in 'Undergraduate Math' started by actuarynrt, Aug 26, 2007.

  1. actuarynrt

    actuarynrt Guest

    I've been getting great help here in this discussion forum.

    The question I have is:
    Find a rational number between 3.14159 and π.

    I am thinking i could just select some number slightly larger than 3.14159 like 3.141591 and that would be it; because π is approximately equal to 3.14592654...

    Am I correct in selecting 3.141591 as an answer? My text doesn't give an answer (I'm assuming because there could be many).

    Thanks.

    John
     
    actuarynrt, Aug 26, 2007
    #1
    1. Advertisements

  2. actuarynrt

    actuarynrt Guest

    Oops...

    the ? should actually be Pi

    Looking for a number between 3.14159 and Pi
     
    actuarynrt, Aug 26, 2007
    #2
    1. Advertisements

  3. actuarynrt

    Virgil Guest

    Your value is one of infinitely many rationals between them.
     
    Virgil, Aug 26, 2007
    #3
  4. actuarynrt

    Jeremy Watts Guest


    Do you mean 'irrational' rather than rational? Because as Virgil pointed
    out there will be infinitely many rationals between 3.141591 and pi, you
    could take your pick...

    If you meant irrational then 15001/15000 * cuberoot (31) is approximately
    equal to 3.14159008, and is irrational and lies between 3.14159 and pi.


    Jeremy Watts
     
    Jeremy Watts, Aug 26, 2007
    #4
  5. As others have stated, there are an infinite number of solutions for
    this sort of problem. One way to get an answer is the way you did it:
    1) List the digits of the decimal expansion of both the given rational
    and irrational numbers, until there is a difference.
    2) Extend your number by choosing digits until it is between the
    givens (generally this should only take 2 more digits at most, but you
    might have a 49999999 vs. 50000000 situation that needs more digits)
    3) If you need a rational, stop here. If you need an irrational,
    append here any irrational sequence of digits (e.g.
    101001000100001....)
     
    The Qurqirish Dragon, Aug 26, 2007
    #5
  6. actuarynrt

    Dana Guest

    Find a rational number between 3.14159 and ?.
    I would say "no", as 3.14... is not a Rational Number.
    The obvious answer is any number between 0 and Pi - 3.141591.
    Here's a different approach at a typical computer' precision.

    Here's your number:
    num = 314159/100000

    Let' turn the difference into a ratio:
    Rationalize[N[Pi - num]]

    199421 / 75151404527

    This should be the thightest ratio we can get on our computer.
    This works:
    num + 199420./75151404527 < Pi
    True

    But this is a hair too high.
    num + 199421./75151404527 < Pi
    False

    Since there are an infinit solutions to a/b < Pi-num,
    what if we picked "b", and found the range for a?
    (n-Numerator, d-denominator)

    Reduce[n/d < 199421/75151404527, n][[2]]
    n < (199421*d)/75151404527

    Here, given a denominator 'd, we can find the range for 'a
    Suppose we want to use a denominator of 1,000,000.


    (199421*d)/75151404527 /. d -> 1000000

    2.6535897932333796

    Our numerator must be <2.6, or basically, 1 or 2.
    Hence, 2/1,000,000 is ok.

    num + 2./1000000 < Pi

    True

    but we now know this should be too high.
    num + 3./1000000 < Pi

    False

    If we reverse the above equation, we find that the denominator must be >=
    376,848
     
    Dana, Aug 26, 2007
    #6
    1. Advertisements

Ask a Question

Want to reply to this thread or ask your own question?

You'll need to choose a username for the site, which only take a couple of moments (here). After that, you can post your question and our members will help you out.