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Posers and Puzzles

Posers and Puzzles

  1. Subscriber sonhouse
    Fast and Curious
    29 May '14 19:00 / 1 edit
    So PI is irrational, and on a game show they said the 5th root of 64 is irrational. Is that root really irrational?

    How can you tell if a number like that is going to be irrational or not?

    Is there some kind of math test or function that can predict irrationality?

    Last question:

    Is there some function or procedure that can turn any irrational number into a rational one?
  2. 30 May '14 13:49
    Originally posted by sonhouse
    So PI is irrational, and on a game show they said the 5th root of 64 is irrational. Is that root really irrational?

    How can you tell if a number like that is going to be irrational or not?

    Is there some kind of math test or function that can predict irrationality?

    Last question:

    Is there some function or procedure that can turn any irrational number into a rational one?
    The last one is easy:
    Say that the number x is irrational, then the function f(x) = 0*x is not only rational but also natural.
  3. Subscriber sonhouse
    Fast and Curious
    30 May '14 14:42 / 5 edits
    Originally posted by FabianFnas
    The last one is easy:
    Say that the number x is irrational, then the function f(x) = 0*x is not only rational but also natural.
    Or f(x)=x/0 maybe also

    I was however, thinking of maybe real numbers greater than zero

    So I found this, proof that the square root of 2 is irrational.

    The proof that square root of 2 is irrational:

    Let's suppose sqr root of 2 were a rational number. Then we can write it sqr root 2 = a/b where a,b are whole numbers, b not zero.
    We additionally assume that this a/b is simplified to the lowest terms, since that can obviously be done with any fraction. Notice that in order for a/b to be in its simplest terms, both a and b must be not be even. One or both must be odd. Otherwise, you could simplify.

    From the equality sqr root 2 = a/b it follows that 2 = a^2/b^2, or a^2 = 2 * b^2. So the square of a is an even number since it is two times something. From this we can know that a itself is also an even number. Why? Because it can't be odd; if a itself was odd, then a * a would be odd too. Odd number times odd number is always odd. Check if you don't believe that!

    Okay, if a itself is an even number, then a is 2 times some other whole number, or a = 2k where k is this other number. We don't need to know exactly what k is; it won't matter. Soon is coming the contradiction:

    If we substitute a = 2k into the original equation 2 = a^/b^2, this is what we get:

    2 = (2k)^2/b^2
    2 = 4k^2/b^2
    2*b2 = 4k^2
    b2 = 2k^2.

    This means b^2 is even, from which follows again that b itself is an even number!!!
    WHY is that a contradiction? Because we started the whole process saying that a/b is simplified to the lowest terms, and now it turns out that a and b would both be even. So the square root of 2 cannot be rational.

    I'm still digesting this!
  4. 31 May '14 11:04
    Is there some kind of math test or function that can predict irrationality?
    Not a known test that works for everything, nobody has yet proved whether e+pi is irrational or not.
  5. Subscriber sonhouse
    Fast and Curious
    31 May '14 17:46 / 1 edit
    Originally posted by iamatiger
    Not a known test that works for everything, nobody has yet proved whether e+pi is irrational or not.
    What about e*pi?

    I found this about those numbers: Dr. Math site.

    http://mathforum.org/library/drmath/view/51617.html
  6. 23 Jul '14 19:49
    Proof that the 5th root of 64 is irrational.

    Suppose not and that it is a/b where a and b are coprime ie have no factor in common. Now factorise both a and b into prime factors. As a and b are coprime then they must have no prime factor in common and hence a^5 and b^5 also have no prime factors in common and a^5=(2^6)*(b^5). Comparing powers of 2 says that 2 divides a and hence not b. But then the power of 2 appearing on the LHS must be a multiple of 5 but the power of 2 appearing on the RHS is 6. This is a contradiction.