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Math Help - Rational & Irrational Numbers Problem

  1. #1
    Member iPod's Avatar
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    Rational & Irrational Numbers Problem

    I came across this question in my exam, however I have exactly no clue on how to come across it.

    ''If the numbers
    e,\pi ,\pi^2, e^2,e\pi, are irrational,
    prove that at most one of the numbers
    \pi+e,  \pi-e,  \pi^2-e^2,  \pi^2+e^2 is rational.''

    I think the person who gets a solution to this deserves a prize
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  2. #2
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    Seems begging for contradiction.

    Suppose two are rational. Let's take \pi + e and \pi - e.

    Those can be expressed as ratios of integers.

    \pi + e = \frac{a}{b} and \pi - e = \frac{c}{d}

    Solving simultaneously shows e is rational AND \pi is rational, since:

    1) Looking at denominators, they must be divisible by 2bd and
    2) Looking at numerators, integers are closed over addition, subtraction, and multiplication.

    I certainly have not covered the whole proof, but that seems a good piece of it.
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  3. #3
    Member iPod's Avatar
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    seems like a good enough answer actually, well done -
    thanks
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