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Math Help - irrational numbers

  1. #1
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    irrational numbers

    prove that the following are irrational numbers

    1) tan 5

    2) log 6 to the base 7
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  2. #2
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    Hello, fxs12!

    Here's the second one . . .


    \text{Prove that the following are irrational numbers: }

    . . (1)\;\tan 5 \qquad\qquad (2)\;\log_76

    Suppose \log_76 is rational.

    Then: . \log_76 \:=\:\dfrac{n}{d}\:\text{ for some integers }n,d

    . . And we have: . 7^{\frac{n}{d}} \:=\:6


    Raise both sides to the power \,d\!:\;\;7^n \:=\:6^d<br />

    . . And we have: . 7^n \;=\;2^d\cdot3^d


    We have a number that has two distinct prime factorizations.
    This is contradiction of the Fundamental Theorem of Arithmetic.

    Therefore: . \log_76 is irrational.

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  3. #3
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    Quote Originally Posted by fxs12 View Post
    prove that the following are irrational numbers

    1) tan 5

    [snip]
    Since \tan (n \theta^{0}) is a rational function of \tan (\theta) it can be deduced from the fact that \tan 60^{0} = \sqrt{3} that \tan (n^{0}) is irrational for n = 1, 2, 3, 4, 5, 6, 10, 12, 15 or 30.
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  4. #4
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    Is this \displaystyle \tan{(5^{\circ})} or \displaystyle \tan{(5^C)}?
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