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Thread: 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 . . .


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

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

    Suppose $\displaystyle \log_76$ is rational.

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

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


    Raise both sides to the power $\displaystyle \,d\!:\;\;7^n \:=\:6^d
    $

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


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

    Therefore: .$\displaystyle \log_76$ 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 $\displaystyle \tan (n \theta^{0})$ is a rational function of $\displaystyle \tan (\theta)$ it can be deduced from the fact that $\displaystyle \tan 60^{0} = \sqrt{3}$ that $\displaystyle \tan (n^{0})$ 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 $\displaystyle \displaystyle \tan{(5^{\circ})}$ or $\displaystyle \displaystyle \tan{(5^C)}$?
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