prove that the following are irrational numbers

1) tan 5

2) log 6 to the base 7

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- Jan 27th 2011, 03:51 AMfxs12irrational numbers
prove that the following are irrational numbers

1) tan 5

2) log 6 to the base 7 - Jan 27th 2011, 04:39 AMSoroban
Hello, fxs12!

Here's the second one . . .

Quote:

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

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

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.

- Jan 28th 2011, 12:33 AMmr fantastic
- Jan 28th 2011, 12:49 AMProve It
Is this $\displaystyle \displaystyle \tan{(5^{\circ})}$ or $\displaystyle \displaystyle \tan{(5^C)}$?