This expression is irrational. $$\tan(x)=\frac{x}{1-\frac{x^2}{3-\frac{x^2}{5-...}}}$$ But then he used the fact that $\tan{\frac{\pi}{4}}=1$, so $\frac{\pi}4$ is irrational. But how can we use tangent function here if we are proving the irrationality of $\pi$.
And is there any simpler proof also? (Using elementary functions and operations)

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    $\begingroup$ Why should we not be allowed to use the tangent function? $\endgroup$ – Robert Israel Aug 14 '15 at 15:23
  • $\begingroup$ Because, tangent function is taking $\pi$ as the input, and how can we input it when we are not sure if it is irrational or rational, we are proving it. $\endgroup$ – Aditya Agarwal Aug 14 '15 at 15:24
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    $\begingroup$ We use numbers all the time without knowing whether they are rational or irrational. $\endgroup$ – Robert Israel Aug 14 '15 at 15:26
  • $\begingroup$ For other proofs, see en.wikipedia.org/wiki/Proof_that_%CF%80_is_irrational $\endgroup$ – Robert Israel Aug 14 '15 at 15:27
  • $\begingroup$ @RobertIsrael But we are especially proving the irrationality of $\pi$ here. $\endgroup$ – Aditya Agarwal Aug 14 '15 at 15:30

Your first sentence is false. Probably the correct version is that your continued fraction is irrational if $x$ is rational. Therefore, since we know that $\tan(\frac{π}{4})$ is rational, $\frac{π}{4}$ cannot be rational.

  • $\begingroup$ How do we know that $\tan(\frac{\pi}{4})$ is rational, if we don't know if $\pi/4$ is rational. $\endgroup$ – Aditya Agarwal Aug 14 '15 at 15:28
  • $\begingroup$ @AdityaAgarwal: Read my argument carefully. If $\frac{π}{4}$ is rational, what do you know about $\tan(\frac{π}{4})$? $\endgroup$ – user21820 Aug 14 '15 at 15:32
  • $\begingroup$ Ohh, gotcha! Uh..Wait. But then how do we know that $\tan(\frac{\pi}4)=1$? Sorry, I know I am skeptical! $\endgroup$ – Aditya Agarwal Aug 14 '15 at 15:36
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    $\begingroup$ @AdityaAgarwal: Ah that's a good question. It is then a completely different question. The definition of $π$ is what you want. If it is defined as the perimeter of a circle of diameter $1$, or area of circle of radius $1$, then you need a lot of work to prove things like $\sin(π) = 0$. But some authors define $π$ as the first positive root of $\cos$, in which case it would take a lot of work to prove the geometric facts about $π$. So take your pick of definition but know that you get roughly the same amount of work to prove everything. $\endgroup$ – user21820 Aug 14 '15 at 15:39
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    $\begingroup$ @AdityaAgarwal: Sorry I missed your last comment. I recommend starting with the differential equation $f' = f$ with $f(0) = 1$ and approximating $f$ around $0$ using polynomials. Continue to the infinite series and use squeeze theorem to prove that it converges and then that its derivative is itself. If done right the proof is the same even if $f$ is a function on complex numbers. Now using the differential equation it is easy to prove all the properties of the exponential. Similarly $f'' = f$ and suitable initial conditions define $\cos,\sin$ and $\exp(ix) = \cos(x) + i \sin(x)$. [continued] $\endgroup$ – user21820 Aug 22 '15 at 13:36

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