Showing that tan(1) is irrational

  • #1
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Homework Statement


Prove that ##\tan (1^\circ)## is irrational.

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The Attempt at a Solution


Suppose for contradiction that ##\tan (1^\circ)## is rational. We claim that this implies that ##\tan (n^\circ)## is rational. Here is the proof by induction: We know by supposition that the base case holds. So, suppose that ##\tan (n^\circ)## is rational. Then ##\displaystyle \tan (n^\circ + 1^\circ) = \frac{\tan(n^\circ) + \tan(1^\circ)}{1-\tan(n^\circ)\tan(1^\circ)}##, and this is the ratio of two rational numbers, and so is rational. So by mathematical induction ##\tan (n^\circ)## is rational.

However, this implies that ##\tan(30^\circ) = \sqrt{3}## is rational, which is a contradiction.
 

Answers and Replies

  • #2
Orodruin
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Do you have a question?
 
  • #3
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Do you have a question?
I guess I was just attempting a solution. Seems like it's correct though
 
  • #5
RPinPA
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Summarizing, if ##\tan(1)## is rational then it follows by induction that ##\tan(n)## is rational for all ##n \in N##. An inductive proof by contradiction is a structure I haven't seen a lot, in fact I can't offhand recall such a proof. But it certainly seems valid to me.

The only issue is, @haruspex pointed out, is that ##30^\circ## is not the angle you want to use in your final line of argument.
 

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