[Daily puzzle] is this number rational \ stacker news ~science

[Daily puzzle] is this number rational

1053 sats \ 10 comments \ @south_korea_ln 25 Oct science

After this week's hiatus, hope to pick up again more regularly.

Today's question:

Is the tangent of

rational, i.e.

with

and

integers.

Provide a proof for it.

Previous iteration: #732372

Thanks for posting this puzzle too: #734034

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542 sats \ 9 replies \ @Scroogey 25 Oct

We can use the trigonometric addition formula

in an inductive proof by contradiction:

Assumption: is rational.
Step: if is rational then must also be rational, because the above formula is the ratio of two rational numbers.
But which is irrational, hence the assumption must be wrong.

11 sats \ 2 replies \ @south_korea_ln OP 25 Oct

Nice job. This was supposedly an entrance exam question for the University of Kyoto in 2006.

160 sats \ 1 reply \ @Rothbardian_fanatic 25 Oct

I understand the Kyodai and Todai exams are literal nightmares! They have people going to special exam schools for several years on end to get the score high enough to be accepted.

0 sats \ 0 replies \ @south_korea_ln OP 26 Oct

Yeah looks pretty extreme. You really need to be trained on this to be able to solve this kind of problem with tight time constraints.

1 sat \ 5 replies \ @SimpleStacker 25 Oct

Nice. Are you a mathematician, @Scroogey?

0 sats \ 4 replies \ @Scroogey 25 Oct

No, I enjoyed college level math thirty-plus years ago, but was more interested in computer programming than pure math.

1 sat \ 3 replies \ @SimpleStacker 25 Oct

Pulling out an inductive proof by contradiction, along with an obscure (to me) formula, is impressive for someone who doesn't do proofs on a regular basis :)

0 sats \ 2 replies \ @Scroogey 25 Oct

I remember such proofs were considered less elegant or even rejected by some mathematicians in earlier times, but don't know the historic context anymore.

1 sat \ 1 reply \ @SimpleStacker 25 Oct

Proof by contradiction was my favorite method in college. I think it's easier because you know where to start and you just move forward. So even to prove that things are true, I'd just assume the converse and show a contradiction.

Because a non-contradiction based proof seems harder. You know where you want to go, but you don't know where to start.

22 sats \ 0 replies \ @south_korea_ln OP 26 Oct

Yeah, Reductio ad Absurdum were also my favorites.