Comment by bko
4 days ago
I like watching youtube videos solving these problems. They're deceptively simple. I remember reading one:
x+y=1
xy=1
The incredible thing is the explanation uses almost all reasoning steps that I am familiar with from basic algebra, like factoring, quadratic formula, etc. But it just comes together so beautifully. It gives you the impression that if you thought about it long enough, surely you would have come up with the answer, which is obviously wrong, at least in my case.
This is slightly tedious to do by hand but there isn't really anything interesting going on in that problem - it's just solving a quadratic equation over the complex numbers.
That isn't much of an argument; nothing in math is truly interesting if you take that approach. exp(i\pi)+1=0 could be said to be dis-interesting because it is just rotation on the complex plane. But it is the opposite - it is interesting because it turned out to be rotation on the complex plane but approached from summing infinite series.
Similarly you can say that solving a quadratic over complex numbers is dis-interesting, but it is actually an interesting puzzle because it is trying its best to pretend it isn't a quadratic. In many ways succeeding, it isn't a quadratic - there is no "2" in it.
It's "not interesting" because no novel insight has to be used in order to solve this. It's immediately obvious how to solve it, just follow the textbook procedure.
This is distinct both from other typical IMO problems that I've seen and from research mathematics which usually do require some amount of creativity.
> exp(i\pi)+1=0
If your definition of "exp(i*theta)" is literally "rotation of the number 1 by theta degrees counterclockwise", then indeed what you quoted is a triviality and contains no nugget of insight (how could it?).
It becomes nontrivial when your definition of "exp" is any of the following:
- The everywhere absolutely convergent power series sum_{i=0}^\infty z^n/n!
- The unique function solving the IVP y'=y, y(0)=1
- The unique holomorphic extension of the real-valued exponential function to the complex numbers
Going from any of these definitions to "exp(i*\pi)+1=0" from scratch requires quite a bit of clever mathematics (such as proving convergence of the various series, comparing terms, deriving the values of sin and cos at pi from their power series representation, etc.). That's definitely not something that a motivated high schooler would be able to derive from scratch.
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It is not interesting because there is no “real” solution (pun intended).
If you go to the complex plane, you are re-defining the plane. If you redefine the plane, then you can do anything. The puzzle is about confusing the observer who is expecting a solution in a certain dimension.
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