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To be fair, "algebra" means very different things to different people, so I don't think Grue3 meant to be quite as brash as it may have sounded.
by joelg 10y ago
To be fair, "algebra" means very different things to different people, so I don't think Grue3 meant to be quite as brash as it may have sounded.
- nandemo 10y agoWhat's the definition of basic algebra that makes that fact obvious? I took 3 semesters of algebra in college (Linear Algebra, Abstract Algebra I and II), and don't see anything obvious about it. But then, neither does Terry Tao: > "the powers φ, φ2, φ3, … of the golden ratio lie unexpectedly close to integers...
- Grue3 10y agoIt becomes obvious when you know closed form expression for Fibonacci numbers as a sum of powers of phi (which I'm sure Tao is aware of). I was taught that in high school. The only reason this wouldn't be obvious to Tao is if he was working on much harder problems for so long that he forgot the basic stuff.
- adrianratnapala 10y agoIt's not basic algebra in the sense of my high-school "Maths B" class. But it is just the kind of thing we did in "Maths C". Here a hint: use the definition of φ that 1 = φ - 1/φ And use it to prove that if F[k] = φ^k + 1/(-φ)^k is an integer, then F[k+1] = φ^(k+1) + 1/(-φ)^(k+1) Must also be an integer. Now show that this proves by induction that all F[k] are integers and note what happens to 1/(-φ)^k as k grows to infinity.
- recursive 10y agoThat seems like a clever proof, and I understand it and even admire it. But it doesn't look like that the kind of thing that's "obvious" unless you frequently work with this kind of thing.