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zero_call comments on Open Thread: March 2010, part 2 - Less Wrong
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Comments (334)
Think about the context here, though. Having a symbol for 2pi would be much more convenient because it would make things consistent. 2pi is the number that you typically cut into fractions. Let's say we define, say, rho to mean 2pi. Then we have rho, rho/2, rho/3, rho/4... whereas with pi, we have 2pi, 2pi/2, 2pi/3, 2pi/4... the problem is those even numbers. Writing 2pi/4 looks ugly, you want to simplify, but writing pi/2 means that you no longer see the number "4" there, which is what's important, that it's a quarter of 2pi. You see the "2" on the bottom so you think it's half of 2pi. It's a mistake everyone makes every now and then - seeing pi/n and thinking it's 2pi/n. If we just had a symbol for 2pi, this wouldn't occur. Other mistakes would, sure, but as commonly as this one does?
If we were to define, say, xi=pi/2, then 4xi, 2xi, 4xi/3, xi, 4xi/5... well, that's just awful.
What? Like who, 6th graders?
I find that unfair. I have made the mistake Sniffnoy describes many times, all of them after I was in 6th grade.
Easy solution. Pi is half a circle. Pie is the whole one. Then there is a smooth transition from grade 3 to university.
I've been looking for a good thing to call 2*Pi - this might cut it.
Nice one! ;)
No, like anyone who isn't watching out for traps caused by bad notation. It's much easier to copy down numbers than it is to alter them appropriately. If you see "e^(pi * i/3)", what stands out is the 3 in the denominator. Except oops, pi actually only means half a circle, so this is a sixth root of unity, not a third one. Part of why I like to just write zeta_n instead of e^(2pi * i/n). Sure, this can be avoided with a bit of thought, but thought shouldn't be required here; notation that forces you to think about something so trivial, is not good notation.
omega_n is the notation I most often run across.
Hm, I've generally just seen omega for zeta_3.
I've certainly used it for that -- but I pattern it with dropping the subscript n, when it is clear when there is only one particular root of unity we're basing off of. I've never ever seen zeta used.