Someone reminded me of this during our usual mathematical arguments over lunch.
Replacing pi with a value twice it's size would seem to be a zero-sum proposition: it makes circumference calculations "more natural" at the expense of area calculations, and makes radian considerations easier at the cost of complicating the periodicity of the tangent function.
Even so, I say we go with it for two reasons.
First, it will set a precedent whereby I can double the amount of grant money I can apply for, on the grounds that the increased figure "seems more natural".
Second, it will really piss this guy off.
12 comments:
Actually it makes area calculations more natural as well as it falls into line with other quadratic forms. The arguments here are very persuasive: http://tauday.com
I particularly like the one about Euler's identity which has never seemed to me quite as beautiful as people claim. With tau instead of pi, it's much more elegant.
Hmm. That's an interesting article, and I confess that my consideration of the unit of "tau" revolved around how it would affect school teaching rather than general mathematics (apparently, you can take the teacher out of school...)
Of course, that point is addressed here, too. If there was more data on this, if it really does prove the case that using tau helps kids more than pi does, then I'd actually be seriously in favour of trying it.
For what it's worth, my main issue is with the idea that you can make radians and trig easier for A-level students (and presumably undergrads) by removing the multiple of two, but you won't making it much harder for young children when you stick a half in front of the area formula for a circle. That latter group would be my first concern.
Having said that, of course, there need be no such choice. Just throw out pi in A-level maths the way we were told to throw away valency in A-level chemistry.
I'm also not sure that "common" really implies "natural". If anything is natural here is how often one integrates in maths, and I fail to see how int(2x)dx =x^2 + c is any less natural than int(x)dx =x/2 + c.
It's interesting you mention Euler's identity. To me it looks short and squat, because I'm missing my 2. In fact, I too have recognised the prevalence of 2*pi in maths (though pi/2 is exceptionally common as well), which I think has probably led to a situation in which I actually see 2*pi as being almost a single term. Which both explains why I don't like the idea of getting rid of it, and gives more credence to the idea that we should give it it's own name.
This whole tau nonsense has to be the biggest pile of pointless since that nonsense about not-being-able-to-decide-is-also-a-decision in Munich.
We should just have both and use whatever is more convenient for the calculation at hand. Indeed, we should also have another constant that is tau/24, and then we'd be able to use that to get rid of most of the fractions in A level questions. pi/12 is a much more annoying number than 2pi.
God, the infinitely recursive "deciding to not decide between groups of decisions" idea. I still get nightmares about that.
And you're quite right, naturally: pi/12 can fuck off.
SpaceSquid - I defer to your teaching experience but I don't see why addition of a half factor to the area formula makes it much harder for young children. They seem to cope with the half in the area of a triangle without any problems. If I recall correctly at that age I just thought of these as magic formulae which you plugged numbers into and got an answer out. Even if there is a small difference it's surely far outweighed by the conceptual simplicity of thinking of a quarter-turn as tau/4, a half-turn as tau/2, etc.
The reason 1/2 tau r^2 is more natural is because it makes far clearer how the formula is derived.
And the only reason the new Euler identity looks wrong to you is because you're used to how the ugly old one looked. You'll get over it!
BigHead - I was as sceptical as you until I read the arguments at tauday.com. The author even addresses the "use whichever is most convenient" red herring. If you have good reasons why the arguments given there are wrong, I'd love to hear them.
Tomsk, you'd tear at your eyes and rend your garments with horror if you knew how many children forget to multiply by a half when calculating the area of a triangle.
In that case putting a 1/2 in the circle formula too will help remind them...
But that puts the parallelograms in danger! Where will the madness end?
Come to think of it they should just get a pencil case with all the formulae printed on like wot I had.
For someone who has no intention of reading that insane page in its entirity, could someone tell me where he addressed the "use whatever is convenient" argument?
Insane?! When the tau revolution comes, you'll be the first against the wall.
Obvious question - How much effort does it take to change (presumably this would involve reprinting anything that contained the old pi way of doing things, and a small amount of training to the various teachers involve), in what time scale would you need to do this in (i.e could you run both methods for a while or would everything have to switch over at the same time), and is the hassle of doing this outweighed by the benefits in the change?
For the record I can see tau being a better option, but I suspect that the general upheaval required to change to it as being far more costly than any advantage it may bring, and running them both at the same time would just be confusing.
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