Er, I'm not saying that changing pi would mess up calculus. I'm saying that changing the unit we use for angles (in such a way as to make the existing pi "right") would mess up calculus.
What? sin(x) is still sin(x). Just because we have a different value for pi doesn’t mean we have a different sin function. sin(6.28...) = sin(6.28...), etc., etc.
Crucially, we still have exp(x + iy) = exp(x) * (cos(y) + i sin(y)), because we haven’t changed the definitions of any of these functions. If we did change the definitions of sin and cos, that’d really be a bummer, you’re right.
We still measure angles in radians. No diameter-ians in sight. Our old x or new y (notice, those have the same value) is just a different fraction of newpi than it is of oldpi, is all.
It amazes me that amalcon’s being voted up and aston is being voted down. People clearly aren’t thinking it through for themselves.
Just because we have a different value for pi doesn’t mean we have a different sin function.
Which is exactly what I just got done saying that I'm not saying. Reading comprehension, much?
To be entirely clear: All I was saying is that the reason we use radians in the first place (instead of, say, cycles) is that it fixes calculus. It has little to do with 2pi. It only relates to the comment it was said in reply to.
Yes, I’ve re-read your original post 4 times, and your intended meaning is quite confusing, because you’re talking about a different way of changing our notation than the link is, but without clearly stating that, and your notation change, which you criticize, is something of a non-sequitur in context of the parent comment and the article, as far as I can tell.
Thus, you seemed to be implying† that the new definition of pi results in messing up calculus. To clear things up: “We could measure angles in any arbitrary units we want, but using radians makes calculus work, and if we’re using radians, the circumference is 2 pi of them, which is why pi as a unit is not ideal, and newpi = 2*pi would be better. If we wanted we could have an angle of pi ‘diametrans’ in a complete circle instead, using our existing definition of pi ~ 3.14, but that would be stupid, because it would break all kinds of symmetries in calculus.”
†: This is apparently a misinterpretation though (mine and also aston’s, who wrote “a radian is a radian”), and you don’t actually mean to be implying that.
> Just because we have a different value for pi doesn’t mean we have a different sin function.
Although amalcon seems to disclaim this point of view (http://news.ycombinator.com/item?id=1450919), I think that this is exactly what it does mean—because we are used to viewing sine as a function that takes numbers (unitless), rather than measurements (with units).
The grand(^n)parent (http://news.ycombinator.com/item?id=1450467) talked about trigonometric functions of cycles, with the understanding that sin(x) now means sin(x cycle) = sin(2πx radian), so that
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(d/dx)sin(x) = (d/dx)sin(x cycle) = (d/dx)sin(2πx radian) = 2πcos(2πx radian) = 2πcos(x cycle) = 2πcos(x).
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Note that, with this convention, sin(6.28…) does not* equal sin(6.28…)—because, as you can tell, the first 6.28… is clearly measured in cycles, and the second in radians. I'm pretty sure that this is all that amalcon was saying.
