Pi is Wrong
math.utah.edu
math.utah.edu
2/3 * NewPi = (2/3's around the circle in radians).
Probably the only drawback is when doing wicked tricks on a snowboarding game such as SSX. Doing a 1080 just sounds cooler then a 3, but which is more intuitive?=)
The only problem with these units is that, like with degrees, d/dx sin(x) != cos(x). It messes with calculus.
d/dy sin(y) = d/dx sin(2x) d/dx (2x) = 2 cos(2x) = 2 cos(y)
It's the same reason why you use radians for angles in calculus: degrees mess it up.
http://en.wikipedia.org/wiki/Trigonometric_functions#The_sig...
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.
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.
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 --- (d/dx)sin(x) = (d/dx)sin(x cycle) = (d/dx)sin(2πx radian) = 2πcos(2πx radian) = 2πcos(x cycle) = 2πcos(x). --- 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.
Reminds me of electrical engineering "mistakes" such as the convention establishing electrons as negatively charged; or the ohm being very small/amp being very large compared to everyday usage.
There have always been a small number of people who claim that electromagnetic charges should be reversed: that is, that the positively charged particle should logically be what "moves" in the circuit.
Most people just look at it, shrug, decide it doesn't really matter, and get back to doing it the way they've always done.
The only real relationship between the charges is that they are opposite. The terms positive and negative have no other meaning. As for the formulas, either way will work just fine, as long as everyone is consistent. Pick something and stick with it.
My view of the problem is that it's hard to grasp what current actually is. The water in a pipe analogy works, but it's confusing because the electrons flow in the opposite direction of the hcurrent, making the "water" positively charged electron holes.
Charged ions moving in an electrolytic cell is charge flowing in a circuit. Electrons are not the only way to have a current.
If an electric charge of -4e doesn't represent a surplus of 4 electrons, what does it represent? Surely not a deficit of 4 protons, since they are much harder to move around?
Perhaps you feel it represents a transient property of matter that just so happens to be closely related to, but not defined by, the movement of elementary particles? That's a reasonable position to take, if you don't have to effect a charge.
And the north magnetic pole is at the geographic south / vice versa.
And horsepower != 1000 Watts.
And Farads are bigger than I'd like them to be.
Ampere = Coulomb/second
Farad = Coulomb/volt
Amps being too large is unfortunate, but it's not a big deal.
That has to class as irony.
Out of curiosity: Does anyone here genuinely believe that Pi should be the circumference÷radius and hold that dates should be written Month/Day/Year ?
I've heard one reasonable defence of American date ordering based on actual priority of information (roughly: "you want the month first to broadly narrow down the locus but the year will be assumed"). But I still go with English or just [truncated] ISO dates.
on an unrelated note, when will the scribd links switch to html5?
carterschonwald is commenting that the Scribd link uses Flash, rather than the recently released HTML5 Scribd implementation.
http://mathoverflow.net/questions/20960/why-is-the-gamma-fun...
pi/4 = 1 - 1/3 + 1/5 - 1/7 + 1/9 - 1/11 ...
That's arguing from pure mathematics. Judging from the comments, I think the article argues (from kind of an engineering point of view) that 2pi would be more convenient.
e^(i pi/2) = i
e^(i pi/4) = sqrt(i)
both of which are arguably even more expressive.
For instance if we assume that half the time we need to divide into halves, 1/4 of the time into thirds, 1/8 into quarters and so on, then over 90% of the time you wish to split something into 1/2, 1/3, 1/4, 1/6 or 1/12. All of which are trivial in a duodecimal system. By contrast only 56% of the time do you wish to split into the similarly easy 1/2, 1/5 or 1/10 in decimal. Even if we say that 1/4 and 1/8 are OK in decimal, we still wind up with inconvenient repeating fractions over 3 times as often as duodecimals do.
However, I was impressed by the far-reaching implications of this fact. The regularity of the base 12 times table was particularly compelling. You know how easy the 2's and 5's row is to learn on the times table, right? There's an obvious pattern that's easy to memorize? Lots of numbers are like that in base 12.
http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Babyl...
But it seems unlikely as the origin of the Babylonian system cited. The 60 symbols are written in base 10; if it grew out of a culture that counted to twelve on their fingers, I'd expect to see five groups of twelve, not six groups of ten.
I can think of a more natural explanation. When I am counting something on my fingers (in the conventional way), I often want a way to store the tens digit. Perhaps the Babylonians did, too. That the tens digit is drawn as two groups of three is suggestive that whatever method they used stored a trinary and a binary state.
So, when I count off (palms up), I get to ten on the thumb of my right hand. It has a little more dexterity than my other fingers and can store more states. It's pretty straightforward to get to six by leaving it bent/straight and pointing out/up/in. It's natural to do while counting, too. (Bent and pointing in is a little uncomfortable if I want to use the other fingers, but then . . . you don't ever need to actually store 'six').
I remember the moment as a kid when I realized that the number 10 wasn't some magical number, but just some arbitrary number we've decided to use as a base. Definitely a stupid choice IMHO, but it's what we're stuck with.
It would have actually been more useful to use 8 fingers = octal.
But then who counts with their fingers these days?
(2^10 - 1 : you have 10 digits on your hands. This also assumes you have a value for 0.)
Sadly, I know this from experience.
However, if you bother thinking about the word arbitrary too hard (as your comment inspired me to do), you end up in the realm of existentialism and down the proverbial rabbit hole.
Base 10 may not be completely arbitrary, but is is undoubtedly lazy.
Well, it’s certainly too hard to practically switch, and so we won’t. But that doesn’t necessarily mean it’s not worth the cost. We’ve apparently had π in its current form for for 300 years; we’re going to have to live with it much longer than that.
The way to actually make a change like that though, assuming we wanted to, would be to just make up a new symbol for 2π, and start using it. For a while, the two symbols would coexist, and then someday the π symbol would just start to fade out.
(Probably about as easy to change in practice as switching to metric dates and times, as the French tried to do after the revolution.)
On a related not, check out the book "Negative Math": http://www.amazon.com/Negative-Math-Mathematical-Rules-Posit...
This book builds up a mathematics in which multiplying two negative numbers gives you a negative, not a positive. The results are very interesting, particularly this: There are no complex numbers. The root of -1 is -1.
This is another example of something you never think to question, you always assume is "just the natural way", but which was a somewhat arbitrary choice and can be changed.
(-1 + 1) * -1 = -1 * -1 + 1 * -1 = -1 + -1 = -2
or
(-1 + 1) * -1 = 0 * -1 = 0.
If -1 * -1 = -1, this algebra disallows distributivity rule and the calculation only proves that.
http://blog.jgc.org/2006/12/midas-number-or-why-divide-by-ze...
Relieved to see it was actually this article, which I've actually referenced a number of times in discussions with engineers to make myself sound like I know more than I actually do!
In radians, sin = x - x^3/3! + x^5/5! - x^7/7!...
In 'Double radians', we have a 2^n factor in front of each term.