When I read this article a while back the part that impressed me was how in intuitive radians become:
2/3 * NewPi = (2/3's around the circle in radians).
It is a much more intuitive way to think about angles. Ask a third grader, how much cherry pie is missing. "About a two-thirds" they will say. They don't mention PI, and right now no one does. This is the way people think about angles naturally. NewPi makes this more intuitive by allowing you describe angle as a number between 0 and 1 (which is usually the way to go, see splines, animation, etc). 0.25NewPi just makes sense. It is a fourth of a circle, and this way of thinking would help kids understand radians instantly.
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?=)
You can do this with regular pi; you only need to use diameter-ans instead of radians. You can do it without a pi at all; there's an actual (used) unit called the "cycle". 1 Hertz is defined as 1 cycle/second.
The only problem with these units is that, like with degrees, d/dx sin(x) != cos(x). It messes with calculus.
There's also radians/sec, which comes up in network analysis. Radians correspond directly with things in the real world, like analog filters (cutoff frequency omega of a single-pole filter in rad/s=1/RC). Radians/sec are used because the math is a lot simpler. Switching to Hz puts pi into the equations, and switching to three-legged pi (thri?) would put 1/2 in the equations.
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
---
(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.
... or the 0.5 pir^2 and 0.5 gt^2 link, it's really eye opening to people who did not quite /get/ the math. Math, in educational sense, (not in the purist proofs sense) is all about seeing the links between things. Geometry and calculus are fundamentally related. We're still stuck teaching kids "sohcahtoa".
Yup, we’d be better off if we had a name for cosine that clearly implied "x coordinate", and a better name for sine that implied "y coordinate", (or at the very least, were shorter to spell and easier to say) and just did away with tangent, secant, etc. altogether, at least for basic pedagogy (1/sin(x) is almost always way clearer than csc(x)). Then we could make up a good name for the 2-argument arctangent function that inverts the function f(x) = cos(x), sin(x), and call it something much more intuitive than arctan2. Everyone would be much better off.
The pdf claims that for simplicity sake, the constant pi should have been a factor of two larger.
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 small minority of people may have a point, but it's the wrong point, since the real problem with the notation of electric charge has nothing to do with the formulas which are used, but with the connotation that people give them of somehow representing a surplus or deficit of electrons, neither of which is true.
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.
somehow representing a surplus or deficit of electrons, neither of which is true.
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.
Are you saying people confuse charged ions with flowing current in a circuit?
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.
That's the thing. When we are talking about current, we are not talking about how many electrons are moving, but about the transfer of energy from one place to another. The water analogy just doesn't work, because while the current itself flows at nearly the speed of light, the electrons themselves move very slowly. In an AC current, the electrons don't flow at all. They vibrate. And in a liquid, the energy transfer can just as well be by means of anion flowing backwards, as cations flowing forward. Yet we think of the current as only flowing in one direction. You see the problem? Trying to think of electrical energy as a "flow of electrons" OR a "flow of protons" completely skews one's understanding of what is taking place.
He has a point. I've always found the diameter:circumference ratio so much more natural for thinking about geometry that when I used to program regularly I had a macro called 'tupi' which would expand to (2 * pi) at compile time. Whatever mental blind spot I have about it was causing me to introduce a bug about a quarter of the time I needed to reference pi, probably because some part of my brain was continually asking what point there was to a constant that needed doubling in order to be useful.
I just starting futzing around with a project using OpenGL and literally just last night wrote myself up a pi2 constant after the third time I found myself writing 2*pi. And three's the charm for refactoring.
Whole amps and handfuls of ohms aren't that rare, they actually match pretty well to what comes out of the wall socket. And that 1.8v or 0.85v or whatever it is these days space heater that runs your computer takes really quite a lot of amps.
Frankly, I have always appreciated that Ohms are almost never partial. milliohms are a rare occurrence, and I just think that's great and works out well, IMHO.
Amps being too large is unfortunate, but it's not a big deal.
Coulombs are my favorite, because it's the fundamental amount of "something" (as opposed to "something per time") that most of these can be traced back to:
So you're going to celebrate refactoring a mathematical constant to give a neater presentation on a day chosen based an apparently illogical ordering scheme for dates.
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.
I see where you M/D/Y people are coming from, but can you at least agree that "June 28, 2010" -> "[Month 6] 28, 2010" -> "6/28/2010" is at least founded in some kind of logic?
Wouldn't it be awesome if somebody already had this planned, complete with a domain name, a compelling candidate for the new constant, and even a whole freaking manifesto written just to promulgate proper pi? I for one wouldn't want to miss out on that, so if I were you I'd hint hint be watching Hacker News quite closely on June 28. ;-)
As long as the html5 has buggy dancing scrollbars of doom on firefox, I will be grateful for flash scribd links. Everyone can hate on flash all they want, but the flash version in this link was a big relief compared to the html5 version.
Unfortunately the article is down at the moment, but there's a beautiful identity that suggests that maybe pi/4 is a constant of nature, not pi:
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.
Old news. This ranks up there with the fact that it was a bad idea to use base 10 instead of base 12. Absolutely and utterly true, but completely not worth the switching costs.
It's still important for everyone doing math to know that the convention is bad, though, even if it's not worth switching. Then it becomes clear in several instances that slight blemishes in the beauty of equations are of human origin. It also help you think of 2pi as an object.
Teaching this in the classroom could engage a lot of students that might otherwise be bored sh*tless of the regular curriculum. For those that do pay attention it could open their minds to a new way of looking at maths as a human tool rather than some universal law. It's a shame its taken me till freshman year and reading this article to make the connection that a 6th of a radian is a third of one quadrant. I can't imagine how much simpler this would have been to learn if it was 2pi.
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.)
It's still important for those studying mathematics to realize that things like this are completely arbitrary.
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.
And we have 12 non-thumb phalanxes on each hand. Point each phalanx of one hand with your thumb (of the same hand), and you can count up to 12. With 5 fingers on the other hand, you can count 5 dozen: 60.
Base 10 may not be completely arbitrary, but is is undoubtedly lazy.
When demonstrating that to others, be wary of counting as a demonstration. It might make be memorable, but people can get upset when you get to the number 4.
Base 10 is arbitrary in the sense that it is "based on or determined by individual preference or convenience rather than by necessity or the intrinsic nature of something."
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.
The reasoning is that when we divide something evenly, we usually want to divide it into a small number of parts. 12 divides evenly into small parts better than 10 does, and so more often provides a convenient division.
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.
This seemed pedantic and silly to me until I read the reasoning in the wikipedia article. After all, we use numbers to do far more things than division, don't we?
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.
A slight tangent, but I recently ran across a fascinating theory on the origin of base-12 systems. If you use your thumb as a marker, you can touch four finger joints on each hand for a total of 12 positions. I tried it and it's a surprisingly natural way of keeping count. Here's the source paper:
Hmm. That is a very natural way to count quite high on your fingers.
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').
Anecdotal evidence of real life application: my mother (Indian/Pakistani, born in the '40s) counts this way. I never asked, but presumably she was either taught by her parents or in school.
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.
This reminds me of the whole thing about the guy who invented a way of dividing by zero some time back. There's nothing at all wrong with coming up with mathematical systems, and nothing wrong with defining -1 * -1 to be -1. But you pay the price in terms of the consequences.
Your'e right, but you don't have to assume that multiplication is distributive over addition. You can build math without that assumption. It gets weird, but it's very interesting to see that so many things we take for granted about mathematics are really just conventions.
Very true, but I like my numbers to behave like... well, numbers. You can define consistent algebras but I'd argue with you if you tried to call them numbers.
I remember you can define algebras however you like with arbitrary commutativity and associativity rules but in this case you should also apply them and not assume they work in a certain way.
If -1 * -1 = -1, this algebra disallows distributivity rule and the calculation only proves that.
I seriously have always suspected this. If you're going to use a value to represent the perfection of a circle, why not take the derivatives of its properties (area, then perimeter, and once more) until you get a constant? 2pi is that constant.
One of my math professors pointed out that to be consistent, "π, φ, χ, ψ, ξ, and ι" should actually be pronounced "pee, fee, khee, psee, ksee and ee-ota," which I found kind of funny and fascinating. Of course, common convention beats out pedantic propriety and rightly so, I think. This sort of thought-provocation is quite valuable nonetheless, if for no other reason than to keep us aware of that about which we do not readily think.
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!
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?=)