π = 3 (sometimes) for Nobel laureate(ask.metafilter.com)
ask.metafilter.com
π = 3 (sometimes) for Nobel laureate
http://ask.metafilter.com/182210/Is-this-where-Im-supposed-to-use-the-phrase-selfappointed-defender-of-the-orthodoxy#2621825
14 comments
>Our teacher always said she accepted all result with an error of less than 10% if we did that by making the calculus easier.
In English, calculus and calculation are two different things. And there are a number of grammatical errors in that sentence that make the meaning unclear.
I'm noting this because if the teacher allowed you to do simpler calculus so long as you maintained an error margin of 10%, that's different from allowing you to do simpler calculations provided you keep an error margin of 10%. Calculus is specifically the branch of mathematics involving limits, functions, derivatives, and integrals, and from the English it sounds like she, if we are being perfectly pedantic, was giving you the option do do approximations with your integrals and derivatives, not with any value you like.
In English, calculus and calculation are two different things. And there are a number of grammatical errors in that sentence that make the meaning unclear.
I'm noting this because if the teacher allowed you to do simpler calculus so long as you maintained an error margin of 10%, that's different from allowing you to do simpler calculations provided you keep an error margin of 10%. Calculus is specifically the branch of mathematics involving limits, functions, derivatives, and integrals, and from the English it sounds like she, if we are being perfectly pedantic, was giving you the option do do approximations with your integrals and derivatives, not with any value you like.
Newton's calculus is just one example of a calculus. A calculus is any set of mechanical rules for manipulating mathematical symbols. Calculation is the process of applying those rules. Cf. the lambda calculus. The branch of mathematics involving limits, functions, derivatives, and integrals is analysis.
>The branch of mathematics involving limits, functions, derivatives, and integrals is analysis.
Everywhere I've studied, it's been referred to as simply calculus. Wikipedia says the same. I did take a class called analysis of functions in high school as an advanced track pre-calculus, but your definition really does not apply in the States at any rate. Where did you hear Analysis used to specifically apply to what we call calculus in the States?
http://en.wikipedia.org/wiki/Analysis#Mathematics
Everywhere I've studied, it's been referred to as simply calculus. Wikipedia says the same. I did take a class called analysis of functions in high school as an advanced track pre-calculus, but your definition really does not apply in the States at any rate. Where did you hear Analysis used to specifically apply to what we call calculus in the States?
http://en.wikipedia.org/wiki/Analysis#Mathematics
My usage is that which is universally accepted in any university math department. As a freshman in a math/science/engineering major, you take (or finish after beginning in high school) three semesters of Calculus, always referred to with a capital "C" to mean specifically Newton's calculus. Here you learn to calculate integrals and derivatives so that you can use them as a tool. If you major in math, then as a junior or senior you will take analysis. Here is where you rigorously study integrals and derivatives. An introductory analysis course typically begins by defining Cauchy sequences, and then defining the real numbers as equivalence classes of Cauchy sequences of rationals. Building on this foundation, you then construct the basic theory of continuous functions, derivatives, and Riemann integrals. Finally, you use this theory to prove that the transformation rules of Newton's calculus are sound.
You're being pedantic in English in response to an exchange that presumably happened in French. :)
Hmm, whilst it's true physicists often simplify equations to really understand the problem, I can't think of a variable in quantum mechanics we "don't know to within 2 orders of magnitude". Especially one called alpha, which usually denotes the fine structure constant, which we know to better than 1 in a billion...
Anyone have some thoughts on what it could be, if true?
http://en.wikipedia.org/wiki/Fine-structure_constant
Anyone have some thoughts on what it could be, if true?
http://en.wikipedia.org/wiki/Fine-structure_constant
You're overthinking this. Yes, in principle we do known the fine structure constant to something like 9 orders of magnitude both experimentally and theoretically.
However, when would you use the pi = 3 approximation? Certainly not when you're in front of a computer, or if you were preparing some experimental results for publication. But, if you're in the lab and need to quickly make some calculations, or just to see if something is feasible and worth spending more time on, pi = 3 isn't so bad.
Example, measuring the fine structure. Sure, you can predict where these energy levels are supposed to be to probably whatever our error on knowing the mass of an electron is. And because you know the fine structure so precisely, you should be able to make a very accurate prediction on where that is. However, throw most of those digits out the door, because a lot will be hidden behind doppler broadening. So when you make your measurement in your fabry perot etalon, you'll probably make a precise measurement
http://en.wikipedia.org/wiki/File:Fabry_Perot_Etalon_Rings_F...
but how accurately can you really measure the position of those fringes? Sure, free spectral range probably lets you get down to about MHz region or so, but the doppler broadened linewidth is probably an order larger than that. Which brings us back to, you've got these experimental errors, why care about 9 digits of precision if you just want a quick and dirty calculation to get things set up?
Anyways, that's all he's trying to say. In most experiments, there will be some sort of experimental error hurting you. Be sloppy in the beginning just to get a feel for things.
However, when would you use the pi = 3 approximation? Certainly not when you're in front of a computer, or if you were preparing some experimental results for publication. But, if you're in the lab and need to quickly make some calculations, or just to see if something is feasible and worth spending more time on, pi = 3 isn't so bad.
Example, measuring the fine structure. Sure, you can predict where these energy levels are supposed to be to probably whatever our error on knowing the mass of an electron is. And because you know the fine structure so precisely, you should be able to make a very accurate prediction on where that is. However, throw most of those digits out the door, because a lot will be hidden behind doppler broadening. So when you make your measurement in your fabry perot etalon, you'll probably make a precise measurement
http://en.wikipedia.org/wiki/File:Fabry_Perot_Etalon_Rings_F...
but how accurately can you really measure the position of those fringes? Sure, free spectral range probably lets you get down to about MHz region or so, but the doppler broadened linewidth is probably an order larger than that. Which brings us back to, you've got these experimental errors, why care about 9 digits of precision if you just want a quick and dirty calculation to get things set up?
Anyways, that's all he's trying to say. In most experiments, there will be some sort of experimental error hurting you. Be sloppy in the beginning just to get a feel for things.
Two aspects spring to mind. First, we don't know how long ago this course was, so it's quite possible some of the story is either dated or a slightly misremembered anecdote for a physicist who was brought up with slide rules for his calculations. Second, the quote specifies "experimentally", which implies measurements limited to two orders of magnitude, i.e., measured vs theoretical. Alas, while I have a few slipsticks in the closet, I am not now nor have I ever been a physicist, so I could be totally wrong on this.
> I can't think of a variable in quantum mechanics we "don't know to within 2 orders of magnitude".
Maybe the neutrino masses differences? In my intro QM class, we did a problem where we calculated the distance it took for one neutrino flavor to mix into another, which depends on the mass differences. You wouldn't want to take a square root of eV, though.
Or it could be it's one of the CKM angles. The latter are known to 1 part in 10^4, but only because of the relatively recent BaBar and BELLE experiments. Maybe back when this guy was an undergrad they only know them to 1 part in 100. Seems kinda advanced for 3rd semester of QM...
Maybe the neutrino masses differences? In my intro QM class, we did a problem where we calculated the distance it took for one neutrino flavor to mix into another, which depends on the mass differences. You wouldn't want to take a square root of eV, though.
Or it could be it's one of the CKM angles. The latter are known to 1 part in 10^4, but only because of the relatively recent BaBar and BELLE experiments. Maybe back when this guy was an undergrad they only know them to 1 part in 100. Seems kinda advanced for 3rd semester of QM...
Off topic, but why is the "pi" character in most common fonts so poorly designed? It always seems to have a straight line at the top rather than a tilde-like stroke and it confuses me just about every time.
I had the same problem, I thought the author mistakenly used ∏ but I see in my browser tab that it's correctly lowercase.
I read n = 3. I thought I was coming into a discreet calculus thread. Then i find it's continuous and got upset. Then I find that it's not even american calculus which makes me even more upset. I'm going to have to hammer out a few combinatoric problems regarding the probability of picking a McNuggets box with TWO boot-shaped nugs just to get my credibility back. /silly
It's a little unrelated, but if you ever need to know the number of seconds in a year, it's pi x 10^7 to within .5%.
Or, as stated in Bertley's "Programming Pearls":
Pi seconds is a nanocentury.Proof for the curious: http://www.google.com/search?q=number+of+seconds+in+a+year+d...
I understand what you did, but the mathematician in me cringed at the word "proof" in that comment.
That looked like a mathematically rigorous proof to me. What was wrong with it?
Proof by construction is perfectly valid, if usually too difficult to be useful.
May be the word "fact" would be better to describe these kind of numbers.
proof by googlization
This guy sounds like every microelectronics prof ever. :) I took that to heart when doing the exams and it make the impossible possible.
I really enjoyed it when my EE professors would simply erase components of a circuit diagram if they didn't matter and they made things complicated. I think that had an impact on my approach to solving problems.
Another trick of the trade is working back from the desired output by stages. For example, for an amplifier delivering 200 W to loudspeakers (or 1 kW to an antenna, or ...), first design the final stage, then work back from that adding the stages needed until the range is OK for the input signal you have.
My favorite one of these comes from physics.
Simple harmonic motion impossible to deal with? Voila, assume |sin(x) - x| is small and just deal with x instead of sin(x)--magic!
Simple harmonic motion impossible to deal with? Voila, assume |sin(x) - x| is small and just deal with x instead of sin(x)--magic!
Pi ~ 3, but we also often need Pi^2, which is better approximated as 10. (I actually got into an argument about rounding Pi^2 to 10 once with a person who liked to round Pi to 3 ...)
I remember using Pi^2 to be approximately g (the acceleration due to gravity near the earth's surface). This comes in real handy when simplifying the time period of a pendulum.
Yep- and rounding g to 10 m/s^2 when there's no pi involved- so you can actually talk conversationally about free-fall because you can then do the math in your head.
Pi is about 3. Pi^2 is about 10. But better than both of these approximations is Pi^2 + Pi = 13, which gives Pi = (-1 + sqrt(53))/2 = 3.1400...
(It's too bad that the combination Pi^2 + Pi occurs basically never.)
(It's too bad that the combination Pi^2 + Pi occurs basically never.)
Every russian professor I had loved gaussian units.
I learned to love them too. Everything is simpler when you don't have 3-5 constants staring you in the face.
I learned to love them too. Everything is simpler when you don't have 3-5 constants staring you in the face.
That's rather close to a smbc comic: http://www.smbc-comics.com/index.php?db=comics&id=1777#c...
(it was previously on Hacker News: http://news.ycombinator.com/item?id=1093703)
(it was previously on Hacker News: http://news.ycombinator.com/item?id=1093703)
As my programming teacher used to say when talking about floating numbers and their errors, π is 5 in agriculture and 3.14159 in civil engineering.
I got exactly the same lesson in almost similar situation, but from a guy who was not even close to (someone who is close to) a Nobel's prize. He did a sum of a problem from the test using a small constants reference book and his head. Despite the numerous simplifications in the calculation, error of his result was less then 5%.
How was it funny? It seems like a sound reasoning, depending on whatever alpha represents.
It's funny because the students thought he was being cheeky, when in fact he was making a logical leap that few individuals would EVER make, much less in a random junior level physics course.
He basically Nerd-burned them, and that is something I can get behind.
He basically Nerd-burned them, and that is something I can get behind.
Ha, in one of my physics classes the professor actually set pi = 1 :)
[deleted]
Any person with a marketing degree will tell you this is false. Try to bill for a circular billboard (say a circular podium at a conference booth) where ~4.7% of the surface lacks proper advertisements or necessary background color.
Our teacher always said she accepted all result with an error of less than 10% if we did that by making the calculus easier.
So I started by "let's define pi=3"
I got a bad grade on that one, even after I protested that the answer was within the required specifications, so afterwards as a nag, I was always over-precise with her assignments - like always included electron mass in nucleus calculations :-)