Wikipedia:Reference desk/Archives/Mathematics/2010 February 4

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February 4

Scientific Notation

If 6.02x10e23 atoms of carbon have a mass of 12g, then what is the mass of 1 atom? Express your answer in scientific notation.

I don't even know where to start on this one. I'm pretty sure it has something to do with dividing the exponent and 6.02.

Explaining how you got your answer would be great.

174.112.38.185 (talk) 01:58, 4 February 2010 (UTC)

If two cars weigh 2 tonnes, how much does one car weigh? —Preceding unsigned comment added by 129.67.39.49 (talk) 02:07, 4 February 2010 (UTC)

1 atom has mass = total_mass / number_of_atoms = 12g / 6.02x10e23. So all you have to do is that calculation and then ensure your result is presented as a normalised scientific notation. (By normalised I mean an integer part in the range 1 .. 9.999 here) - So one half might be "0.5x10^0" which normalises to "5x10^-1". -- SGBailey (talk) 10:17, 4 February 2010 (UTC)

Another way to look at it... the scientific notation is a product, 6.02 × 10²³. So, how does one divide a product? Well, 12 is 3 × 4; when we divide that by 2, we do not get (3/2)·(4/2), nor do we usually think of the product as (3/2) × 4, but rather as 3 × (4/2).—PaulTanenbaum (talk) 12:04, 4 February 2010 (UTC)

Note that the notation "6.02x10e23" is nonstandard: you want "6.02x10^23" (or, more nicely formatted, $6.02\times 10^{23}$) or "6.02e23" (E notation). With the "x10e" notation, the reader could suppose that you mean $6.02\times (10\times 10^{23})$ (a product of a normal number and a number in E notation) and so be off by an order of magnitude. --Tardis (talk) 16:51, 4 February 2010 (UTC)

Statistics course (titled Stochastic processes)

I'm taking a statistics course (titled stochastic process). It's like no other stats course I've taken previously because the prof covers in lecture many proofs and mathematical theorems. I've taken only calculus 1 to 3 and I don't have any background in proof. I don't know why but I also have proof-phobia. Proofs just never appealed to me or they were never possible for me to understand reproduce by myself. I don't know how I should ace this course. Even the homework is really hard. In my past stats courses, I prepared for exams by doing chapter review questions at the end of every chapter. But the prof's questions are nothing like the ones in the text. What should I do? —Preceding unsigned comment added by 142.58.129.94 (talk) 02:29, 4 February 2010 (UTC)

Is the course a requirement? If not, drop it. If the course is required, do any other professors teach it? Check to see if they are better suited to your skills. If so, switch courses. If it is required and he is the only professor, meet him after class - as much as possible - and ask tons and tons of questions. The more questions you ask, the more answers you will get. -- kainaw™ 02:31, 4 February 2010 (UTC)

It's not a requirement. But if I drop the course now, I'll get no refund back for the course tuition (around $400). I'll also have "W" mark in my transcript. I don't have a single "W" right now, but I heard it doesn't look good. —Preceding unsigned comment added by 142.58.129.94 (talk) 02:41, 4 February 2010 (UTC)

In any case, there is little chance that we can assist you (effectively) regarding this issue; you make your own future. But we can offer you some advice, and with enough perseverance on your part, this advice may be useful for you later on. Firstly, it is a big mistake to practice "reproducing proofs"; proofs should come naturally to you, but if they do not, reproducing them is a bad habit and can be detrimental to your understanding of the subject in question. The best option available if you do not appreciate proofs in stochastic calculus, is to attempt to appreciate them in "lower-level analogous". For instance, since a proof is nothing but a series of logical implications, practice "logical implications" by manipulating some basic trigonometric identities (take out a calculus book, and try to appreciate some theor

etical proofs, such as that of the fundamental theorem of calculus, as well; look at the underlying intuition of the proof rather than the proof itself). Personally, you do not necessarily have to "know how to do proofs in university" to become a great mathematician, but your professor will probably tell you otherwise.

When you say that your professor's questions are nothing like those in the test, it is likely (but not necessarily the case) that the professor's questions test an understanding of the material rather than a routine memorization of the material. Thus, instead of attempting to have the ability to "reproduce the textbook in exams", attempt to understand the textbook; be in the position where you have a feel for the material that would permit you to engage in a 1 hour discussion with any expert of stochastic calculus, and be interested in that which is being discussed! It is difficult to attain good grades if you are not interested in what you are doing, but some students do have the ability to do exactly this (and this is an extremely difficult alternative; I am sure that many professors of mathematics would fail mathematics exams if they followed this procedure).

At the end of the day, you make your own future; instead of feeling helpless about your course, try to take it step by step. You do not necessarily need to attain an A; thoroughly understand whatever material you can and enjoy what you are doing! If there happens to be a few concepts that you do not understand, try to enjoy thinking about these concepts, and maybe you will understand them eventually. Finally, when the final exam is imminent, do not spend too much time solving textbook problems; if you have taken the course, you have probably solved enough of those types of problems anyhow. Rather, try to discuss stochastic calculus with a friend or fellow student, and by this I mean engage in a lengthy discussion covering most of the topics in the syllabus (use paper and pen as well, if necessary) (and forget memorized speeches; discuss the material as if you were discussing what you did on the weekend). To summarize, the most important advice in this instance is to enjoy what you are doing; if you are enjoying it, everything else will come naturally to you. Try to determine the aspects of stochastic calculus that interest you the most and develop your appreciation of the entire subject from there. PST 03:43, 4 February 2010 (UTC)

Not much I can add except proofs aren't really something you can learn overnight. Most math majors take an elementary analysis course to get the basics. But there aren't any easy recipes otherwise there wouldn't be any unproven conjectures, and there are plenty of those. There are books devoted to the basic mechanics, e.g. How to Read and Do Proofs: An Introduction to Mathematical Thought Processes by D. Solow comes to mind. Keep in mind also that proofs come in all levels of difficulty, so with this course the proofs may just be a matter of applying rules of algebra to known formulas.--RDBury (talk) 04:03, 4 February 2010 (UTC)

I would guage carefully how much time you have to withdraw with or without substituting another course. You might, if you still have time to make a substitution, get up to speed on a course that requires more proof than you're yet used to, but less than the specific one you've chosen. One thing is that understanding what's going on during the lecture should take a backseat to taking good notes and reviewing them later, if you have to choose one over the other. You may be able to handle the course you are currently taking just fine if you manage this part of it, but you should probably not be optimistic about that if you really don't feel that's the case.Julzes (talk) 07:10, 4 February 2010 (UTC)

I think taking notes is not as important as actually listening to what the professor says; very often, students make the mistake of taking notes without actually thinking about what the notes mean (they note down the statements made by the professor but not the explanations). One difference between reading a textbook and attending a lecture is that a real person is actually telling you something in a lecture. If students spend too much time on notes, they will turn the "real person" into a textbook, and that makes attending the lectures very pointless. Taking notes can be very helpful if done right, but if done wrong, they can have a negative effect on the student. PST 10:33, 5 February 2010 (UTC)

Taking notes on what the professor says was part of what I was indicating as important. Realistically, it would be impossible to take anything resembling good notes in a majority of courses without understanding what is said to some degree. Good notes would enable the student to grasp what was said at a later point, while remembering everything said without good notes would be awfully difficult unless the subject is already easy to the student. One other thing is important for this OP (and others) to understand: Mathematics majors shouldn't rely upon their past experience with grades. If you actually want to be a mathematician per se, you should be able to master the undergraduate mathematics curriculum and get mostly high grades; but if you just want the major's degree and to use mathematics professionally, fighting your way through one or a few grades of C won't hurt you.Julzes (talk) 17:58, 5 February 2010 (UTC)

If that course covers what I usually think of the title as meaning (e.g. functional integrals, brownian motion, etc) and you haven't already had a good real-analysis class, you are doomed. Switch to another course. 66.127.55.192 (talk) 08:46, 4 February 2010 (UTC)

I tend to think of proofs as something you're supposed to learn long before you reach calculus. Otherwise what are you taking math for? You don't compete on a football team before a crowd of spectators before you've learned to walk. Situations like this seem like a serious case of the educational system lying to students. Michael Hardy (talk) 01:04, 5 February 2010 (UTC)

If you think that is a serious case of the educational system lying to students, you have yet to see the worst... PST 10:28, 5 February 2010 (UTC)

Based on the IP address, I guess the asker is talking about this course. Not much beyond calculus is assumed, it seems; no functional integration is there as it's an introductory course to the concepts of Markov chains etc. Learning such things, I guess it is most important to memorize the definitions well and think about them pictorially to develop some intuition about what is going on at all. Theorems take the definitons and give statements about the things involved. If you've been thinking about the subject long enough and have the intuition about the definitions, then, in a good maths text, the statements of theorems will, one after another, start to look plausible (although maybe ingenious), even before you've read the proofs. Proving a theorem, you just have to fill in the space in between the definitions (first plug them into the assumptions and the statement!) and something stated as the result to be proved. Often a proof (not anyhow uniquely) can be achieved by doing just plausible transformations to the assumptions with the definitons plugged in. When you have several transformations to choose from, take the one having something in common with the result you are expected to get at.

There are some special techniques like proof by contradiction and mathematical induction, it will certainly be helpful to know what they are about. Try to read the proofs presented to you and see what techinques are used (what is plugged where) and in what lines it is economical to think when proving anything about stochastics. When you have seen what is usually done to a Markov chain, you can try something similar yourself, e. g. in the homework. — Pt _(T) 02:25, 5 February 2010 (UTC)

Your proof-phobia is the central issue. Consider some elementary formula such as Heron's formula or the formula for the volume of a cone. Do you understand the meaning? Do you appreciate the usefulness? Can you check the correctness in special cases? Do you wonder if it is generally true? Only when you answer Yes to such questions will a proof be interesting to you. Bo Jacoby (talk) 11:58, 5 February 2010 (UTC).

Your proof-phobia is the central issue.

I'd think the underlying cause of a phobia is ignorance of what proofs are, and that would be the central issue. Trying to learn prerequisite material while taking the course is of course an up-hill battle. Michael Hardy (talk) 03:35, 6 February 2010 (UTC)

Let's try not to convince the OP that s/he is flatly not good at mathematics. S/he likely overstated how awful s/he finds proofs to be if the experience s/he had was of getting A's (I infer from the question) through a third semester of calculus. At the time the question was asked, there was apparently a sense of being overwhelmed from being at a higher level, but I would guess that this is not irrevocable. Writing up good proofs is ultimately just an extension of more routine problem solving. It's just a more refined process of getting from one place to another.Julzes (talk) 04:53, 6 February 2010 (UTC)

To make an obvious analogy, becoming good at proofs rather than just routine problems is like going from being a cook--with a cookbook--to being a chef--with a good memory of what works with a cookbook but without its use.Julzes (talk) 05:01, 6 February 2010 (UTC)

Really simple algebra problem.

The surface area of a sphere is A = 4πr² . Now I want to replace the radius in the formula with diameter. Since r = d/2, I can rewrite the question as A = 4π(d/2)². Correct? Then reducing it follows A=2πd²... Is this correct? Assuming it is, why am I being told that the answer is A = πd² ??? 198.188.150.134 (talk) 04:57, 4 February 2010 (UTC)

• Careful... don't forget to square the 2 in the denominator of the replacement fraction as well: $A=4\pi \left({\dfrac {d}{2}}\right)^{2}=4\pi \left({\dfrac {d^{2}}{4}}\right)=\pi d^{2}$. --Kinu ^t/_c 04:59, 4 February 2010 (UTC)

aaah, gosh darnit! You're right! Thanks for pointing that out, what a silly mistake... 198.188.150.134 (talk) 05:01, 4 February 2010 (UTC)

Plane sections of an ellipsoid

Let u, v be the axes of the ellipses obtained as intersection of a plane with an ellipsoid (x/a)²+ (y/b)²+(z/c)²=1. Taking all planes by the origin, what is the resulting curve with Cartesian coordinates (u,v)? What about the analog in higher dimension? I've few time to think about it, but I'd be curious to know the answer. Maybe somebody already knows the result? --pma 08:14, 4 February 2010 (UTC)

I suppose it won't be a curve but rather a 2-dimensional set. It will certainly contain points (a,b), (b,a), (a,c), (c,a), (b,c) and (c,b). It will also contain some lines joining these points (imagine the cutting plane rotating e.g. around the X axis from XY to XZ plane)... --CiaPan (talk) 09:57, 4 February 2010 (UTC)

Fisher information for dependent variables

Hi all,

Fisher information has the property that it is additive if the two variables are independent:

${\mathcal {I}}_{X,Y}(\theta )={\mathcal {I}}_{X}(\theta )+{\mathcal {I}}_{Y}(\theta ).$

Does anyone know of a similar result which works for dependent variables, or some papers I could read on this?

Thanks in advance. x42bn6 Talk Mess 15:06, 4 February 2010 (UTC)

Certainly nothing that satisfies any reasonable definition of "information content" satisfies that rule for all dependent variables: we would have $A_{X}(\theta )=A_{X,X}(\theta )=A_{X}(\theta )+A_{X}(\theta )$, so $A_{X}(\theta )\equiv 0$. But see mutual information, which isn't precisely the same (it works for information entropy) but might be useful. --70.56.219.67 (talk) 15:49, 4 February 2010 (UTC)

Standard deviation

Why is standard deviation(which is more complicated)used in statistics instead of taking average of absolute values of deviation from the mean? —Preceding unsigned comment added by 113.199.182.172 (talk) 15:35, 4 February 2010 (UTC)

The most important reason is that the square of the sd, the variance, is additive - if X and Y are independent, then $V(X)+V(Y)=V(X+Y)$. If a variable is the sum of many independent variables (which is a good model for many phenomena), then it is completely determined by their individual means and variances, and not by their average-absolute-deviations. If you're only starting to learn statistics, you will in the future encounter many applications for sd\variance that just wouldn't work if you used anything else. -- Meni Rosenfeld (talk) 15:57, 4 February 2010 (UTC)

That's good question, and Meni Rosenfeld nailed it. I'd have guessed a half-dozen people would have written rambling discussions skirting the issue before someone did that; I see that here. Michael Hardy (talk) 17:46, 4 February 2010 (UTC)

Thanks Meni Rosenfeld for such a nice answer, I had been searching for the answer for a long time and had not found any satisfactory answer. —Preceding unsigned comment added by 116.90.224.116 (talk) 05:57, 5 February 2010 (UTC)

Question

Hello my name is hursday. Can you please explain to me the difference between a cup cake and a muffin. thank you. —Preceding unsigned comment added by Dr hursday (talk • contribs) 23:10, 4 February 2010 (UTC)

They seem identical to me, they are both closed balls. I can see the difference from a doughnut which is a toroid though. Have you looked at the articles cup cake and muffin? And by the way this is the mathematics reference desk, perhaps you wanted the Miscellaneous reference desk? Dmcq (talk) 23:21, 4 February 2010 (UTC)

Self-correct - replaced torus with toroid, the torus is strictly only the surface. Some people cut off the crust from bread an eat what could be considered an open ball, very few only eat the crust. Dmcq (talk) 10:20, 5 February 2010 (UTC)

It seems to me that a muffin is not as sweet as a cupcake. —Preceding unsigned comment added by 74.130.183.36 (talk) 02:07, 5 February 2010 (UTC)

<Joke>Doctor, as you will tell your patients, muffins and cup cakes are detrimental to the health, and should only be taken occassionally. I just realized you asked for the difference! Well, from a doctor's (or a topologist's) point of view, they are equivalent. Disclaimer: This post is intended to be humorous and should not be taken seriously. </Joke> PST 02:58, 5 February 2010 (UTC)

I think the disclaimer should be outside the joke tag. -- Meni Rosenfeld (talk) 08:27, 5 February 2010 (UTC)

The disclaimer is actually part of the joke... PST 10:24, 5 February 2010 (UTC)

Then I guess unlike Homer Simpson, I don't get jokes. -- Meni Rosenfeld (talk) 14:30, 5 February 2010 (UTC)

How is an elephant like a plum? They're both purple except for the elephant. --Trovatore (talk) 03:04, 5 February 2010 (UTC)

actually, cupcakes are smooth, simple, everywhere differentiable volumes, except for the coronal surface, which often has a planar discontinuity to another smoothly differentiable, but much sweeter, volume of material (barring the presence of sprinkles...). Muffins, by contrast, have a much more complex density function full of coarse variations, often with numerous non-differentiable discontinuities of the 'fruit' or 'nut' variety, and generally lack the coronal layering present in cupcakes. There are also distinct differences in their responses to shearing stressors, differences in their capacities to absorb warm liquids, and distinct patterns of plastic/elastic deformation when they impact on a solid surface - say, the back of someone's head. If you like, I'm happy to demonstrate the last point. --Ludwigs2 05:55, 5 February 2010 (UTC)

How is this mathematical? Nyttend (talk) 14:19, 5 February 2010 (UTC)

I think the real question is how isn't this mathematical. Do you want me to actually define the density functions for cupcakes and muffins? If so, I'll need to do research, and you're paying. meet me at starbucks. --Ludwigs2 18:46, 5 February 2010 (UTC)