Wikipedia:Reference desk/Archives/Mathematics/2009 November 25

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November 25

True or false?

For every Lebesgue integrable function f, there exists a Riemann integrable function g such that f=g almost everywhere.

Is the previous statement true or false? If false, can you give a counterexample? This is not homework; I'm just curious. Thanks, --COVIZAPIBETEFOKY (talk) 00:32, 25 November 2009 (UTC)

Our article Riemann integral has the answer: the statement is false, and the indicator function of the Smith–Volterra–Cantor set is a counterexample. Algebraist 01:57, 25 November 2009 (UTC)

ideas or resources for math in everyday life

I don't know much about math so I would appreciate ideas. Are there any magic formulas that can be used for everyday life that would just improve (purposefully vague) life?

Tableornament (talk) 02:06, 25 November 2009 (UTC)

"Five-eleven's your height, one-ninety your weight, you cash in your chips around page eighty-eight." 67.117.145.149 (talk) 06:35, 25 November 2009 (UTC)

Let me give the usual lecture given to people who ask this sort of question (which is based on a common misconception). Firstly, mathematics has absolutely nothing to do with "magic formulas", and is not solely done for "daily life". Most mathematicians research abstract structures which are not restricted to numbers or equations. Number theory is indeed an area of mathematics but it has little to do with manipulating numbers around - most mathematicians do not need or have a "human calculator ability" (although admittedly, some do; my point is that it is completely irrelevant to most of mathematics). Number theory subsumes (and has subsumed) an abstract understanding of general properties of the integers. Theorems such as Fermat's last theorem cannot be obtained by sheer "calculations" (and if I have not already mentioned, mathematics is not about arithmetical calculations). They must be solved via deep intuitive thinking and an abstract understanding of various concepts. In fact, research in number theory has shifted to the theory of algebraic number fields - structures which encompass a variety of more abstract objects than the integers.

On the other hand, there are branches of mathematics which have little to do with number theory, as well (although they may have non-trivial connections with number theory). For instance, topology is an area of mathematics which studies notions of "nearness" on higher dimensional analogues of the universe, as well as abstract objects which do not live in our universe at all (including infinite-dimensional objects). For instance, on some of the complex objects a mathematician studies, the distance from "A" (a point) to "B" (another point) may be different from the distance from "B" to "A" (that is, the distance depends on the order in which one considers the two points)! Some objects need not possess a proper notion of a distance at all.

Many "basic and advanced" branches of mathematics, such as topology, ring theory, group theory, field theory (and so forth) are axiomatic theories. That is, objects within these fields are described by a set of axioms (requirements, in a basic sense), and mathematicians study these objects and classify them. Mathematics is not a closed system in that it will never end; there is no shortage of open problems. In fact, there are open problems which mathematics do not know yet exist and are waiting to be discovered. Little (if not nothing) can be solved by applying formulas - mathematicians have to constantly invent new techniques (ways of thinking - not formulas) to develop intuition about abstract structures. Said differently, mathematics is an art and requires artistic thinking; it requires years to develop such thinking (memorization can never achieve this). In fact, I shall say (and I think most people will agree) that no-one can ever develop such thinking to perfection. Succintly, my point is that magic formulas do not exist anywhere. Even in applications of mathematics (which do not undermine the purpose of mathematics), deep thinking such as that which I have described, is required. I hope I have not said too much (remember it though, especially when you hear someone claiming that mathematics is about basic arithmetic or formulas (if they do, they are completely incorrect)!)... --PST 10:07, 25 November 2009 (UTC)

Many fields of maths have immediately useful applications. For many consumers, understanding exponential functions (as in compound interest) would be useful. Simple formulas for mass, size, and volume are useful when planning things like papering walls or painting floors. Concepts of game theory and the difference between cooperative and zero-sum games would be useful in political debates. To quote Heinlein (who attributes it to his fictional character, Lazarus Long, but let's be serious - he's just using him as a mouthpiece ;-): Anyone who cannot cope with mathematics is not fully human. At best he is a tolerable subhuman who has learned to wear shoes, bathe, and not make messes in the house. --Stephan Schulz (talk) 18:55, 25 November 2009 (UTC)

Short answer: Yes, Bayes' theorem.

Longer answer: Proper knowledge and understanding of mathematics can enrich and improve all aspects of life for individuals and society as a whole. It can change (for the better) the way you think, perceive the world, learn, and solve everyday problems. However, generally such knowledge cannot be reduced to a few mindless "magic formulas" in which you just plug in some variables and suddenly your life gets better.

That said, there are some examples of simple formulas or concepts that are commonly unknown or misunderstood, and familiarity with them can really turn your life around. The above mentioned Bayes' law - which establishes how you can utilize new knowledge and evidence to correct your beliefs about what is true and false - along with the basic probability notions surrounding it, is one such formula. Others can be found in the realm of mathematical logic, and of course you have the examples given above by Stephan (and below by the people who will no doubt post after this). -- Meni Rosenfeld (talk) 20:52, 25 November 2009 (UTC)

OP, if you are interested in learning about the practical necessity of math is daily life, I'd recommend books by John Allen Paulos, such as Innumeracy. On the other hand, if you wish to learn a bit about the joys and beauty of math as a purely intellectual discipline, G. H. Hardy's A Mathematician's Apology is a good starting point. Books by Martin Gardner, Howard Eves, Ralph P. Boas, Jr. etc. lie somewhere in between and provide a window into the "less serious aspects" of mathematics. Abecedare (talk) 21:24, 25 November 2009 (UTC)

A Mathematician's Apology is a beautiful and sad book, and gives a very interesting insight to the personality of an outstanding mathematician as Hardy, and to the rules of an intellectual society as Cambridge. But, outside these aspect, I do not see what interest it may have for a layman interested in math. And, sorry, recreational mathematics is funny, but it's definitely not the answer to the question what is maths for? --pma (talk) 13:54, 26 November 2009 (UTC)

The OP asks for magic formulas. A formula appears magic when the proof is not understood. Heron's formula for example appears magic because it is the square root of a four-dimensional volume, which has no geometrical interpretation. Bo Jacoby (talk) 16:14, 26 November 2009 (UTC).

The Micawber Principle: "Annual income twenty pounds, annual expenditure nineteen nineteen six, result happiness. Annual income twenty pounds, annual expenditure twenty pounds ought and six, result misery." Mitch Ames (talk) 13:16, 27 November 2009 (UTC)

Euler-Maclaurin formula

I was playing with the Euler-Maclaurin formula for sums, and wondering if it'd be possible to sum up a "fractional" number of terms with it. It seems to work for some functions, but not others, e.g. summing $\sum _{n=0}^{x}\log(1+n)$ to get $\log((x+1)!)=\log(\Gamma (x+2))$. But the remainders given don't seem to work there at fractional x. Is there some generalization of the formula to handle this, or some way to compute fractional sums of general (analytic at least) functions? mike4ty4 (talk) 02:57, 25 November 2009 (UTC)

The series defines a function on integer points and you want to somehow interpolate between them. There isn't going to be a unique way to do that, even if you require the final function to be analytic (eg. the zero function and sin(πx) agree on all integer points, both are analytic but they are not the same function). I can't think of a stricter condition than being analytic that would give interesting results. --Tango (talk) 03:09, 25 November 2009 (UTC)

Hmm. Yes, it is about the interpolation problem. Yet even though there exist multiple solutions, I was wondering if there was some "natural" interpolation between them, akin to how one "naturally" interpolates the factorial via the Gamma-function, or the exponentiation via the exponential function. Faulhaber's formula provides a "natural" way to interpolate the sum for a power, and this can be applied to some Taylor series, but not all, giving interpolation for those, e.g. it can be used to interpolate the sum of "exp", but not of "log", and I was wondering if this could be used to expand this notion to more functions. mike4ty4 (talk) 03:23, 25 November 2009 (UTC)

IMO an interpolation is more natural if it is smoother, in the sense of having high-order derivatives of lower magnitude. This way the zero function clearly beats the sine. Maybe this can be generalized to a unique natural interpolation for any function bounded by a polynomial (such as $\log \Gamma (x)$). -- Meni Rosenfeld (talk) 09:12, 25 November 2009 (UTC)

Going to insert my opinion here: I would not call the Gamma function a "natural" interpolation of the factorial. Rather, the Gamma function is a highly interesting function in its own right, that just happens to be equal to the factorial on the positive integers. To me, calling the Gamma function an interpolation of the factorial is only technically true but mathematically not so important (although I'd be interested to see examples of useful extensions of formulae that involve factorials). Eric. 131.215.159.171 (talk) 18:02, 26 November 2009 (UTC)

Here is one nice example. -- Meni Rosenfeld (talk) 13:48, 27 November 2009 (UTC)

Thanks for the link, however (based on my limited knowledge of fractional derivatives) this is not an example of what I'm looking for. The definition of a fractional derivative using the Gamma function only uses the fact that the Gamma function is an extension of the factorial that satisfies the same functional relation on the reals that the factorial satisfies on the integers; it looks to me like any other such extension of the factorial would work just as well, and still give a consistent definition of fractional derivative. So this does not convince me that the Gamma function is still the one-and-only natural choice for an extension of the factorial. Eric. 131.215.159.171 (talk) 04:40, 30 November 2009 (UTC)

So what are you looking for, then? mike4ty4 (talk) 08:11, 30 November 2009 (UTC)

I'd say there you should want more than just an interpolation of $S(n):=\sum _{k=1}^{n-1}\log k;$ you want a solution of the functional equation $S(x+1)=S(x)+\log x$ (say for all x>0). This reduces the undeterminacy to just the unit interval. Work on s(x):=S'(x), that is, the functional equation $s(x+1)=s(x)+1/x.$ The homogeneous associated equation is $s(x+1)=s(x),$ meaning that, of course, you can add a 1-periodic function to a solution and get a solution, and any two solutions differ from a 1-periodic function. You can immediately check that a special solution of this is

$s(x):=\sum _{k=0}^{\infty }\left({\frac {1}{k+x}}-{\frac {1}{k+1}}\right),$

and what makes it special is that it is, up to a constant, the only monotonic solution (easy: for any monotonic solution, necessarily s(x+1)-s(x)=o(1) as x→∞ so adding a non-constant 1-periodic would destroy the monotonicity). You can then check that it is even analytic, and expand it in power series. Translated in terms of the equation F(x+1)=xF(x) and exp(S(x)), the above gives you the Bohr-Mollerup characterization of the Gamma function (you just have to compute the exact constant to get F(1)=1, wich turns out to be minus the Euler-Mascheroni constant).

PS: More generally, the telescopic trick to get a unique monotonic solution of s(x+1)=s(x)+f(x) also works if f(x) is positive, decreasing, and vanishing at infinity. Also, some cases, as the above $f(x)=\log x$, reduce to the latter after derivations. --pma (talk) 11:22, 25 November 2009 (UTC)

For the general theory of a summation operator that is not confined to an integral number of terms, see indefinite sum. Gandalf61 (talk) 11:44, 25 November 2009 (UTC)

In fact that article has a lot of summations, but no theory: at most a bit of metaphysics ;-) --pma (talk) 11:58, 25 November 2009 (UTC)

Yes, which mentions the Faulhaber formula when discussing the case of a non-specific analytic function, which does not converge for the log. In fact, I don't think it even converges for the Gaussian(!!!) function $f(x)=e^{-x^{2}}$ either. mike4ty4 (talk) 20:39, 25 November 2009 (UTC)

Trigonometry / college level algebra (Pre-calculus)

I am considering taking these two classes concurently next semester. Will the material covered in these two classes coincide with each other? Thank you.161.165.196.84 (talk) 06:46, 25 November 2009 (UTC)

If you haven't already had a reasonable high school algebra class, you should probably take the algebra class first and the trigonometry class afterwards. Trig classes usually require you to be be fairly adept at algebraic calculation. 67.117.145.149 (talk) 06:54, 25 November 2009 (UTC)

The classes at my high school weren't exactly labeled this way, but the one that covered most of the trig stuff came before the class called "pre-calculus" which covered what I assume is "college level algebra." I imagine it varies a lot how these topics are grouped and what order they're taught in. It's probably best to ask someone like a math teacher at your school, who knows exactly what the curriculum is. 67.100.146.151 (talk) 21:49, 25 November 2009 (UTC)

You say "college level algebra" - do you mean you are studying algebra at college or is this some kind of AP course at high school? The word "algebra" is used very differently at school than at college. At school, it just means manipulating equations which have an "x" in them, a college it is a field of mathematics in itself that involves all kinds of mathematical objects (eg. groups, rings and fields). You don't need any knowledge of the latter type of algebra to study trig, but some basic knowledge of the former is required for anything but the most basic trig (as it is required for anything but the basics in any other field of maths). --Tango (talk) 22:17, 25 November 2009 (UTC)

You wish, Tango :-). Well, actually it depends on the college or university. At sufficiently selective schools, there may be no course called algebra that doesn't mean abstract algebra. But at schools, even research universities, that are maybe in the second or third tier of selectivity, there may very well be a course called College Algebra, the purpose of which is to teach the kids what they ought to have learned when they were maybe thirteen. It does go faster than algebra courses taught in high school, but that's about the extent of the difference. --Trovatore (talk) 22:35, 25 November 2009 (UTC)

I'm primarily familiar with the UK system, which is very different. In the UK universities have courses like that, but they aren't for maths students, they are for science and engineering students. I know the US doesn't distinguish like that so much (you have "majors" where we have the subject of our entire degree). Would a "maths major" be taking courses that teach you how to solve "5x+3=0" (which, if I recall correctly, was about the level of algebra I was doing aged 13 - well, that's what the rest of the class were doing, I'd taught myself that kind of stuff several years before and just sat there being bored)? --Tango (talk) 22:45, 25 November 2009 (UTC)

Ordinarily you will not find a math major in such a class, although I don't know of anything formal that makes it impossible. It would be kind of alarming for a science or engineering major to be there also (though, certainly, some students get serious late and have to play catch-up). Usually all students are required to take some mathematics as a requirement for graduation, and that's usually the reason they're there.

Teaching these classes is an interesting experience. I found that the students were willing to work very hard indeed. Unfortunately no one had ever introduced them to the idea that mathematics involves thinking — they expected to get through by memorizing algorithms. The curriculum was designed to make it possible for them to do just that. What benefit this was supposed to provide anyone, beyond getting some money from the state to support grad students and junior faculty who taught the classes, I never figured out. --Trovatore (talk) 22:58, 25 November 2009 (UTC)

At top tier universities in the UK there would be a requirement for science and engineering students to have maths A-level (which includes algebra up to around the level of solving quadratic equations), so the Uni maths modules aimed at them would be slightly more advanced (they generally assume you didn't actually learn anything doing A-level and repeat it all, but very quickly, before moving on to more advanced things). In the UK, if you don't need maths for your degree then you don't have to study any maths. Compulsory subjects stop at age 16. I've never understood the reasons for the US system of requiring all college students to take compulsory modules in subjects unrelated to their degree - it just means they can't reach the same depth or breadth in the subject they actually intend to use. --Tango (talk) 23:11, 25 November 2009 (UTC)

However, if you are an exceptional student in a particular course, you may be exempted completely from taking other courses. Ordinary students are required to take a variety of courses at college because they may discover later that they wish to specialize in something completely different to that in which they thought they would specialize. On the other hand, within particular subjects such as mathematics, for instance, one is often required to attain a breadth of knowledge in many fields of study. In some universities, one must have a strong depth of experience in differential geometry, algebraic topology, algebraic geometry and complex analysis, while having an expert knowledge in algebra and real analysis. This is, of course, out of the ordinary, but it is a good system in that it allows graduate mathematics students to appreciate almost all fields of mathematics, even if they specialize in something totally different (and often other branches of mathematics can aid a student in his speciality). --PST 02:52, 26 November 2009 (UTC)

There is a big difference between requiring all maths students to take a module in linear algebra (as most UK unis do) and requiring all students to take a module in literature (as I believe US unis do). In the UK, the basic knowledge that everyone ought to have is supposed to complete by age 16, after which you only have to study things you are interested in. --Tango (talk) 03:32, 26 November 2009 (UTC)

There's also a pretty wide range of what can be considered "algebra" (referring not to abstract algebra). For example, during my education (in the US and pretty recent) we covered things like how to solve "5x+3=0" in 6th grade. On the other hand, a lot of what was in the class I took in 10th grade that was called "pre-calculus" could be considered algebra too (67.100.146.151 is me by the way). Things like how to solve a system of linear equations with matrices, the fundamental theorem of algebra, strategies for finding roots of nth degree polynomials, synthetic division, rational functions, classifying conic sections, etc. At the college I went to, people were expected to know that sort of stuff already and mostly started at calculus or beyond (math majors with an introduction to analysis and proofs), but a linear algebra was part of the curriculum also. I can imagine some colleges might teach that "pre-calculus" range of material to people who didn't get it in high school, and still need to learn some math for whatever they're doing. I'm pretty sure the New York State mandated minimum high school curriculum stopped short of that material even though everyone is (hopefully) exposed to "5x+3=0". Rckrone (talk) 23:28, 25 November 2009 (UTC)