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December 3[edit]

Natural logarithm[edit]

Why is the natural logarithm ln rather than nl or something else? --70.250.212.95 (talk) 00:47, 3 December 2011 (UTC)[reply]

Our article on it says that it was once called "logarithmus naturalis", which sounds like Latin.--121.74.125.249 (talk) 01:05, 3 December 2011 (UTC)[reply]

...which illustrates that English is only one of the (obviously many) languages in which mathematics is expressed, and many of them place adjectives after the nouns they modify. In any event, there is a strong, broad convention throughout mathematics and science to build notation that way—from the outside in, if you will. Two examples:

-If a statistician wished to invent variables to represent the respective probabilities that a tossed coin might result in heads and tails, he'd likely use P_H and P_T rather than, say H_P and T_P.

-When mathematicians wish to index a set of variables sequentially, they write things like x₁ and x₂, rather than ₁x and ₂x.

—PaulTanenbaum (talk) 17:55, 3 December 2011 (UTC)[reply]

Integrability[edit]

hello. today in the calculus class that I AP (along with one other grad student) we introduced Riemann integration and the other said something to the effect that for a function to be integrable, it must not have "too many holes" (informally speaking), that is it must only be discontinuous on a finite number of points. I countered from my memory of real analysis (which may be faulty, I do stats, not a mathematician proper :S) a theorem that as long as the function's discontinuities are countable set of points, even if they are infinite, the funcion is integrable. But then I thought about the Dirichlet function, which is defined (one of many ways) as f(x)=x for rational x, 0 for irrational x. Since the rationals are countable the function should be integrable if the theorem I recalled is valid but the DIrichlet function is clearly not Riemann integrable (it is Lebesgue integrable, but that is something else altogether). I think I am misstating the theorem- I knew it had something to do with countability, can anyone help me out? THanks. 24.92.85.35 (talk) 01:21, 3 December 2011 (UTC)[reply]

A function is Riemann integrable if and only if the set of discontinuities has Lebesgue measure zero. Measure zero sets include finite sets, countable sets, and even some uncountable sets (like the cantor set). The set of all irrational numbers in [0,1] has lebesgue measure one. The dirichlet function is discontinuous on all the irrational numbers and therefore NOT Riemann integrable. For the longest time everyone worked hard on this to figure what exactly are the necessary and sufficient conditions on a function to be integrable. Riemann came along and fixed up the integral (meaning defined it properly) and put it upon solid ground and proved many of its properties from this new definition, which everyone had previously taken for granted since Newton. He made the question well posed and hence the integral is named in his honor. And then Henri Lebesgue was the first one who successfully answered this question and also generalized the Riemann integral.71.211.145.44 (talk) 03:02, 3 December 2011 (UTC)[reply]

To be precise, a function on an interval [a,b] is Riemann integrable if and only if it is bounded and the set of discontinuities has Lebesgue measure zero. 78.13.145.217 (talk) 22:54, 6 December 2011 (UTC)[reply]