Wikipedia:Reference desk/Archives/Mathematics/2013 January 1

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January 1

Archimedes uses time machine to beat Lambert

Sphere#Surface_area_of_a_sphere says:

This formula was first derived by Archimedes, based upon the fact that the projection to the lateral surface of a circumscribed cylinder (i.e. the Lambert cylindrical equal-area projection) is area-preserving

However Lambert cylindrical equal-area projection says:

The invention of this projection is attributed to the Alsatian mathematician Johann Heinrich Lambert in 1772.

Which article is wrong? Hcobb (talk) 01:27, 1 January 2013 (UTC)

I don't have any knowledge about this, but I'd guess that Archimedes used this projection in a non-rigorous way, while Lambert formalized it, analyzed it in more detail, and described its specific applicability to mapmaking. -- Meni Rosenfeld (talk) 09:24, 1 January 2013 (UTC)

Yup, as a non-mathematician who had to exercise his brain in an unaccustomed way to get to the point of what Archimedes and Lambert were each saying, I have to agree - Archimedes seems to have said it first, but Lambert described it better, or more rigorously. I think that the best solution is to replace 'i.e.' with 'as later defined by', or something similar. AndyTheGrump (talk) 09:35, 1 January 2013 (UTC)

All it seems to be saying to me is that Archimedes never thought of using the fact for cartography, it was just an interesting incidental lemma on the way as far as he was concerned. Dmcq (talk) 10:32, 1 January 2013 (UTC)

Archimedes was a bright guy. He did beat most of us. Bo Jacoby (talk) 12:30, 1 January 2013 (UTC).

SOLVING ALL POLYNOMIALS BY RADICALS

It is over 350 years now since Mathematicians started searching for the general formula for solving all equations and the search was believed to be impossible as some famous mathematicians have proved. The proof outlines vividly why it is always impossible to find the general formula. Formulae(solutions) for quadratic , cubic and quartic equations are well-known but quintic equations and above have since the start of mathematics (Classical algebra) troubled mathematicians rendering the general formula to be impossible. A young Ghanaian mathematician has found it to be possible to have a general formula for all equations whose work is about to be published in the JOURNAL OF GHANA SCIENCE ASSOCIATION. The question is, why young mathematicians always make great contributions to the development of Algebra or generally mathematics? — Preceding unsigned comment added by 41.215.160.218 (talk) 14:07, 1 January 2013 (UTC)

Yeah, we know: [1], [2], [3]. We do not believe this result, and neither will anyone whose opinion matters. You ignored my advice in 2012, and deserve the crackpot label that you will soon earn. Congratulations. Sławomir Biały (talk) 15:00, 1 January 2013 (UTC)

You can read more here as well. It is surprising how far these things go. It would be very difficult for someone to answer all the questions that have been put to him, even if he were correct (such as showing the exact flaw in Niels Hendrik Abel's proof). It would be easy for someone to show the correct answer to some equations that have been given, however. Some people call this trolling, which I regard as a bit out of bounds. I would call it simply time-wasting, and encourage the OP to work at something that has a better practical chance of success. He may in fact be very gifted, and just in need of some direction. IBE (talk) 17:21, 1 January 2013 (UTC)

As he graduated with a "Second Class Upper" in Mechanical Engineering, I doubt that his strengths will lie in that field, so maybe maths is the likelier bet.→31.54.246.112 (talk) 17:04, 2 January 2013 (UTC)

See Crank (person). Bo Jacoby (talk) 04:53, 4 January 2013 (UTC).

matrix

is it possible to divide a number by a matrix?

As in if i have a matrix m, does 1/m have any meaning? — Preceding unsigned comment added by 86.151.178.51 (talk) 14:39, 1 January 2013 (UTC)

No, you can't divide a number by a matrix or write 1/M to get a reciprocal for the matrix M. But you can write M^-1 for the inverse of the matrix. This means that M times M^-1 equals the identity matrix, and M^-1 times M also equals the identity matrix. Duoduoduo (talk) 15:40, 1 January 2013 (UTC)

Someone might like to add more/ correct me here, but a rough way to consider what Duoduoduo wrote is that there is some analogy between division and multiplication by an inverse. They are not quite the same thing here, largely because matrix multiplication is not commutative. If it were, you could do the operations in any order, and you might think of M^-1 as equivalent to 1/M. In fact, the use of 1/M is technically an issue regarding notation only, but the analogy with ordinary division is not as strong as it might at first appear. For the abstract generalisation of these operations, you might like to read up on fields IBE (talk) 19:22, 1 January 2013 (UTC)

I think fields is too strong - properly speaking the matrices form a ring under entrywise addition and RC-style multiplication. The critical difference is that elements of a ring are not required to have inverses under the "multiplication" (second) operation, merely under the first "addition" operation, whereas as you suggest elements of a field are. 72.128.82.131 (talk) 05:28, 4 January 2013 (UTC)

[ec] I think it's perfectly acceptable to use $1/M$ to refer to $M^{-1}$, and more generally, $a/M=aM^{-1}$. Scalar matrices are identified with scalars, and indeed $(1/M)M=I=1$.

A harder problem is to ascribe meaning to $A/B$ for matrices. One way is to have "right division" $A/B=AB^{-1}$ and "left division" $A/B=B^{-1}A$. Another possibility (which may or may not be useful), for a positive definite matrix B which has a unique positive definite square root, is to use a symmetric $A/B=B^{-0.5}AB^{-0.5}$. -- Meni Rosenfeld (talk) 19:31, 1 January 2013 (UTC)

It may be perfectly acceptable to use $1/M$ to refer to $M^{-1}$, after defining it as such, in the sense that there would seem to be no other way to interpret it. But I've never seen such notation, and I think if you tried to use it in a journal article the referee would probably say that the author seems unfamiliar with standard math notation. Or am I wrong about this? -- Can you give a link to someplace that actually uses it this way? Duoduoduo (talk) 20:45, 1 January 2013 (UTC)

I haven't seen it either and I don't think this notation is particularly useful. But it's healthy to think about how mundane concepts can be generalized.

Maybe it can be handy when talking about substituting matrices in polynomials and other scalar functions. -- Meni Rosenfeld (talk) 20:06, 2 January 2013 (UTC)