Thanks for all the replies. I guess strictly logical and consistent conventions are not as important as making sense and not using up tons of paper writing ()'s. In fact, now that I think about it, I’ve been writing (3 + ½) as (3½) for years, and nobody would see that and try to multiply it out to 1.5.
Another little bit of ignorance fought and conquered!
In continuum mechanics, we use u[sub],x[/sub] to denote the partial derivative of u with respect to x.
d[sup]2[/sup]u/dydx is then u[sub],yx[/sub]
It’s much more compact, unambiguous, and does not perpetuate the erroneous notion that the derivative is a ratio.
Incidentally, there have been other notational conventions that arose from scarcity. The famous “Einstein summation convention” came about because in Einstein’s papers on relativity, every other formula would involve at least one (sometimes two, three, or more) summation and the typesetters very quickly ran out of sigmas. The convention arose that whenever the same index showed up twice, it would be summed over the appropriate range. Often it was restricted to an upper and lower pair of indices, reflecting that one would be indexing coordinates on a tangent space and the other on a cotangent space. Real mathematicians now work without coordinates as much as possible, but you’ll still find physicists and engineers index-juggling away.
As an example, the Riemann curvature tensor is (as far as GR is concerned) a rank four tensor, which has four indices.
R[sub]ijkl[/sub]
Often it’s written with the first index raised
R[sup]i[/sup][sub]jkl[/sub] = g[sup]im[/sup]R[sub]mjkl[/sub]
Where g is the “metric tensor” and the summation convention means that the right hand side should be evaluated for m = 0, 1, 2, and 3 and all four of these values added together. Then one can define the Ricci curvature as
R[sub]jl[/sub] = R[sup]i[/sup][sub]jil[/sub]
Then raise the first index and contract again to get the Ricci scalar curvature
R = R[sup]l[/sup][sub]l[/sub] = g[sup]lj[/sup]R[sub]jl[/sub]
So already I’ve saved five summation symbols by using the convention (see if you can see where they should go).
Actually, you’ve got it backwards. Again using D for \partial:
D[sup]2[/sup]u/DyDx = D/Dy D/Dx u = D/Dy u[sub],x[/sub] = u[sub],xy[/sub]
It’s only in flat manifolds that mixed partials commute.
Anyhow, I’d be interested to see what notion of derivative you use in continuum mechanics that isn’t defined by a limit of a ratio. Yes, there’s a functorial definition, but that’s so abstract it’s almost meaningless to calculate with in applications.
I for one am so used to writing 3 + ½ as 7/2 that I am always tempted to interpret 3½ as 31/2. Especially because some people actually type three halves as 31/2, without bothering to include a space after the three.
Can anyone tell me why high schoolers are taught that a fraction whose numerator exceeds its denominator is “improper”? Is there any technical field where this convention is actually followed?
High schoolers? If they haven’t seen fractions by high school, something’s dreadfully wrong. Anyhow, the immediate benefit is seeing where a given fraction falls with respect to the integers. 3678943578956/23718567423 doesn’t tell as much at a glance as 155 2565628391/23718567423. Admittedly though, this is far outweighed in practice by the algorithmic benefits of “improper” fractions and the name “improper” is rather a loaded one.
But to a mathematician ‘proper’ isn’t really loaded - it basically means ‘non-pathological’ or ‘a normal sort of one’, yes? Of course, that doesn’t really explain it since mathematicians never and school children do use ‘improper’ and anyhow, improper fractions aren’t pathological in the usual sense, but that could have been in the mind of whoever coined the term, couldn’t it?
This just aggravates me when I get a question on, say, derivatives and it is written sinx[sup]2[/sup] and when I get confused, the teacher tells me to look at it as sin[sup]2[/sup]x. Why not just do it that way in the first place? I mean, everyone sees it differently, but why isn’t there a standard?
Well, it tends more to be used as “nontrivial”. A proper subset is a subset other than the whole set itself. A proper ideal of a ring is an ideal which is neither the zero ideal or the entire ring. I’d be willing to bet that “improper fraction” has always been used in teaching basic arithmetic and never in “real mathematics”.
It may seem pretty silly to have the people setting the basic math curricula and the people actually doing mathematics be separate groups, but remember that the last time mathematicians were allowed to decide how arithmetic should be taught we got “new math”: so simple that only a child can do it.
OK, if you want to get picky, in classical continuum mechanics a body is a three-dimensional differentiable manifold which must be viewed in its configurations. A configuration is a smooth homeomorphism of the body onto a region of plain old Euclidean 3-space. So we never have to deal with anything but a flat manifold. That is why we can use the subscript notation without getting confused, and we can always assume that mixed higher partials commute. Researchers in relativistic continuum mechanics, of whom there are a very, very few, use a different notation.
Also, a limit of a ratio is not the same thing as a ratio.
And the thread went over my head (again) at Mathochist’s last post, but I would like to say something about the word “improper.” The way I understood it, it was just a name, the same way we call numbers irrational, complex, transcendental, etc. We’re not saying that 3+2i is really a “made up” number, we just call it imaginary because it’s useful to call it that. In the same way, I never got the impression that improper fractions were bad or undesirable. We just called 3/2 improper, and 1 1/2 mixed. I think every teacher I’ve had would accept either form as a correct answer, unless they specifically said to write one or the other. So I suppose I’d agree that “improper” is a “loaded” name, but so are several other names that mathematicians actually do use outside of high school.
What I meant was not that they learn the term “improper fraction” in high school. (Even though, looking back at my comment, that is how I said it.) What I mean is why are they still using mixed numbers in high school. I see what you’re saying about seeing where fractions lie relative to the integers, but typically in high school math classes students are allowed to use calculators (if I remember correctly), so they can just type the fraction in and instantly see it in decimal form. I’m just a bit bothered by the fact that students get to college and are still worrying about things like converting improper fractions to mixed numbers or rationalizing their denominators, rather than just putting the answers in the simplest looking form.
I think by high school the students should be learning to write mathematical expressions the way mathematicians, physicists, etc. would actually write them.
This sounds nothing like any math class beyond “pre-algebra” that I’ve seen. No algebra teacher I’ve ever talked with has insisted on one form or another. Further, I’ve been teaching college freshmen for quite a few years now and with the exception of some heavily remedial students I tutored as an undergraduate, none of them have “worried” at all about such things. Where are you getting this impression of yours?
I think some people don’t like the form 124/51 just as it was considered bad form when I was at the University of Iowa to leave a radical in the denominator.
Now I must ask which course this was and when it was taught. Both could be very enlightening.