Often engineers seem to find inspiration in the natural world. Could this be inspired in the bone matrix? Looks like it to me. If it is, it'd be nice and interesting if they were up front about it.
...is the same. I totally second the suggestion to "hang out with lots of different types of people". I like to think that I've managed to avoid the greater part of the arrogance that we're talking about here, and I put it down to the fact that in addition to my own nerdy activities, I've also maintained involvement with groups like musical societies, sporting groups and a drawing club, I've used vacations to do labour like in a mine and in a sawmill, and I work constantly with kids. It's the breadth of experience and human interaction that keeps one well-rounded and open-minded.
Hear, hear! Thank you. It's such a relief to know I'm not alone.
As for the reason, I think there isn't a good one at all, but it's probably nothing more than ignorance (of the Accept-Language header) on the part of web developers. But that is just "what I think", with no supporting evidence whatsoever.
But maybe someone actually knows the reason and is about to enlighten us.
It's not a matter of there being not enough time in the school year to get learning done. It's a case of the pace of learning being too low (essentially zero in some cases).
I agree.
And I think a risk in increasing the quantity, as proposed by Obama, is that the quality could actually be reduced, for increased school time will almost surely only reduce the appeal of the teaching profession (sufficient increase in reward is, I fear, highly unlikely) and thereby reduce the number / motivation / commitment of teachers.
He may be onto a good thing, but then he might not be. It's certainly not clear cut.
We'll find out soon whether Envion's process works as well as the company claims --- the $5 million inaugural plastic-to-fuel plant opened today in Washington, DC, and an undisclosed company has already agreed to buy Envion's product to blend into vehicle fuel.
Even more than merely vetoing a law, as recently as 1975 she sacked an entire government. Admittedly, that was via her representative in Australia, the Governer-General, but it is nonetheless a recent example of her power being exerted, demonstrating that she does have "real power" still.
And let's not forget this Roma. It also has (oldish) stone buildings and it also results in name confusion. For instance, I've encountered several Australians who thought that the "Roma tomatoes" that they get in the supermarket are called that because they come from Queensland!
Maybe they did mess up by getting pictures from this Roma. That would explain why all their images seem to be full of flies.
Yep, I see where you're coming from. I'd like to let you know though that with regards negatives causing confusion, I guess that would depend on the context, for certainly from the algebraic point of view, excluding them is what would actually be problematic! That's why in mathematics we include them. But one thing's for certain: if we're only discussing natural numbers in the first place, there can be no confusion at all.
With regards some of your questions:
- What is the prime factorization of -60? (-2)(2)(3)(5) will do, and consistent with the Fundamental Theorem of Arithmetic, this is unique up to the order of the factors and the appearance of units (i.e. 1 or -1 in Z)
- Are we going to allow -1 to be prime? No, a prime is a non-unit.
- Is -3 a prime factor of 6? Yes.
- Why are the prime factors 2 and 3 instead of -2 and -3? When you're only talking about natural numbers, then the definite article in "the prime factors" is totally unambiguous. However, if you're working in Z, then 6 = (2)(3) = (-2)(-3) are both prime factorizations of 6, and consistent with the Fundamental Theorem of Arithmetic, the factorization is unique up to order of factors and appearance of units.
For example, first paragraph, Chap 4, Stewart & Tall, "Algebraic Number Theory", 2nd ed., 1987, Chapman & Hall: (I write \pm for "plus or minus")
In Z we can factorize into prime numbers and obtain a factorization which is unique except for the order of factors and the presence of units \pm 1. Such a notion of unique factorization does not carry over to all rings of integers, but it does hold in some cases. As we shall see this caused a great deal of confusion in the history of the subject. The nub of the problem turned out to be the definition of a prime. In Z a prime number has two basic properties:
(1) m | p implies m = \pm p or \pm 1,
(2) p | mn implies p | m or p | n.
Either of these will do as the definition of a prime number in Z, and we usually take the former. In an arbitrary domain, it turns out that property (2) is what is required for uniqueness of factorization and in general (2) does not follow from (1). Property (1) is simply the definition of an irreducible element in Z. We will reserve the term prime for an element which satisfies (2) and is neither zero nor a unit. A prime is always irreducible, but not vice versa.
It is interesting to note the bit that says "we usually take the former". On this point, I stand corrected. When authors are restricting their attention to cases where all irreducibles are primes (for instance, when only considering Z like in the original discussion here on/.), they do seem to prefer property (1) as their definition. (e.g. Birkhoff and MacLane, "A Survey of Modern Algebra", 3rd ed, 1965, Macmillan, Ch 3, Sec 6: "An element not a unit with no proper divisors is called prime or irreducible.")
When more general algebraic theory is to be discussed, the distinction is made. However, it seems common that authors use property (1) in preliminary sections when talking about prime numbers, but then they redefine prime later as the need arises. For example, Ireland & Rosen, "A Classical Introduction to Modern Number Theory", 2nd ed, 1990, Springer-Verlag:
page 1: "... we say that a number p is a prime if its only divisors are 1 and p." At this point, only positive integers are being discussed.
page 2: "It will be more convenient to work with Z [where] the notion of divisibility carries over with no difficulty [...]. If p is a positive prime, -p will also be a prime."
page 9: "An element p is said to be irreducible if a|p implies that a is either a unit or an associate of p. A nonunit p is said to be prime if p \neq 0 and p|ab implies that p|a or p|b."
So in summary: property (1) is frequently used as the definition (my mistake), but only when it turns out that properties (1) and (2) are equivalent anyway; when the distinction does matter, it is property (2), not property (1) that defines a prime (on that note, I was correct); and still, as was ultimately the point, the "positive" part isn't redundant, but the "nonzero" part is.
Every prime number is a natural number, and every natural number is a positive/non-negative (depending on which definition you choose) integer. "Positive prime" is redundant.
The "positive" part is not the redundant part... it is the "nonzero" part that is. You have started with "every prime number is a natural number", which is a false premise... you can't rely on wikipedia for everything.
More precisely, that definition taken from wikipedia is closer to that for an irreducible, not a prime.
A nonzero element p in a ring is a prime if when p divides a product "ab", then p must divide one of the factors "a" or "b". A nonzero element p is irreducible if whenever you write p = st then either s or t must be a unit (in the case of integers, 1 or -1).
It just so happens that in the case of integers, the concepts of prime and irreducible turn out to be equivalent, which results in endless confusion. This means that "definition" of primes that people usually give is more correctly a "theorem". Anyhow, in the ring of integers, we have both positive *and* negative primes (i.e. 2 and -2 are both primes). In common speech though, we restrict ourselves to natural numbers (as the wikipedia article appears to do, sacrificing mathematical correctness for vulgarity).
So as I said to start with, the "positive" part isn't redundant; it's just being more precise than people normally bother to be. However yes, the "nonzero" part is redundant.
... was "Dive Into Python" (http://www.diveintopython.org/). I don't remember how I came across the book in the first place, but I did, I set and used the text for the course, and the publishers probably got some sales out of it, too, from those who like to have a bound copy for the bookcase. So perhaps you could have a look at that book's publisher for another alternative.
Precisely! And I wish more multi-lingual sites would pay attention to my preferences as given in the accept-language header rather than demanding that I make my selection from their language menu.
...is Principles of Network and System Administration by Mark Burgess. Very enjoyable read, good for IT info and also good for the "people" side of sysadmin. After having read the book from beginning to end, I now find myself frequently going back to read bits and pieces.
Some point out that Word is already dominant in various parts of academia, like biology. Yes... on the desktop, but not for making the final print-ready copy in a quality journal. The publishers will transfer the content of the Word file to another system (e.g. InDesign, QuarkXPress, something in-house, or who knows, maybe even TeX) to produce the print copy. The point is that Word may be dominant on the desktop, but it isn't in the publishing house... and nor do I believe it ever will be because it's just not what it's made to do.
Documents created with programs like Word, while adequate in some situations, simply never look professional. Making a beautiful page is more complex than what word processors do. It concerns issues beyond merely identifying ligatures, like identifying aesthetically "optimal" positioning of characters, words and lines (kerning, line-breaking, etc.). Now apart from its utility in formatting equations, which is surely the reason for its ubiquity in mathematically oriented fields, what is ultimately special about TeX lies in such things as its line- and page-breaking algorithms (Have you ever noticed how changing one character can change a line-break ten lines earlier as TeX takes a holistic view of the page's aesthetics?).
Basically, comparing the publishing process to earlier times, word processors are the typewriters and typesetting systems like TeX are, well, the typesetters who skillfully place the type for the printing press. Making a newer, better typewriter is a great thing for the authors who use them, but it won't displace the typesetters. With a better typewriter, people might be more inclined to simply circulate their (comparatively ugly) typewriter copy, but serious publication will still demand typesetting.
As a final remark, several people have commented that TeX separates content from presentation. That is really quite far from the truth; if anything, I'd say this is really more true of word processors like Word (if used well)! Rather, that is where LaTeX comes in, defining lots of macros to essentially support this content / presentation separation.
Word is a word processor; TeX is a typesetting system.
- Most any of the books from the MAA, especially the New Mathematics Library (now Anneli Lax New Library?), e.g. Geometric Inequalities, Geometric Transformations, Graphs and Their Uses, An Introduction to Inequalities, Uses of Infinity, Continued Fractions, The Mathematics of Choice, etc.
- Ian Stewart's books, especially Nature's Numbers.
- Loren C. Larson's book, Problem Solving Through Problems.
- Many of the smaller Dover books (e.g. Excursions in Geometry)
Slashdot Mirror
Often engineers seem to find inspiration in the natural world. Could this be inspired in the bone matrix? Looks like it to me. If it is, it'd be nice and interesting if they were up front about it.
...is the same. I totally second the suggestion to "hang out with lots of different types of people". I like to think that I've managed to avoid the greater part of the arrogance that we're talking about here, and I put it down to the fact that in addition to my own nerdy activities, I've also maintained involvement with groups like musical societies, sporting groups and a drawing club, I've used vacations to do labour like in a mine and in a sawmill, and I work constantly with kids. It's the breadth of experience and human interaction that keeps one well-rounded and open-minded.
Same bug in Opera, too. I think your "that don't render right anywhere" just might be true.
But you're going to have to install Windows on your router, too, to really make sure you're not vulnerable.
That was quick. Yipppeeeeee!
Four clicks into the menu? She was probably doing it intentionally; she just wanted a reason to talk to you and maybe even get you to visit.
Hear, hear! Thank you. It's such a relief to know I'm not alone.
As for the reason, I think there isn't a good one at all, but it's probably nothing more than ignorance (of the Accept-Language header) on the part of web developers. But that is just "what I think", with no supporting evidence whatsoever.
But maybe someone actually knows the reason and is about to enlighten us.
And still Google thinks that Dutch (the minority language) is the best choice to use.
Er... I believe it is actually the majority language. At least it was still when I left Belgium in 2005, but maybe you'd trust a web reference more.
But yes, I wouldn't doubt if more people can speak French, simply because I believe more Flemish people learn French than Walloons learn Dutch.
The bit:
'I recently wrote to the minister [Minister McDonald] regarding my concern...'
has McDonald writing to himself. Absurd! It should read:
'I recently wrote to the minister [Minister Brendan O'Connor] regarding my concern...'
This is made quite clear in TFA in their own correction at the end in boldface.
...it's quality.
It's not a matter of there being not enough time in the school year to get learning done. It's a case of the pace of learning being too low (essentially zero in some cases).
I agree.
And I think a risk in increasing the quantity, as proposed by Obama, is that the quality could actually be reduced, for increased school time will almost surely only reduce the appeal of the teaching profession (sufficient increase in reward is, I fear, highly unlikely) and thereby reduce the number / motivation / commitment of teachers.
He may be onto a good thing, but then he might not be. It's certainly not clear cut.
...the key part of TFA for me was:
We'll find out soon whether Envion's process works as well as the company claims --- the $5 million inaugural plastic-to-fuel plant opened today in Washington, DC, and an undisclosed company has already agreed to buy Envion's product to blend into vehicle fuel.
So yes, we'll find out soon, I guess.
Even more than merely vetoing a law, as recently as 1975 she sacked an entire government. Admittedly, that was via her representative in Australia, the Governer-General, but it is nonetheless a recent example of her power being exerted, demonstrating that she does have "real power" still.
And let's not forget this Roma. It also has (oldish) stone buildings and it also results in name confusion. For instance, I've encountered several Australians who thought that the "Roma tomatoes" that they get in the supermarket are called that because they come from Queensland!
Maybe they did mess up by getting pictures from this Roma. That would explain why all their images seem to be full of flies.
Yep, I see where you're coming from. I'd like to let you know though that with regards negatives causing confusion, I guess that would depend on the context, for certainly from the algebraic point of view, excluding them is what would actually be problematic! That's why in mathematics we include them. But one thing's for certain: if we're only discussing natural numbers in the first place, there can be no confusion at all.
With regards some of your questions:
- What is the prime factorization of -60? (-2)(2)(3)(5) will do, and consistent with the Fundamental Theorem of Arithmetic, this is unique up to the order of the factors and the appearance of units (i.e. 1 or -1 in Z)
- Are we going to allow -1 to be prime? No, a prime is a non-unit.
- Is -3 a prime factor of 6? Yes.
- Why are the prime factors 2 and 3 instead of -2 and -3? When you're only talking about natural numbers, then the definite article in "the prime factors" is totally unambiguous. However, if you're working in Z, then 6 = (2)(3) = (-2)(-3) are both prime factorizations of 6, and consistent with the Fundamental Theorem of Arithmetic, the factorization is unique up to order of factors and appearance of units.
For example, first paragraph, Chap 4, Stewart & Tall, "Algebraic Number Theory", 2nd ed., 1987, Chapman & Hall: (I write \pm for "plus or minus")
In Z we can factorize into prime numbers and obtain a factorization which is unique except for the order of factors and the presence of units \pm 1. Such a notion of unique factorization does not carry over to all rings of integers, but it does hold in some cases. As we shall see this caused a great deal of confusion in the history of the subject. The nub of the problem turned out to be the definition of a prime. In Z a prime number has two basic properties:
(1) m | p implies m = \pm p or \pm 1,
(2) p | mn implies p | m or p | n.
Either of these will do as the definition of a prime number in Z, and we usually take the former. In an arbitrary domain, it turns out that property (2) is what is required for uniqueness of factorization and in general (2) does not follow from (1). Property (1) is simply the definition of an irreducible element in Z. We will reserve the term prime for an element which satisfies (2) and is neither zero nor a unit. A prime is always irreducible, but not vice versa.
It is interesting to note the bit that says "we usually take the former". On this point, I stand corrected. When authors are restricting their attention to cases where all irreducibles are primes (for instance, when only considering Z like in the original discussion here on /.), they do seem to prefer property (1) as their definition. (e.g. Birkhoff and MacLane, "A Survey of Modern Algebra", 3rd ed, 1965, Macmillan, Ch 3, Sec 6: "An element not a unit with no proper divisors is called prime or irreducible.")
When more general algebraic theory is to be discussed, the distinction is made. However, it seems common that authors use property (1) in preliminary sections when talking about prime numbers, but then they redefine prime later as the need arises. For example, Ireland & Rosen, "A Classical Introduction to Modern Number Theory", 2nd ed, 1990, Springer-Verlag:
page 1: "... we say that a number p is a prime if its only divisors are 1 and p." At this point, only positive integers are being discussed.
page 2: "It will be more convenient to work with Z [where] the notion of divisibility carries over with no difficulty [...]. If p is a positive prime, -p will also be a prime."
page 9: "An element p is said to be irreducible if a|p implies that a is either a unit or an associate of p. A nonunit p is said to be prime if p \neq 0 and p|ab implies that p|a or p|b."
So in summary: property (1) is frequently used as the definition (my mistake), but only when it turns out that properties (1) and (2) are equivalent anyway; when the distinction does matter, it is property (2), not property (1) that defines a prime (on that note, I was correct); and still, as was ultimately the point, the "positive" part isn't redundant, but the "nonzero" part is.
Every prime number is a natural number, and every natural number is a positive/non-negative (depending on which definition you choose) integer. "Positive prime" is redundant.
The "positive" part is not the redundant part... it is the "nonzero" part that is. You have started with "every prime number is a natural number", which is a false premise... you can't rely on wikipedia for everything.
More precisely, that definition taken from wikipedia is closer to that for an irreducible, not a prime.
A nonzero element p in a ring is a prime if when p divides a product "ab", then p must divide one of the factors "a" or "b". A nonzero element p is irreducible if whenever you write p = st then either s or t must be a unit (in the case of integers, 1 or -1).
It just so happens that in the case of integers, the concepts of prime and irreducible turn out to be equivalent, which results in endless confusion. This means that "definition" of primes that people usually give is more correctly a "theorem". Anyhow, in the ring of integers, we have both positive *and* negative primes (i.e. 2 and -2 are both primes). In common speech though, we restrict ourselves to natural numbers (as the wikipedia article appears to do, sacrificing mathematical correctness for vulgarity).
So as I said to start with, the "positive" part isn't redundant; it's just being more precise than people normally bother to be. However yes, the "nonzero" part is redundant.
... was "Dive Into Python" (http://www.diveintopython.org/). I don't remember how I came across the book in the first place, but I did, I set and used the text for the course, and the publishers probably got some sales out of it, too, from those who like to have a bound copy for the bookcase. So perhaps you could have a look at that book's publisher for another alternative.
Name them all Bruce, just to avoid confusion.
I'm interested in news that matters...
Obviously! You are here on /.
Precisely! And I wish more multi-lingual sites would pay attention to my preferences as given in the accept-language header rather than demanding that I make my selection from their language menu.
...is Principles of Network and System Administration by Mark Burgess. Very enjoyable read, good for IT info and also good for the "people" side of sysadmin. After having read the book from beginning to end, I now find myself frequently going back to read bits and pieces.
Some point out that Word is already dominant in various parts of academia, like biology. Yes... on the desktop, but not for making the final print-ready copy in a quality journal. The publishers will transfer the content of the Word file to another system (e.g. InDesign, QuarkXPress, something in-house, or who knows, maybe even TeX) to produce the print copy. The point is that Word may be dominant on the desktop, but it isn't in the publishing house... and nor do I believe it ever will be because it's just not what it's made to do.
Documents created with programs like Word, while adequate in some situations, simply never look professional. Making a beautiful page is more complex than what word processors do. It concerns issues beyond merely identifying ligatures, like identifying aesthetically "optimal" positioning of characters, words and lines (kerning, line-breaking, etc.). Now apart from its utility in formatting equations, which is surely the reason for its ubiquity in mathematically oriented fields, what is ultimately special about TeX lies in such things as its line- and page-breaking algorithms (Have you ever noticed how changing one character can change a line-break ten lines earlier as TeX takes a holistic view of the page's aesthetics?).
Basically, comparing the publishing process to earlier times, word processors are the typewriters and typesetting systems like TeX are, well, the typesetters who skillfully place the type for the printing press. Making a newer, better typewriter is a great thing for the authors who use them, but it won't displace the typesetters. With a better typewriter, people might be more inclined to simply circulate their (comparatively ugly) typewriter copy, but serious publication will still demand typesetting.
As a final remark, several people have commented that TeX separates content from presentation. That is really quite far from the truth; if anything, I'd say this is really more true of word processors like Word (if used well)! Rather, that is where LaTeX comes in, defining lots of macros to essentially support this content / presentation separation.
Word is a word processor; TeX is a typesetting system.
- Most any of the books from the MAA, especially the New Mathematics Library (now Anneli Lax New Library?), e.g. Geometric Inequalities, Geometric Transformations, Graphs and Their Uses, An Introduction to Inequalities, Uses of Infinity, Continued Fractions, The Mathematics of Choice, etc. - Ian Stewart's books, especially Nature's Numbers. - Loren C. Larson's book, Problem Solving Through Problems. - Many of the smaller Dover books (e.g. Excursions in Geometry)