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.
'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 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).
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
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.
You can not get anything worthwhile done without raising a sweat. -- The First Law Of Thermodynamics