I was curious to know what comes after:     Primary, secondary, tertiary, ...    says it is "quartenary, quinary, ..."  But they are already taken!     Unary, binary, ternary, quaternary, quinary, ...   And according to  deriving from the Latin originals should give:     Primary, secondary, tertiary, quartary, quintary, ...    says that "quartary" (from the ordinal) is correct, but "quaternary" (from the distributive number) is common especially in biology.  So, what should it really be in English?
120431
I looked on Oxford's online dictionary and was able to find the names identifying orders of a given degree:   primary secondary tertiary quaternary quinary senary septenary octonary nonary denary -- no term for 11th degree?? duodenary   I am curious as to what would be the sequence of terms regarding a set of 'n' items? I have up to four:   unary binary ternary quaternion   but I cannot seem to find anything beyond that. Does anyone know where this list may be?
8060
I have troubles writing a fraction whose items are words not numbers.  In a nutshell I should write:  minaccia/vulnerabilità   --&gt; this is the numerator ----------------------  contromisure             --&gt; this is the denominator   I simply do this, but it doesn't work.  \[ \frac{minaccia/vulnerabilit\'a}{contromisure}  \]   Can anyone help me?
11020
Could you please remind me what term linguists and the grammar people use to call it when it's used as a subject pronoun, but the funky thing is that it doesn't really refer back to anything in particular like he, she or we usually do which do refer back to a person or an animal that they stand for.  In grammar-speak, when it is used like that they say that it has no antecedent. I think I knew the term, but I unfortunately forgot it. I did a Google search, but the only thing I could find was empty subject. But the truth is that I don't think that's what they actually use in linguistics. Would you please refresh my memory?  Examples:        It's quite cold today outside. So, I don't really wanna go out.   Did I solve that math problem? Yes. Actually, it was very easy to get it cracked. There was really nothing to it!   Don't mention it! It's nothing. I really like to help people who are in need.
506
In the decimal system: $...-3, -2, -1, 0, 1, 2, 3...$ In the binary system: $...-11, -10, -1, 0, 1, 10, 11...$ In the unary system: $-111, -11, -1, \text{ ??? }, 1, 11, 111...$   I was only able to find a related question but none with an answer so far:
94997
Suppose we want to define a first-order language to do set theory (so we can formalize mathematics). One such construction can be found . What makes me uneasy about this definition is that words such as "set", "countable", "function", and "number" are used in somewhat non-trivial manners. For instance, behind the word "countable" rests an immense amount of mathematical knowledge: one needs the notion of a bijection, which requires functions and sets. One also needs the set of natural numbers (or something with equal cardinality), in order to say that countable sets have a bijection with the set of natural numbers.  Also, in set theory one uses the relation of belonging "$\in$". But relation seems to require the notion an ordered pair, which requires sets, whose properties are described using belonging...  I found the following in Kevin Klement's,  (pages 2-3).  "You have to use logic to study logic. There’s no getting away from it. However, I’m not going to bother stating all the logical rules that are valid in the metalanguage, since I’d need to do that in the metametalanguage, and that would just get me started on an infinite regress. The rule of thumb is: if it’s OK in the object language, it’s OK in the metalanguage too."  So it seems that, if one proves a fact about the object language, then one can also use it in the metalanguage. In the case of set theory, one may not start out knowing what sets really are, but after one proves some fact about them (e.g., that there are uncountable sets) then one implicitly "adds" this fact also to the metalanguage.  This seems like cheating: one is using the object language to conduct proofs regarding the metalanguage, when it should strictly be the other way round.  To give an example of avoiding circularity, consider the definition of the integers. We can define a binary relation $R\subseteq(\mathbf{N}\times\mathbf{N})\times(\mathbf{N}\times\mathbf{N})$, where for any $a,b,c,d\in\mathbf{N}$, $((a,b),(c,d))\in R$ iff $a+d=b+c$, and then defining $\mathbf{Z}:= \{[(a,b)]:a,b\in\mathbf{N}\}$, where $[a,b]=\{x\in \mathbf{N}\times\mathbf{N}: xR(a,b)\}$, as in  question or  on Wikipedia. In this definition if set theory and natural numbers are assumed, then there is no circularity because one did not depend on the notion of "subtraction" in defining the integers.  So my question is:     Question Is the definition of first-order logic circular?   If not, please explain why.   If the definitions are circular, is there an alternative definition which avoids the circularity?   Some thoughts:   Perhaps there is the distinction between what sets are (anything that obeys the axioms) and how sets are expressed (using a formal language). In other words, the notion of a set may not be circular, but to talk of sets using a formal language requires the notion of a set in a metalanguage. In foundational mathematics there also seems to be the idea of first defining something, and then coming back with better machinery to analyse that thing. For instance, one can define the natural numbers using the Peano axioms, then later come back to say that all structures satisfying the axioms are isomorphic. (I don't know any algebra, but that seems right.) Maybe sets, functions, etc., are too basic? Is it possible to avoid these terms when defining a formal language?
101572
What's the difference between describing a number or place as cardinal versus ordinal?  They both seem to have to do with the order of things.  For example, when saying "He is the first in his class", is first ordinal or cardinal?  What are the usage differences here?
106213
In the following example, I have one wife, one daughter and one son.  (1) My wife, Angie; my son, Mike; and my daughter, Danielle, will be at the picnic.  I think the semicolons are very technically correct, but they are an eyesore.  Other possibilities:  (2) My wife, Angie, my son, Mike and my daughter, Danielle, will be at the picnic.  (3) My wife Angie, my son Mike, and my daughter Danielle will be at the picnic.  Which is preferred -- 1, 2 or 3 above?
40348
This question is also open on EL&amp;U:   Considering the amount of controversy it aroused, I believe it's past "Learners'" level.    An edit was suggested to my sentence.     There was were an orange, some grapes, two apples and a small pile of cherries on the plate.   In my native language plurality of the verb always follows plurality of the first element on the list. There were an orange,... sounds awkward to me, no matter what follows up. My simple solution was reordering:     There were some grapes, an orange, two apples and a small pile of cherries on the plate.   But that's not the first time I faced this situation and I'd like to know what the rules of grammar say about that — was my editor overzealous or am I trying to copy rules of my language that don't apply in English?
17258
What are the differences in mathematical notation around the world? I know that in some other countries they write 1,2 meaning 1.2, but what else can be confusing in an academic environment (when people are doing math on a board or on paper).
67531
The question gives the following: ¬(p ∨ ¬q) → (¬r ∧ (p → r))  With my own working, I have come to the answer of (p ∨ ¬q) ∧ (¬r ∧ ¬p) ∧ (¬r ∧ r). Can anyone tell me if this is correct?  Also, please tell me what Disjunctive Normal Form really means (without complicated terminology). The answer I got was from trying to break down what other similar questions have done. I still don't know what it really means.
42119
Consider any integer, suppose a number 3. We can break any such integer down into a sequence of 1's. So, we can write: $$ 3\equiv1+1+1 $$ 3 represents what we get when we add 1 to 1 to another 1. Conceptually, what I am getting confused with, is the fact that notionally, at least according to this definition, there is one unit at the beginning, to which we add another, and yet a third. There seems to be a sequence here, which would imply there being an order. Am I missing something obvious?
63261
Recently I was looking at the X-Men box set and saw that currently five have been released.  I had it in my head that these would be called a quintology but I have seen them being called a pentalogy.  I assume both words are interchangeable, but is one of the terms preferred over the other, or is it a matter of taste?
79557
I'm having a little argument with my friend. He says that "$a + 0i$" is, in every way, absolutely equal to "$a$" (e.g.: $2 + 0i = 2$).  I say this is practically the case, so in every calculation you just assume that "$a + 0i$" is equal to the real number "$a$" and you always get the right results (this might not even be true, but I am no expert and as far as I can see this is the case). But then again, I believe that just cutting away "$+ 0i$" makes this number is not the same kind of number anymore, it's changed its structure and is not totally equivalent anymore. This is where he says I am absolutely wrong.  So, is it mathematically (strictly speaking!) perfectly fine to say that every complex number with an imaginary part of $0$ is just a real number? Or does this change the mathematical structure so much that it cannot be mathematically-valid but just something that happens to work (which is what I believe)?  I must say that I believe a complex number is one entity, whilst he believes that it is just a conglomerate of other entitites. So, as one said here, "$(a + bi) - bi = a$", I believe is barely a good argument. But then again: is it correct to view $a + bi$ as a single entity (like "$5$" or "$2$", just consisting of multiple symbols)? I'm (as a non-mathematician) not sure there either.
86064
1) "The book 'The Three Musketeers' is a wonderful example of..." Here we have a proper noun, a title that happens to end in a plural, and I have no sense that the verb should be plural. "Musketeers"/is seems proper/natural/correct, and 'are' not at all.   However, such certainty begins to wane in other examples: 2) "'The Heidi Chronicles' is a wonderful example of..." doesn't completely shake my confidence, but it moves the needle, at least a little bit. But, practically speaking, I wouldn't have a second thought about correcting a student who used 'are' instead of 'is'.  Here's the one that really bothers me: 3a) " 'The Basement Tapes' are a wonderful example of..." I don't think this sounds any less proper/natural/correct than 3b) " 'The Basement Tapes' is..." In fact, 3a seems less awkward than 3b.  Why?
12721
We've just been learning about complex numbers in class, and I don't really see why they're called numbers.  Originally, a number used to be a means of counting (natural numbers).  Then we extend these numbers to instances of owing other people money for instance (integers).  After that, we consider fractions when we need to split things like 2 pizzas between three people.  We then (skipping the algebraic numbers for the purposes of practicality), we use the real numbers to describe any length; e.g. the length of a diagonal of a unit square.  But this is when our original definition of a number fails to make sense- when we consider complex numbers, which brings me to my main question: what is a rigorous definition of 'number'?  Wikipedia claims that "A number is a mathematical object used to count, label, and measure", but this definition fails to make sense after we extend $\mathbb{R}$.     Does anyone know of any definition of number that can be generalised to complex numbers as well (and even higher order number systems like the quaternions)?
97159
I have a question about using the phrase "some kind of _".  Is it incorrect to say "some kind of + plural noun"?     For example, is it wrong to say "are these rock-like things some kind of chocolates?"   I didn't use "kinds" and that kind of adds to my question. Is "kind" always plural, like "some kinds of chocolates"? Because although I'm using "some", it feels like "kind" shouldn't be a countable noun.  Anyway, I don't think I've heard of "some kind of + plural", but I think it makes more sense then using singular, since I think chocolates are countable. Please help!
74736
Is there a prefix or adjective that means "one and a half", as "tri-" or "triple" is for "three"?  The exact usage I have is to describe "18" in terms of a dozen. Where I live they've started making 18-egg cartons, and the local grocer was wondering what name to give them in his computerised cash register, hoping for a single word.  Even slang, informal or invented term would do if there isn't such a term already.