Questions on the usage and meaning of words in mathematics, the names for mathematical entities, and other such questions.

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64
votes
10answers
3k views

Why 1 is not considered to be a prime number?

Why $1$ is not considered to be a prime number? Or why definition of prime numbers is given for integers greater than $1$?
30
votes
7answers
4k views

Is $0$ a natural number?

Is there a consensus in the mathematical community, or some accepted authority, to determine whether zero should be classified as a natural number? It seems as though formerly $0$ was considered in ...
23
votes
4answers
6k views

What is the term for a factorial type operation, but with summation instead of products?

(Pardon if this seems a bit beginner, this is my first post in math - trying to improve my knowledge while tackling Project Euler problems) I'm aware of Sigma notation, but is there a function/name ...
60
votes
14answers
3k views

Are “if” and “iff” interchangeable in definitions?

In some books the word "if" is used in definitions and it is not clear if they actually mean "iff" (i.e "if and only if"). I'd like to know if in mathematical literature in general "if" in definitions ...
16
votes
6answers
2k views

Alternative ways to say “if and only if”?

There are some scenarios about which I would like to get some confirmation: when defining a concept A, We call A, if ... [definition of concept A] Does "if" here mean equivalence instead of ...
33
votes
3answers
7k views

difference between class, set , family and collection

In school I have always seen sets. But I was watching a video the other day about functors and they started talking about any set being a collection but not vice-versa and I also heard people talking ...
23
votes
3answers
2k views

Why doesn't $0$ being a prime ideal in $\mathbb Z$ imply that $0$ is a prime number?

I know that $1$ is not a prime number because $1\cdot\mathbb Z=\mathbb Z$ is, by convention, not a prime ideal in the ring $\mathbb Z$. However, since $\mathbb Z$ is a domain, $0\cdot\mathbb Z=0$ is ...
34
votes
4answers
3k views

What do Algebra and Calculus mean?

I sometimes see phrases like 'the relational algebra' or 'the lambda calculus'. What is the difference between an algebra and a calculus?
31
votes
6answers
4k views

Lemma vs. Theorem

I've been using Spivak's book for a while now and I'd like to know what is the formal difference between a Theorem and a Lemma in mathematics, since he uses the names in his book. I'd like to know a ...
18
votes
6answers
10k views

Is there any difference between mapping and function?

I wonder if there is any difference between mapping and a function. Somebody told me that the only difference is that mapping can be from any set to any set, but function must be from $\mathbb R$ to ...
95
votes
1answer
8k views

Why are rings called rings?

I've done some search in Internet and other sources about this question. Why the name ring to this particular object? Just curiosity. Thanks.
55
votes
5answers
3k views

Why “characteristic zero” and not “infinite characteristic”?

The characteristic of a ring (with unity, say) is the smallest positive number $n$ such that $$\underbrace{1 + 1 + \cdots + 1}_{n \text{ times}} = 0,$$ provided such an $n$ exists. Otherwise, we ...
6
votes
7answers
836 views

Fundamental Theorem of Trigonometry

This is a pretty open ended question and I apologize, in advance, if this is not the place for it. But what do you recommend should be given the title of the Fundamental Theorem of Trigonometry and ...
90
votes
5answers
9k views

What is the Riemann-Zeta function?

In laymen's terms, as much as possible: What is the Riemann-Zeta function, and why does it come up so often with relation to prime numbers?
5
votes
1answer
873 views

Motivation for the term “separable” in topology

A topological space is called separable if contains a countable dense subset. This is a standard terminology, but I find it hard to associate the term to its definition. What is the motivation for ...
18
votes
14answers
3k views

Mathematical concepts named after mathematicians that have become acceptable to spell in lowercase form (e.g. abelian)?

I would like to collect a list of mathematical concepts that have been named after mathematicians, which are now used in lowercase form (such as "abelian"). This question is partly motivated by my ...
1
vote
3answers
4k views

why do we use 'non-increasing' instead of decreasing?

In english based math language it seems that non-increasing $\Longleftrightarrow$ less or equal (non-strict decreasing) decreasing $\Longleftrightarrow$ strict less ( strict decreasing) ...
71
votes
5answers
3k views

Why does mathematical convention deal so ineptly with multisets?

Many statements of mathematics are phrased most naturally in terms of multisets. For example: Every positive integer can be uniquely expressed as the product of a multiset of primes. But this ...
41
votes
17answers
5k views

What exactly is a number?

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 ...
53
votes
4answers
5k views

Do we have negative prime numbers?

Do we have negative prime numbers? $..., -7, -5, -3, -2, ...$
12
votes
6answers
417 views

Should every group be a monoid, or should no group be a monoid?

Question: What is more convenient/useful? Writing mathematics as if every group is a monoid, or as if these two classes are disjoint? Additional discussion. Define a monoid as follows. Defn 1. A ...
8
votes
7answers
831 views

What is a number?

A dictionary I consulted said a 'number' is a 'quantity', so I looked up what quantity means and the same dictionary said it is an amount or number of some material or thing. Since quantity and ...
15
votes
4answers
13k views

Is positive the same as non-negative?

I would assume the answer to my question is yes, but I want to make sure because my book uses both terminologies. Please also indicate where zero falls into the mix. UPDATE: Here is an excerpt from ...
3
votes
1answer
137 views

Problem with “tree” definitions

In my studies of choice principles, I've encountered the concept of a tree several times. Frustratingly, no two of the sources I've been working with define it in quite the same way! Notation: Given ...
7
votes
1answer
1k views

Is there a name for $[0,1]$?

When writing software, there are often situations where I need a parameter to be a floating point number $x \in [0,1]$. I don't know of a name for that category, but I think there must be one because ...
6
votes
6answers
469 views

In what sense of “structure” do group homomorphisms “preserve structure”?

It is commonly said that group homomorphisms "preserve the structure of the group", e.g., from Wikipedia: The purpose of defining a group homomorphism as it is, is to create functions that ...
5
votes
3answers
2k views

What is the difference between vector components and its coordinates?

Some mathematitians told me that vector components and coordinates are different things. They say that vector $F^n$ always has N components but coordinates depend on chosen basis and, therefore, it is ...
2
votes
1answer
90 views

In arbitrary commutative rings, what is the accepted definition of “associates”?

In an integral domain, the following are equivalent: $r \mid s$ and $s \mid r$ $r=us$ for some unit $u$ However in arbitrary commutative rings this is no longer the case; in particular, (2) ...
5
votes
3answers
1k views

Notions of equivalent metrics

Let $X$ be a set, and $d,d'$ two metrics on $X$. Consider the identity map $i : (X,d) \to (X,d')$ as a map of metric spaces. There are (at least) three reasonable notions of equivalence for $d$ and ...
14
votes
3answers
3k views

What is a natural isomorphism?

I came across the adjective "natural" many times in my reading and I think this has something to do with category theory. Could someone please illustrate the idea behind this adjective to me? Many ...
43
votes
12answers
5k views

Is it wrong to tell children that $1/0 =$ NaN is incorrect, and should be $∞$?

I was on the tube and overheard a dad questioning his kids about maths. The children were probably about 11 or 12 years old. After several more mundane questions he asked his daughter what $1/0$ ...
16
votes
5answers
2k views

What is the difference between “family” and “set”?

What is the difference between "family" and "set"? The definition of "family" on mathworld (http://mathworld.wolfram.com/Family.html) is a collection of objects of the form $\{a_i\}_{i \in I}$, where ...
21
votes
6answers
3k views

Is there a name for the function $\max(x, 0)$?

Is there a name for the function $ \max(x, 0) $? For comparison, the function $ \max(x, -x) $ is known as the absolute value (or modulus) of x, and has its own notation $ |x| $
21
votes
7answers
8k views

Why does “convex function” mean “concave *up*”?

A function $f : \mathbb{R} \to \mathbb{R}$ is convex (or "concave up") provided that for all $x,y \in \mathbb{R}$ and $t \in [0,1]$, $$f(tx + (1-t)y) \le tf(x) + (1-t)f(y).$$ Equivalently, a line ...
12
votes
2answers
1k views

Name for matrices with orthogonal (not necessarily orthonormal) rows

Is there a name for a matrix whose rows (or columns) are non-zero orthogonal vectors ? It seems to me that "orthogonal matrix" would be a good name, but this is already taken -- it refers to a matrix ...
11
votes
3answers
8k views

What does “isomorphic” mean in linear algebra?

My professor keeps mentioning the word "isomorphic" in class, but has yet to define it... I've asked him and his response is that something that is isomorphic to something else means that they have ...
19
votes
3answers
2k views

When do I use “arbitrary” and/or “fixed” in a proof?

In many proofs I see that some variable is "fixed" and/or "arbitrary". Sometimes I see only one of them and I miss a clear guideline for it. Could somebody point me to a reliable source (best a ...
18
votes
3answers
1k views

where does the term “integral domain” come from?

Self-explanatory title really! A student today asked me why they were called integral domains -- and I realised that the word "integral" seems to be being used in a way totally unlike any other way I ...
4
votes
3answers
2k views

Different ways to express If-Then

What are some different ways to write the conditional statement $p\implies q\,$, but in English? There's the obvious "If p, then q", but are there any other ways to write it? I'm looking for another ...
10
votes
3answers
1k views

What's the difference between 'any', 'all' and 'some'?

There are lots of expressions like, for all x, for any x, for some x, etc. I think 'for some x in R s.t ~' means that there exists at least one point in R s.t ~~. right? However, I can't know the ...
6
votes
2answers
1k views

Precise definition of “weaker” and “stronger”?

If I say that $A$ is stronger than $B$, do I mean that $A \Rightarrow B$, or that $B \Rightarrow A$? (Or something else?) I feel like I have seen both usages in literature, which is confusing. ...
5
votes
3answers
426 views

Generic Elements of a Set.

Mild Motivation: In writing a post about the Baire Category Theorem, I learned the neat fact that a "generic" $f\in C^{0}([a,b], {\mathbb R})$ was nowhere differentiable and not monotone on any ...
9
votes
4answers
1k views

What is the name of the vertical bar?

I've always wanted to know what the name of the vertical bar in these examples was: $f(x)=x^2+1\mid_{x = 4}$ (I know this means evaluate x at 4) $\int_0^4 (x^2+1) dx = (\frac{x^3}{3}+x+c) ...
9
votes
6answers
1k views

How is the codomain for a function defined?

Or, in other words, why aren't all functions surjective? Isn't any subset of the codomain which isn't part of the image rather arbitrary?
7
votes
4answers
195 views

How do you read the symbol “$\in$”?

A variable in an equation may be replaced by any of the numbers in its domain. The resulting equation may be either true or false. Here is another way to show ...
5
votes
7answers
991 views

Dictionaries and resources for translation of mathematical terminology

Nowadays English seems to be the most frequently used language in mathematics. (Although plenty of papers and books are published in other languages, e.g., Russian, French, German and Chinese.) ...
2
votes
1answer
7k views

Complementary Solution = Homogenous solution?

I have calculated solutions to homogenous equations but is the complementary solution mentioned here the same as the homogenous solution? Let's take example $y''-3y'+2y=\cos(wx)$ and now ...
9
votes
3answers
3k views

What does a “convention” mean in mathematics?

We all know that $0!=1$, the degree of the zero polynomial equals $-\infty$, the interval$[a,a)=(a,a]=(a,a)=\emptyset$ ... and so on, are conventions in mathematics. So is a convention something that ...
2
votes
2answers
157 views

The use of any as opposed to every.

This is a really basic question, but it is one I never really thought about until now. Let $\mathscr{G}$ be a tree. Then every pair of vertices in $\mathscr{G}$ is connected by a unique walk. We ...
42
votes
6answers
2k views

Why is compactness in logic called compactness?

In logic, a semantics is said to be compact iff if every finite subset of a set of sentences has a model, then so to does the entire set. Most logic texts either don't explain the terminology, or ...