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

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155
votes
11answers
14k views

In simple English, what does it mean to be transcendental?

From Wikipedia A transcendental number is a real or complex number that is not algebraic A transcendental function is an analytic function that does not satisfy a polynomial equation However these ...
124
votes
1answer
11k 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.
105
votes
5answers
13k 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?
104
votes
13answers
11k views

Why do we use the word “scalar” and not “number” in Linear Algebra?

During a year and half of studying Linear Algebra in academy, I have never questioned why we use the word "scalar" and not "number". When I started the course our professor said we would use "scalar" ...
88
votes
6answers
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 ...
76
votes
38answers
12k views

Unusual mathematical terms

From time to time, I come across some unusual mathematical terms. I know something about strange attractors. I also know what Witch of Agnesi is. However, what prompted me to write this question is ...
71
votes
12answers
8k 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 $1$?
71
votes
14answers
5k 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 ...
70
votes
6answers
8k views

Do we have negative prime numbers?

Do we have negative prime numbers? $..., -7, -5, -3, -2, ...$
67
votes
19answers
19k views

What is the difference between a point and a vector

I understand that a vector has direction and magnitude whereas a point doesn't. However, the course note that I am using states that a point is the same as a vector. Also, can you do cross product ...
65
votes
5answers
4k 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 ...
64
votes
6answers
4k views

Why is a geometric progression called so?

Just curious about why geometric progression is called so. Is it related to geometry?
57
votes
12answers
6k views

I need mathematical proof that the distance from zero to 1 is the equal to the distance from 1 to 2 [closed]

I didn't know how to phrase the question properly so I am going to explain how this came about. I know Math is a very rigorous subject and there are proofs for everything we know and use. In fact, I ...
57
votes
7answers
3k 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 ...
52
votes
10answers
8k 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 ...
49
votes
6answers
8k views

Why does the Cauchy-Schwarz Inequality even have a name?

When I came across the Cauchy-Schwarz inequality the other day, I found it really weird that this was its own thing, and it had lines upon lines of proof. I've always thought the geometric definition ...
47
votes
3answers
12k 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 ...
47
votes
4answers
86k views

What are the numbers before and after the decimal point referred to in mathematics?

Sorry for asking such a basic question - but is there an actual term for the numbers that appear before and after the decimal point? Example: 25.18 I know the 1 ...
46
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 $1/0$ ...
43
votes
13answers
15k views

What does the term “undefined” actually mean?

I have read many articles on many sites and in many books to understand what undefined means? On some sites of Maths, I read that it could be any number. and on some sites, I read that it may be some ...
43
votes
17answers
6k 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 ...
42
votes
12answers
5k views

Does “Doing a thing to both sides of an equation” have a name?

A two part question. 1 True or False: when working with an equation or inequality, everything that you do is either: a substitution, or an operation performed on each side Note that algebraic or ...
42
votes
6answers
3k views

What do mathematicians mean by “equipped”

I am a mathematical illiterate so I do not know what people mean when they say equipped. For example, I say that Hilbert space is a vector space equipped with a inner product. What does that ...
40
votes
4answers
54k views

Probability density function vs. probability mass function

I've a confession to make. I've been using pdf's and pmf's without actually knowing what they are. The idea that I've been having so long is that density = area under the curve but if I look at it ...
37
votes
10answers
2k views

Definition of “well defined” in mathematics

I have encountered this term "well defined" in many places in maths like well-defined set, well-defined function, well-defined group, etc. What are the contexts in which we can talk about well ...
37
votes
1answer
518 views

Sign Language and Deaf Mathematicians

Something I've often wondered (and I suppose this goes for all kinds of technical terminology, not just that of mathematics) is what kind of sign language exists for practising professional ...
36
votes
6answers
2k views

Why is a linear transformation called linear? [duplicate]

$T(av_1 + bv_2) = aT(v_1) + bT(v_2)$ Why is this called linear? $f(x) =ax + b$, the simplest linear equation does not satisfy $f(x_1 + x_2) = f(x_1) + f(x_2)$. Thank you.
36
votes
4answers
4k 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?
35
votes
11answers
3k views

What is exactly the difference between a definition and an axiom?

I am wondering what the difference between a definition and an axiom. Isn't an axiom something what we define to be true? For example, one of the axioms of Peano Arithmetic states that $\forall n:0\...
33
votes
3answers
1k views

Where does the word “torsion” in algebra come from?

Torsion is used to refer to elements of finite order under some binary operation. It doesn't seem to bear any relation to the ordinary everyday use of the word or with its use in differential geometry ...
33
votes
2answers
2k views

How did “one-to-one” come to mean “injective”?

How did a "one-to-one" function come to mean an injective one? I find it so non-intuitive that I often have to backtrack when reading texts that use "one-to-one" because I suddenly discover that I ...
32
votes
5answers
12k 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 ...
32
votes
6answers
5k 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 ...
32
votes
6answers
2k views

Why are integrals called integrals?

What is the historical background for this term? I cannot quite see what is integral about an integral, even if we go back to the viewing it as the area under a curve. It seems a strange choice of ...
31
votes
1answer
6k views

Why are invertible matrices called 'non-singular'?

Where in the history of linear algebra did we pick up on referring to invertible matrices as 'non-singular'? In fact, since the null space of an invertible matrix has a single vector an ...
31
votes
2answers
2k views

Word origin / meaning of 'kernel' in linear algebra

It may be the dumbest question ever asked on math.SE, but... Given a real matrix $\mathbf A\in\mathbb R^{m\times n}$, the column space is defined as $$C(\mathbf A) = \{\mathbf A \mathbf x : \...
30
votes
5answers
21k views

What are the differences between rings, groups, and fields?

Rings, groups, and fields all feel similar. What are the differences between them, both in definition and in how they are used?
29
votes
5answers
13k views

Use of “without loss of generality”

Why do we use "without loss of generality" when writing proofs? Is it necessary or convention? What "synonym" can be used? Thanks.
28
votes
3answers
3k views

Why is lambda calculus named after that specific Greek letter? Why not “rho calculus”, for example?

Where does the choice of the Greek letter $\lambda$ in the name of “lambda calculus” come from? Why isn't it, for example, “rho calculus”?
28
votes
10answers
11k 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 ...
28
votes
7answers
18k 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 $\...
28
votes
4answers
46k views

Difference between axioms, theorems, postulates, corollaries, and hypotheses

I've heard all these terms thrown about in proofs and in geometry, but what are the differences and relationships between them? Examples would be awesome! :)
28
votes
2answers
2k views

What comes after length, area and volume? [duplicate]

The length of the unit is 1. The area of the unit square is 1. The volume of the unit cube is 1. The $\color{red}{???}$ of the unit tesseract is 1. The $\color{blue}{???}$ of the unit 5-cube ...
27
votes
6answers
3k views

The logarithm is non-linear! Or isn't it?

The logarithm is non-linear Almost unexceptionally, I hear people say that the logarithm was a non-linear function. If asked to prove this, they often do something like this: We have $$ \ln(x + ...
27
votes
10answers
15k views

What's the difference between stochastic and random?

What's the difference between stochastic and random? I've read in the Portuguese Wikipedia that there's a difference, but I still didn't see this point on English Wikipedia.
27
votes
4answers
7k views

What does it really mean for something to be “trivial”?

I see this word a lot when I read about mathematics. Is this meant to be another way of saying "obvious" or "easy"? What if it's actually wrong? It's like when I see "the rest is left as an exercise ...
26
votes
7answers
3k views

Is there such a thing as a matrix of functions?

Do we ever put functions as entries of a matrix? If so, are these matrices used in linear algebra or do they have some other special use? There have been minor not neccessarily conflicts per se, but ...
26
votes
3answers
4k views

What is this operator called?

If $x \cdot 2 = x + x$ and $x \cdot 3 = x + x + x$ and $x^2 = x \cdot x$ and $x^3 = x \cdot x \cdot x$ Is there an operator $\oplus$ such that: $x \oplus 2 = x^x$ and $x \oplus 3 = {x^{x^x}}$? ...
26
votes
6answers
121k views

What does it mean to have a determinant equal to zero?

After looking in my book for a couple of hours, I'm still confused about what it means for a $(n\times n)$-matrix $A$ to have a determinant equal to zero, $\det(A)=0$. I hope someone can explain this ...
26
votes
4answers
2k views

Why is “mathematical induction” called “mathematical”?

One of my whims is that I never write "mathematical induction" but just "induction". We are doing maths, so what is the point about precising? We don't say "Let $f$ be a mathematical function from the ...