Question on the meaning, history, and usage of mathematical symbols and notation. Please remember to mention where (book, paper, webpage, etc.) you encountered any mathematical notation you are asking about.

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11
votes
4answers
17k views

What is 48÷2(9+3)? [duplicate]

There is a huge debate on the internet on $48÷2(9+3)$. I figured if i wanted to know the answer this is the best place to ask. I believe it is 2 as i believe it is part of the bracket operation in ...
15
votes
6answers
2k views

What does $dx$ mean?

$dx$ appears in differential equations, such us derivatives and integrals. For example, a function $f(x)$ its first derivative is $\dfrac{d}{dx}f(x)$ and its integral $\displaystyle\int f(x)dx$. But ...
11
votes
1answer
763 views

$\arcsin$ written as $\sin^{-1}(x)$

I know that different people follow different conventions, but whenever I see $\arcsin(x)$ written as $\sin^{-1}(x)$, I find myself thinking it wrong, since $\sin^{-1}(x)$ should be $\csc(x)$, and not ...
18
votes
2answers
612 views

Notation: Why write the differential first?

From reading answers here, I've noticed that some people write integrals as $\int dx \; f(x)$, while other people write them as $\int f(x)\;dx$. I realize that there is no mathematical difference ...
14
votes
3answers
3k views

If $\frac{dy}{dt}dt$ doesn't cancel, then what do you call it?

I have $y$ is a function of $t$. I have reached a situation here where I need to evaluate $$\displaystyle \int_0^b{\frac{dy}{dt}dt}$$ Now clearly $y$ has dependence on $t$, otherwise $\displaystyle ...
13
votes
7answers
813 views

What does $\ll$ mean?

I saw two less than signs on this Wikipedia article and I was wonder what they meant mathematically. http://en.wikipedia.org/wiki/German_tank_problem EDIT: It looks like this can use TeX commands. ...
150
votes
22answers
8k views

Why do mathematicians use single-letter variables?

I have much more experience programming than I do with advanced mathematics, so perhaps this is just a comfort thing with me, but I often get frustrated trying to follow mathematical notation. ...
13
votes
2answers
1k views

Is $A \times B$ the same as $A \oplus B$?

When $A, B$ are $K$-modules, then $A \times B$ is the same as $A \oplus B$. Let $A, B$ be two $K$-algebras, where $K$ is a field. Is $A \times B$ the same as $A \oplus B$? Thank you very much. ...
11
votes
5answers
1k views

When should I use $=$ and $\equiv$?

In what context should I use $=$ and $\equiv$? What is the precise difference? Thanks! (I wasn't sure what to tag this with, any suggestions?)
10
votes
5answers
6k views

Backwards epsilon

What does the $\ni$ (backwards element of) symbol mean? It doesn't appear in the Wikipedia list of mathematical symbols, and a Google search for "backwards element of" or "backwards epsilon" turns up ...
28
votes
2answers
2k views

What are the rules for equals signs with big-O and little-o?

This question is about asymptotic notation in general. For simplicity I will use examples about big-O notation for function growth as $n\to\infty$ (seen in algorithmic complexity), but the issues that ...
12
votes
4answers
1k views

Notation question: Integrating against a measure

Suppose $\mu$ is a measure. Is there any difference in meaning between the notation $\int f(x)d\mu(x)$ and the notation $\int f(x) \mu(dx)$? Many thanks.
15
votes
3answers
70k views

What does E mean in 9.0122222900391E-5?

I am not a mathematician(IANAM), however I wish I could be. My question: I often find this at the bottom of pages. Page generated in 0.00013899803161621 Sometimes, I come across Page ...
5
votes
2answers
930 views

Is there a common symbol for concatenating two (finite) sequences?

Say we have two finite sequences $X = (x_0,...,x_n)$ and $Y = (y_0,...,y_n)$. Is there a more or less common notation for the concatenation of these sequences, like $\sum (X,Y) = ...
5
votes
3answers
162 views

Using $p\supset q$ instead of $p\implies q$

I saw that a use for the notation $p\supset q$ instead of $p\implies q$ that got me a bit confused. One occurrences is in this Wikipedia link. It seems to me opposite than what it should be, let me ...
64
votes
5answers
2k 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 ...
16
votes
10answers
1k views

Challenge: Demonstrate a Contradiction in Leibniz' differential notation

I want to know if the Leibniz differential notation actually leads to contradictions - I am starting to think it does not. And just to eliminate the most commonly showcased 'difficulty': For the ...
14
votes
6answers
6k views

What software and/or language to use to take Math lecture notes?

I have a terrible hand-writing and I'm very good at typing, so I had an idea about taking my math lecture notes using a computer. I've tried using a simple syntax (using purely ASCII) but it's ...
16
votes
11answers
1k views

Alternative notation for exponents, logs and roots?

If we have $$ x^y = z $$ then we know that $$ \sqrt[y]{z} = x $$ and $$ \log_x{z} = y .$$ As a visually-oriented person I have often been dismayed that the symbols for these three operators ...
7
votes
2answers
603 views

Notation to work with vector-valued differential forms

What it the standard notation used while working with vector-valued differential forms? I tried using abstract index notation, for example denoting a $1$-form valued $2$-form as $P_{i[bc]}$, but I'm ...
21
votes
7answers
1k views

Notation for repeated application of function

If I have the function $f(x)$ and I want to apply it $n$ times, what is the notation to use? For example, would $f(f(x))$ be $f_2(x)$, $f^2(x)$, or anything less cumbersome than $f(f(x))$? This is ...
11
votes
6answers
1k views

In written mathematics, is $f(x)$a function or a number?

I often see notation/wording like "let $f(x)$ be a continuous function" or "let $f(x) \in C^0(\mathbb{R})$". I would say that $\sin$ and $x \mapsto \sin(x)$ are functions, while $\sin(x)$ is a real ...
2
votes
3answers
500 views

What does | mean?

I found this symbol on Wolfram|Alpha. Does it mean "or"? $\displaystyle \large \cos^{-1}(-1)=\mathrm{cd}^{-1}(-1\mid 0)$
17
votes
5answers
21k views

How does one denote the set of all positive real numbers?

What is the "standard" way to denote all positive (or non-negative) real numbers? I'd think $$ \mathbb R^+ $$ but I believe that that is usually used to denote "all real numbers including infinity". ...
5
votes
4answers
829 views

$\subset$ vs $\subseteq$ when *not* referring to strict inclusion

Inspired by the confusion in the comments on this question: I always thought that the standard was to read $\subset$ as "is a strict subset of", and $\subseteq$ could mean proper or improper ...
8
votes
5answers
1k views

use of $\sum $ for uncountable indexing set

I was wondering whether it makes sense to use the $\sum $ notation for uncountable indexing sets. For example it seems to me it would not make sense to say $$ \sum_{a \in A} a \quad \text{where A is ...
4
votes
1answer
79 views

What is the proper notation for integer polynomials: $\Bbb Y=\{p\in\Bbb Q[x]\mid p:\Bbb Z\to \Bbb Z\}$?

I would like to write down some of my thoughts on "the set of polynomials $p\in\Bbb Q[x]$ which map the integers to the integers" and I would like to know what the proper notation is for discussing ...
4
votes
2answers
362 views

Meaning of $\mathbb{R}[x]$

I ran into this expression in a paper I was reading, and I'm confused about part of the meaning. Here $u$ and $v$ are two polynomials. $$u, v \in \mathbb{R}[x]$$ I'm not really familiar with usage ...
2
votes
2answers
626 views

What is the difference between $d$ and $\partial$?

After seeing the following equation in a lecture about tensor analysis, I became confused. $$ \frac{d\phi}{ds}=\frac{\partial \phi}{\partial x^m}\frac{dx^m}{ds} $$ What exactly is the difference ...
29
votes
11answers
3k views

What could be better than base 10?

Most people use base 10; it's obviously the common notation in the modern world. However, if we could change what became the common notation, would there be a better choice? I'm aware that it very ...
15
votes
5answers
2k views

Usage of dx in Integrals

All the integrals I'm familiar with have the form: $\int f(x)\mathrm{d}x$. And I understand these as the sum of infinite tiny rectangles with an area of: $f(x_i)\cdot\mathrm{d}x$. Is it valid ...
6
votes
3answers
835 views

What's the difference between “$\to$” (implication) and “$\vdash$” (therefore)?

In Wikipedia, here in the last axiom of the Natural deduction system, it says "From [accepting $p$ allows a proof of $q$], infer $(p \to q)$." Isn't that a tautology? In the big table "Basic and ...
4
votes
3answers
1k views

Notation for image and preimage

Let $X$, $Y$ be sets and $f:X\rightarrow Y$ be a map. Denoting the image of $D\subset X$ under $f$ by $f(D)$ can sometimes be confusing. As for preimages, I've seen unambiguous notation like ...
6
votes
5answers
673 views

Incorrect notation in math?

Does math have an incorrect notation / syntax? I don't mean writing misaligned notation (google), but when you take something like a number to powers to powers to powers, $${{2^2}^2}^3$$ (I was told ...
12
votes
3answers
12k views

Element-wise (or pointwise) operations notation?

Is there a notation for element-wise (or pointwise) operations? For example, take the element-wise product of two vectors x and y (in Matlab, x .* y, in numpy x*y), producing a new vector of same ...
9
votes
2answers
674 views

Notation for intervals

I have frequently encountered both $\langle a,b \rangle$ and $[a,b]$ as notation for closed intervals. I have mostly encountered $(a,b)$ for open intervals, but I have also seen $]a,b[$. I recall ...
6
votes
2answers
912 views

Limit of integration can't be the same as variable of integration?

I am told that an expression like $$ \int_a^x f(x)dx $$ is not well formed, i.e. it should be $$ \int_a^xf(t)dt $$ or similar. Why is it that the limits of integration can't depend on the ...
4
votes
2answers
366 views

Different standards for writing down expressions in a formal way

What are standard ways to write mathematical expressions in a (semi)formal way ? In different posts of mine concerning similar question I have encountered for a generic expression of the type "for all ...
3
votes
4answers
391 views

A short way to say f(f(f(f(x)))) [duplicate]

Is there a short way to say $f(f(f(f(x))))$? I know you can use recursion: $g(x,y)=\begin{cases} f(g(x,y-1)) & \text{if } y > 0, \ \newline x & \text{if } y = 0. \end{cases}$
10
votes
3answers
503 views

What is the purpose of the $\mp$ symbol in mathematical usage?

Occasionally I see the $\mp$ symbol, but I don't really know what it is for, except in conjunction with the $\pm$ symbol thus: $a \pm b \mp c$ which (I believe) means $a+b-c$ or $a-b+c$ (please ...
8
votes
4answers
257 views

Mathematical notation around the world

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 ...
8
votes
5answers
242 views

Notation for infinite product in reverse order

This question is related to notation of infinite product. We know that, $$ \prod_{i=1}^{\infty}x_{i}=x_{1}x_{2}x_{3}\cdots $$ How do I denote $$ \cdots x_{3}x_{2}x_{1} ? $$ One approach ...
3
votes
3answers
8k views

What is “multiplication by juxtaposition”?

I was reading http://www.purplemath.com/modules/orderops2.htm it shows = 16 ÷ 2[2] + 1 (**) ... = 5 The general consensus among math people is that ...
2
votes
1answer
651 views

Conventions for function notation

Hey there world, I'm in year 11 right now and I just had a brief discussion with my maths teacher about function notation, specifically, how to write the result of a function squared, as apposed to ...
1
vote
1answer
238 views

The meaning of a notation from complex analysis

I have read the book Function Theory of Several Complex Variables of Krantz.But there is a notation I don't know what's it meaning. Can somebody give me a definition.Thank you. The notation is ...
44
votes
9answers
4k views

Why is 'abuse of notation' tolerated?

I've personally tripped up on a few concepts that came down to an abuse of notation, and I've read of plenty more on stack exchange. It seems to all be forgiven with a wave of the hand. Why do we ...
23
votes
2answers
438 views

What is the Coxeter diagram for?

I understand that Coxeter diagrams are supposed to communicate something about the structure of symmetry groups of polyhedra, but I am baffled about what that something is, or why the Coxeter ...
19
votes
12answers
4k views

How to represent the floor function using mathematical notation?

I'm curious as to how the floor function can be defined using mathematical notation. What I mean by this, is, instead of a word-based explanation (i.e. "The closest integer that is not greater than ...
20
votes
6answers
597 views

The formalism behind integration by substitution

When you are doing an integration by substitution you do the following working. $$\begin{align*} u&=f(x)\\ \Rightarrow\frac{du}{dx}&=f^{\prime}(x)\\ \Rightarrow ...
26
votes
5answers
2k views

Who invented $\vee$ and $\wedge$, $\forall$ and $\exists$?

I can rather easily imagine that some mathematician/logician had the idea to symbolize "it E xists" by $\exists$ - a reversed E - and after that some other (imitative) mathematician/logician had the ...