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
18k 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 ...
16
votes
7answers
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 ...
19
votes
2answers
692 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 ...
11
votes
1answer
773 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 ...
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 ...
152
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
7answers
825 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. ...
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. ...
12
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?)
11
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 ...
7
votes
2answers
407 views

Different standards for writing down logical quantifiers in a formal way

What are standard ways to write mathematical expressions involving quantifiers in a (semi)formal way ? In different posts of mine concerning similar question I have encountered for a generic ...
29
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.
16
votes
3answers
73k 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 ...
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 ...
5
votes
2answers
1k 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) = ...
9
votes
2answers
749 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 ...
5
votes
3answers
165 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 ...
4
votes
2answers
364 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 ...
67
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 ...
17
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
2k 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
626 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 ...
6
votes
5answers
683 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 ...
2
votes
3answers
510 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
22k 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
3answers
246 views

Why do statisticians like “$n-1$” instead of “$n$”?

Does anyone have an intuitive explanation (no formulas, just words! :D) about the "$n-1$" instead of "$n$" in the unbiased variance estimator $$S_n^2 = \dfrac{\sum\limits_{i = 1}^n ...
5
votes
4answers
901 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
87 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 ...
2
votes
3answers
251 views

Question about set notation

In the following question here the notation $c\in ]a,b[$ is used. What does this mean? I have never seen it before.
2
votes
2answers
663 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
851 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 ...
13
votes
3answers
13k 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 ...
6
votes
2answers
955 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 ...
3
votes
4answers
392 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
516 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
264 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
253 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 ...
4
votes
2answers
216 views

Why do some people place the differential at the beginning of their integral? [duplicate]

I've seen several times on this site people writing integrals like $ \int \! dxf(x) $. This seems confusing to me, especially in an iterated integral or next to a long integrand and it's nonstandard ...
4
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
2answers
290 views

What is the product of the empty set?

Give: $fn(S)=\prod_{x\in S}x$ what is: $fn(\emptyset)$ I can see reason that it would be defined as 1 or 0. Note: I thought about restricting the domain of $S$ but that would make the problem ...
2
votes
1answer
666 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
245 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 ...
83
votes
24answers
7k views

What are some examples of notation that really improved mathematics?

I've always felt that the concise, suggestive nature of the written language of mathematics is one of the reasons it can be so powerful. Off the top of my head I can think of a few notational ...