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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14
votes
4answers
855 views

Set builder notation, left or right of :| convention

Set builder notation which specify a subset such as $Z$ or $R$ tend to put this condition on the left, whereas other conditions go on the right. $$\{ x ∈ Z : x > 0 \}$$ Why is this preferred ...
14
votes
6answers
7k 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 ...
14
votes
4answers
617 views

Solving (quadratic) equations of iterated functions, such as $f(f(x))=f(x)+x$

In this thread, the question was to find a $f: \mathbb{R} \to \mathbb{R}$ such that $$f(f(x)) = f(x) + x$$ (which was revealed in the comments to be solved by $f(x) = \varphi x$ where $\varphi$ is ...
14
votes
2answers
1k views

What does “∈” mean?

I have started seeing the "∈" symbol in math. What exactly does it mean? I have tried googling it but google takes the symbol out of the search.
14
votes
2answers
371 views

The double factorial notation

The double factorial is defined as $$n!! = \begin{cases} n \cdot (n-2) \cdot (n-4) \cdots 3 \cdot 1 = \dfrac{(n+1)!}{2^{(n+1)/2}((n+1)/2)!} & \text{ If $n \in \mathbb{Z}^+$, is odd}\\ n \cdot ...
13
votes
11answers
5k views

Dividing by 2 numbers at once, what is the answer?

Let's say i have 4/1/5. or 4 divided by 1 divided by 5. Are there any rules that i am allowed to use to stop any mistakes?, for example this has 2 solutions, 4/5 , and 20. Edit: Thanks for your ...
13
votes
8answers
866 views

Why does the logarithm require a special notation?

Since the logarithm is the reversed exponentiation, why does it need a distinct notation for it? Why can't we just ask: $$2^x=8$$ Instead of: $$\log_2 8=x$$
13
votes
3answers
2k views

What does the notation $[0,1)$ mean?

I am studying the procedure for bucket sort from Introduction To Algorithms by Cormen et al, which assumes that the input is generated by a random process that distributes the elements uniformly and ...
13
votes
3answers
1k views

Is it possible to write a number in a base of less than 1?

Following on from this question: http://math.stackexchange.com/a/217112/45127 If we take base 10 as an example, the granularity is 1. I.e. we increment the digits in an increment of 1 until we ...
13
votes
3answers
921 views

Symbol for unknown relation?

When solving equations like $$\begin{align} 4x-4 &=\frac{(2x)^2}{x} \\ -4 &= \frac{4x^2}{x} -4x \\ -4 &= 4x -4x \\[0.2em] -4 &= 0\end{align}$$ using the equality-symbol feels like ...
13
votes
7answers
856 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
6answers
822 views

Appropriate Notation: $\equiv$ versus $:=$

With respect to assignments/definitions, when is it appropriate to use $\equiv$ as in $$M \equiv \max\{b_1, b_2, \dots, b_n\}$$ which I encountered in my analysis textbook as opposed to the ...
13
votes
3answers
15k 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 ...
13
votes
3answers
3k views

What does this symbol mean? (looks like a 1 with double vertical line)

I'm studying a course on probability and statistics and at some point this symbol comes up without introduction. It looks like the number one, but slightly bigger and with a double vertical line. ...
13
votes
1answer
824 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 ...
13
votes
3answers
5k views

How did the square root get its shape?

I was wondering when in history did people start use the $\sqrt{}$ sign for square root, what did they use before, and why this curious nomenclature is adopted.
13
votes
2answers
1k views

Etymology of $\arccos$, $\arcsin$ & $\arctan$?

Does anyone know the origin of the words $\arccos$, $\arcsin$ & $\arctan$? That is to say, why are they named like this? What connects "arc" with inverse? Can't seem to find out via Google. ...
13
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?)
13
votes
1answer
14k views

What's the correct notation for log squared?

I ran across these two notations for the log function (squared), which one is more conventional. $\log^2(n)$ or $[\log(n)]^2$
13
votes
1answer
878 views

A user's guide to Penrose graphical notation?

Penrose graphical notation seems to be a convenient way to do calculations involving tensors/ multilinear functions. However the wiki page does not actually tell us how to use the notation. The ...
13
votes
3answers
246 views

Working with subsets, as opposed to elements.

Especially in algebraic contexts, we can often work with subsets, as opposed to elements. For instance, in a ring we can define $$A+B = \{a+b\mid a \in A, b \in B\},\quad -A = \{-a\mid a \in A\}$$ ...
13
votes
1answer
240 views

Did Einstein introduce anything new to mathematics? [duplicate]

Newton introduced calculus, so I am wondering, did Einstein introduce anything important to mathematics?
12
votes
8answers
3k views

How would you count a base > 36 system?

When I am counting in a base greater than ten, I can use the letters of the alphabet. What do I use when I run out of those? What comes after z?: 0, 1, 2, 3... 9, a, b, c, d... x, y, z, (?) And ...
12
votes
5answers
854 views

Should the notation $\int_{0}^{x} f(x) dx$ be frowned upon?

In old mathematics books, I see a lot of notations like $\int_{0}^{x} f(x) dx$. For example, Courant-Hilbert: Methods of mathematical physics. However, when I wrote it in this site, it was sometimes ...
12
votes
7answers
685 views

Does $\,x>0\,$ hint that $\,x\in\mathbb R\,$?

Does $x>0$ suggest that $x\in\mathbb R$? For numbers not in $\,\mathbb R\,$ (e.g. in $\mathbb C\setminus \mathbb R$), their sizes can't be compared. So can I omit "$\,x\in\mathbb R\,$" and ...
12
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 ...
12
votes
9answers
901 views

Strangest Notation? [closed]

While this may be a fruitless pursuit of anecdotes, I still ask: what is the strangest (or most blatantly wrong (at least in the eyes of common notation)) mathematical notation you have ever seen?
12
votes
4answers
2k 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.
12
votes
4answers
984 views

Why doesn't Spivak ever write $dx$ in an integral?

I've noticed that Spivak, and many other analysis books I read like Munkres, do not use $dx$ when they integrate. Why is that? This is a serious question.
12
votes
5answers
1k views

Why does notation for functions seem to be abused and ambiguous?

I really need to clear up a few things about function notation; I can't seem to grasp how to interpret it. As of right now, I know that a function is roughly a mapping between a set $X$ and a set $Y$, ...
12
votes
1answer
634 views

Using $\bigvee$ and $\bigwedge$ instead $\exists$ and $\forall$

My professor of Algebra use some "strange" notation for me. He uses $\bigvee$ instead $\exists$ and $\bigwedge$ instead $\forall$. For example $$\displaystyle\bigwedge_{x\in \mathbb{Z}}\bigwedge_{m\in ...
12
votes
3answers
459 views

Uses for esoteric integral symbols

A while ago, I was searching for a TeX package which would provide a double integral symbol with a circle which I could use to typeset some lecture notes on surface integrals. I happened upon the ...
12
votes
2answers
1k views

Why is “h” used for entropy?

Why is the letter "h" (or "H") used to denote entropy in information theory, ergodic theory, and physics (and possibly other places)? Edit: I'm looking for an explanation of the original use of "H". ...
12
votes
2answers
6k views

What is the difference between kernel and null space?

What is the difference, if any, between kernel and null space? I previously understood the kernel to be of a linear map and the null space to be of a matrix: i.e., for any linear map $f : V \to W$, ...
12
votes
3answers
705 views

Formalizing Those Readings of Leibniz Notation that Don't Appeal to Infinitesimals/Differentials

[disclaimer: I've studied a lot of logic but never been good at analysis, so that's the angle I'm coming from below] in my attempt to find a precise version of the 'definitions' usually given when ...
12
votes
0answers
5k views

What's more common? Re / Im or Fraktur-R / Fraktur-I for real / imaginary part?

Title says it all. What's more common? Is there one to prefere (maybe due to some norm)? This: $\operatorname{\mathfrak{R}} z, \operatorname{\mathfrak{I}} z$ or that: $\operatorname{Re}z, ...
11
votes
4answers
19k 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 ...
11
votes
6answers
1k views

Is there a notation for being “a finite subset of”?

I would gladly use a notation for "A is a finite subset of B", like $$A\sqsubset B \text{ or } A\underset{fin}{\subset} B,$$ but I have never seen a notation for that. Are there any? While ...
11
votes
5answers
2k views

Why does two terms immediately adjacent “mean” multiply?

I am currently teaching a GED math class. While learning about the order of operations, the students asked why does a number next to a parentheses mean multiplication? I understand the rule that two ...
11
votes
7answers
993 views

Is an empty parenthesis a valid mathematical expression? [on hold]

Is using an empty parenthesis valid? For example, $15+()=15$. What is the meaning if it is valid? I need an academic reference to validate this.
11
votes
1answer
237 views

Is the equation $\phi(\pi(\phi^\pi)) = 1$ true? And if so, how?

$\phi(\pi(\phi^\pi)) = 1$ I saw it on an expired flier for a lecture at the university. I don't know what $\phi$ is, so I tried asking Wolfram Alpha to solve $x \pi x^\pi = 1$ and it gave me a bunch ...
11
votes
3answers
771 views

History of notation: “!”

Does anyone know where the factorial "!" symbol came from? I can't decide if it is my favorite or least favorite notation in mathematics...
11
votes
4answers
418 views

Notation problem in integration: write $dx$ or ${\mathrm{d}}x$?

I have a question. When I write the integral of a generic function $f(x)$, do I have to write $$\int f(x) \color{red}dx$$ or $$\int f(x) \color{red}{\mathrm{d}}x \quad ?$$ Why? Thank you!
11
votes
2answers
12k views

Is there an accepted symbol for irrational numbers?

$\mathbb Q$ is used to represent rational numbers. $\mathbb R$ is used to represent reals. Is there a symbol or convention that represents irrationals. Possibly $\mathbb R - \mathbb Q$?
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 ...
11
votes
2answers
21k views

multiplication equivalent of the summation symbol

I was curious (even though this is a very amateur question)... what would the multiplication equivalent of sigma (the summation symbol) be? $$\sum$$ I want to do a series of multiplication of ...
11
votes
1answer
2k views

What is mathematical basis for the percent symbol (%)?

Percent means 1 part of 100 or 1/100 and is indicated with %. Per mille means 1 part of 1000 or 1/1000 and is indicated with ‰, so it seems that these symbols indicate the mathematical operations ...
11
votes
4answers
279 views

The origin of the function $f(x)$ notation

What are the historical origins of the $f(x)$ notation used for functions? That is when did people start to use this notation instead of just thinking in terms of two different variables one being ...
11
votes
1answer
191 views

Why is $e$ the Identity?

Some authors use $e$ to be the identity element of a group instead of $1$. What is the origin of this notation? Was this before or after we used $e$ to represent the base of the natural logarithm? ...
11
votes
4answers
299 views

Why are the order-of-operations conventions good?

Children are sometimes taught silly mnemonics like "PEMDAS" to remember conventions on order of operations. (I never heard of "PEMDAS" until long after graduating from college, as far as I can ...