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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16
votes
3answers
11k views

Symbol for elementwise multiplication of vectors

This is a notation question. Assume one is given two vector $\mathbf{a}$ and $\mathbf{b}$, and one constructs a third vector $\mathbf{c}$ whose elements are given by $$c_k=a_k b_k$$ Is there any ...
16
votes
5answers
721 views

What are reasons why some symbols in mathematical logic are not standardized?

Why is so hard to find a standardisation regarding symbolism and/or terminology in Mathematical Logic ? We see again and again students asking if e.g. $\rightarrow$ and $\implies$ means the same ...
16
votes
1answer
279 views

How and why did Weierstrass $\wp$ get its special symbol?

I kind of always hated drawing the Weierstrass $\wp$ symbol by hand, and it struck me as odd how and why it achieved its special status in the first place. After all, there are tons of other important ...
16
votes
0answers
255 views

Why is $J$ often used to denote $\mathbb{N}$ or $\mathbb{Z}$ in older texts?

In older books, I've noticed that authors tended to use $J$ to denote (usually) the natural numbers and (less commonly) the integers. Does anyone have any idea why that might've been? A few examples ...
15
votes
9answers
1k views

Is there an interval notation for complex numbers?

Just as $$\{x \in \mathbb{R}: a \leq x \leq b\}$$ can be written in the more-compact form $[a,b],$ is there an analogous notation for $$\{z \in \mathbb{C}:z=x+yi, x \in[a,b], y \in[c,d]\} \quad ?$$ ...
15
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 ...
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 ...
15
votes
3answers
15k 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 $\mathbb R - \mathbb Q$?
15
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. ...
15
votes
4answers
958 views

Donald Knuth's summation notation confuses me.

I do not understand a lot of cases of Knuth's summation notation in Concrete Mathematics. In general, I cannot seem to get a grasp on the commutative law as applied to manipulating sums. The ...
15
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.
15
votes
3answers
517 views

Mathematical Notation and its importance

You can see how mathematical notation evolved during the last centuries here. I think everyone here knows that a bad notation can change an otherwise elementar problem into a difficult problem. Just ...
14
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 ...
14
votes
2answers
2k 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. ...
14
votes
4answers
923 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
9k 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
1answer
876 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
6k 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.
14
votes
4answers
655 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
4answers
1k views

What is the symbol for primes?

Although there isn't much difference between $\mathbb{Z},\mathbb{N},\mathbb{I}$, they are well-known, and each one gets its own distinguished symbol. Is there any reason that primes don't get their ...
14
votes
5answers
2k 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?)
14
votes
2answers
398 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 ...
14
votes
1answer
6k 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, ...
14
votes
1answer
272 views

Did Einstein introduce anything new to mathematics? [duplicate]

Newton introduced calculus, so I am wondering, did Einstein introduce anything important to mathematics?
13
votes
4answers
21k 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 ...
13
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 ...
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
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 ...
13
votes
7answers
690 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 ...
13
votes
8answers
890 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

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
925 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
966 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
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.
13
votes
4answers
245 views

How to write $\aleph$ by hand

So far, I've only seen the symbol $\aleph$ in its printed form and am wondering how this symbol could be written by hand on paper or on a board (in mathematical contexts, of course). Whenever I try to ...
13
votes
2answers
24k 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 ...
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
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". ...
13
votes
1answer
15k 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
1k 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
248 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
357 views

Looking for an approach to mathematical notation wherein the universe is divided into disjoint worlds.

Is there a rigorous approach to mathematical notation wherein the "universe" is divided into disjoint "worlds," and the meaning of notation is world-dependent? This would solve a few pesky problems. ...
12
votes
4answers
1k views

Why not write $\sqrt{3}2$?

Is it just for aesthetic purposes, or is there a deeper reason why we write $2\sqrt{3}$ and not $\sqrt{3}2$?
12
votes
5answers
866 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
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 ...
12
votes
9answers
941 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
1k 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
729 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 ...