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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136
votes
22answers
7k 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. ...
72
votes
24answers
6k 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 ...
58
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 ...
56
votes
6answers
2k views

How Do You Actually Do Your Mathematics?

Better yet, what I'm asking is how do you actually write your mathematics? I think I need to give brief background: Through most of my childhood, I'd considered myself pretty good at math, up ...
41
votes
7answers
3k 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 ...
40
votes
14answers
2k views

Notations that are mnemonic outside of English

Some conventional math notations seem arbitrary to English speakers but were mnemonic to non-English speakers who started them. To give a simple example, Z is the symbol for integers because of the ...
40
votes
9answers
1k views

Have there been efforts to introduce non Greek or Latin alphabets into mathematics?

As a physics student, often I find when doing blackboard problems, the lecturer will struggle to find a good variable name for a variable e.g. "Oh, I cannot use B for this matrix, that's the magnetic ...
25
votes
13answers
3k views

Is it possible to have multiple decimal points in a number?

Is it ever possible to have multiple decimal points in a number? If so, how? For example is the value 1.1.2 possible? This is a question posed purely out of curiosity.
24
votes
9answers
2k 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 ...
23
votes
2answers
1k 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 ...
22
votes
4answers
1k 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 ...
20
votes
7answers
897 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 ...
18
votes
8answers
3k views

How did the notation “ln” for “log base e” become so pervasive?

Wikipedia sez: The natural logarithm of $x$ is often written "$\ln(x)$", instead of $\log_e(x)$ especially in disciplines where it isn't written "$\log(x)$". However, some mathematicians ...
17
votes
12answers
3k 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 ...
16
votes
4answers
2k views

Is there a way to denote the calculation $1+2+3+\dots+n$? [duplicate]

Since $n!$ represents $$1\cdot2\cdot3\cdots n,$$ I am wondering if there is a way to represent $$1+2+3+\dots+n?$$ What are some usual notations for the computation of some common sequences? Any other ...
16
votes
3answers
576 views

How to answer a student objection to the use of “of” in pronouncing f(x)?

Once upon a time in elementary school, a student learned how to translate certain English words into math. For example, 'and' usually means 'plus' such as "If John has 3 oranges AND 5 apples, how ...
16
votes
2answers
8k views

Weird E letter? (sigma) [duplicate]

Possible Duplicate: What does the math notation $\sum$ mean? My school's prescribed book uses the weird letter E character without explaining what it is in the first chapter when it talks ...
16
votes
6answers
405 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 ...
15
votes
3answers
1k views

When log is written without a base, is the equation normally referring to log base 10 or natural log?

For example, this question presents the equation $$\omega(n) < \frac{\log n}{\log \log n} + 1.4573 \frac{\log n}{(\log \log n)^{2}},$$ but I'm not entirely sure if this is referring to log base ...
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
2answers
489 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
6answers
1k 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 ...
14
votes
4answers
643 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
7answers
3k views

How to write \mathcal letters by hand? [closed]

I have a formula in a paper that I want to write out by hand, and it contains two "D"s, a normal D, $D$ in latex, and a 'mathcal', caligraphic D, $\mathcal{D}$ in latex. What are some ...
14
votes
11answers
931 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 ...
14
votes
4answers
683 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 ...
14
votes
5answers
13k 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". ...
14
votes
4answers
511 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
3answers
237 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
2answers
630 views

The Gamma function and the Pi function

I have been studying differential equation, in particular special functions. Euler's Gamma function, and Gauss's Pi function are essentially the same, differing only by an offset of one unit. for ...
13
votes
5answers
812 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 ...
13
votes
8answers
833 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
7answers
735 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
577 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
42k 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 ...
13
votes
5answers
594 views

Notation Question: What does $\vdash$ mean in logic?

In a "math structures" class at the community college I'm attending (uses the book Discrete Math by Epp, and is basically a discrete math "light" edition), we've been covering some basic logic. I've ...
13
votes
6answers
3k 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 ...
13
votes
4answers
3k views

What is the meaning of this symbol ($\models$)?

What's the meaning of this symbol in mathematical notation? : $\models$
13
votes
4answers
1k views

Contradiction! Any Symbol for?

Perhaps, this question has been answered ago but I don't aware of its existing answer. Is there any international icon or symbol for showing Contradiction or reacing a contradiction in Mathematical ...
13
votes
2answers
827 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
311 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
3answers
240 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 $$A+B = \{a+b\mid a \in A, b \in B\},\quad -A = \{-a\mid a \in A\}$$ ...
12
votes
7answers
673 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
3answers
1k 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 ...
12
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 ...
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.
12
votes
9answers
579 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
2answers
895 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
1answer
367 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
374 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 ...