Questions tagged [notation]

For questions about the history and development of how symbols and related objects are written.

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57 votes
3 answers
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Why are $X$ and $Y$ commonly used as mathematical placeholders?

I realize that $X$ and $Y$ are relatively popular terms when wanting to use a placeholder for an unknown English or math term. What is the origin of this term, and why was it $X$ and $Y$; why not the ...
Sweet_Cherry's user avatar
44 votes
5 answers
8k views

Writing Mathematical Symbols in 20th century

As I was reading some papers written by Schrödinger and Heisenberg back in 1920s, I noticed that the symbols they use such as the integral or summation sign or calligraphic letters are as if printed ...
Gonenc's user avatar
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44 votes
3 answers
64k views

Why is price on the vertical axis and quantity on the horizontal axis?

In most of science, it is typical to have the independent variable on the horizontal axis and the dependent variable on the vertical axis. But in economics, this is often (traditionally?) flipped ...
user avatar
41 votes
1 answer
4k views

What was Euler's motivation for introducing $i$ for $\sqrt{-1}$?

[Mauro Allegranza has answered the question of who introduced the notation $i$ (Euler, followed later by Gauss), so I have changed the title. I have also edited the question in other ways to make it ...
Michael Weiss's user avatar
32 votes
1 answer
4k views

Why is American and French notation different for open intervals (x, y) vs. ]x, y[?

The Americans and the French use a different notation for open intervals: The Americans use (x, y) while the French use ]x, y[. How did this notational divergence appear?
Franck Dernoncourt's user avatar
31 votes
1 answer
50k views

Who first defined the "equal-delta" or "delta over equal" ($\triangleq$) symbol?

The symbol $\triangleq$ is sometimes used in mathematics (and physics) for a definition. It is instantiated for instance in the Unicode Character 'DELTA EQUAL TO' (U+225C). The notation $t \triangleq ...
Laurent Duval's user avatar
29 votes
2 answers
837 views

How did Isaac Newton write the integral symbol?

Isaac Newton is known as the discoverer of the FTC (Fundamental Theorem of Calculus), so maybe he wrote the integral symbol and derivative symbol. I know he wrote the derivative symbol as $\dot y$ but ...
MIKANkankitsu's user avatar
28 votes
1 answer
2k views

Why did I learn to write the base of the logarithm differently from the rest of the world?

It only occurred to me recently, in connection with this MO posting, that the way I write the base of the logarithm is not shared by the rest of the world. I am Dutch, and I learned at school to write ...
Carlo Beenakker's user avatar
25 votes
6 answers
2k views

When was the first recorded use of subscript in mathematics to represent index?

(Disclaimer: apologies for any incorrect usage of mathematical terminology throughout this question.) In modern mathematical notation, a variable with a subscript can represent a couple of different ...
user avatar
23 votes
2 answers
2k views

When did it become understood that irrational numbers have non-repeating decimal representations?

I know that the notion of irrational number (in one form or another) goes back to the Pythagoreans, and therefore far predates the decimal system, and certainly the representation of non-integer ...
mweiss's user avatar
  • 567
21 votes
1 answer
8k views

Why did angular momentum get the letter L

Note - this question was inspired by this questions on physics.SE. Many (most) physical quantities are denoted with a single letter - latin or greek. For many, the letter chosen makes sense: $t$ for ...
Floris's user avatar
  • 758
20 votes
5 answers
1k views

Who invented the Leibnitz notation $\frac{d^2y}{dx^2}$ for the *second* derivative?

This MSE question made me wonder where the Leibnitz notation $\frac{d^2y}{dx^2}$ for the second derivative comes from. It does not arise immediately as the obvious generalization of $\frac{dy}{dx}$. ...
Federico Poloni's user avatar
20 votes
1 answer
4k views

Why is the radical symbol $\sqrt{}$ called "radical"?

This question arose in a conversation with a teacher who was introducing square roots to her students. I know from the website Earliest Uses of Symbols of Operation that the symbol $\sqrt{}$ has its ...
Joseph O'Rourke's user avatar
19 votes
2 answers
1k views

Who invented the way we write exponentiation?

Why do we write $a^n$ instead of $^n\!a$ for exponentiation? What benefit is there to writing the base before the exponent? With addition and multiplication order doesn't matter since $a + b = b + a$, ...
Frank Vel's user avatar
  • 301
19 votes
1 answer
1k views

Who first introduced the notation $\mathcal{O}$ in algebraic geometry or algebraic number theory

This is my first question for HSM. If it is consider too specialized for HSM, perhaps it can be migrated to MathOverflow. In algebraic number theory, one frequently denotes the ring of algebraic ...
Todd Trimble's user avatar
17 votes
1 answer
2k views

How did mathematicians notate the empty set before $\varnothing$?

Recently, I learned that $\emptyset$ or $\varnothing$ is a relatively new notation for the empty set and was created in 1939. I know $\{\}$ is also used along with $\{\cdot\}$ to denote empty sets. ...
quiet's user avatar
  • 273
17 votes
2 answers
834 views

Is the prime notation for derivatives $f'$ due to Euler?

Cajori, the website on Earliest Uses of Symbols of Calculus and many other sources claim that Lagrange introduced the notation $f'(x)$ for the derivative of $f(x)$ with respect to $x$. But I see Euler ...
Michael Bächtold's user avatar
16 votes
2 answers
13k views

What is the origin of polynomials and notation for them?

This may be quite a broad question, but lately I've been wondering about the history behind polynomials. Nowadays these are pretty much the simplest kind of functions to work with, but I'd like to ...
hjhjhj57's user avatar
  • 1,142
15 votes
2 answers
2k views

Why is the Digamma function always denoted with the letter "psi"?

My question is on the notation of the Digamma function. The Factorial function $n!$ (which is met in secondary school), is conceptually seminal to the Digamma function. The Factorial function is ...
Elements In Space's user avatar
15 votes
1 answer
396 views

First use of "Spur" (trace) for linear maps / matrices

Every student of linear algebra learns about the trace of a linear map. Its easiest (albeit not most conceptual) definition is: write the map as matrix, then the trace is the sum of the diagonal ...
Torsten Schoeneberg's user avatar
14 votes
2 answers
2k views

History of italicising variables and mathematical formatting in general

In the modern day, especially with the advent of $\mathrm\TeX$, it is common practice to italicise variables. This can be seen as far back as Hann, J. (1850). Examples on the Integral Calculus. What's ...
Shaikh Ammar's user avatar
14 votes
5 answers
1k views

Why is the notation $\sin^{-1} x$ so common?

$\sin^{-1} x$ means the inverse of $\sin x$ (it is also often called $\arcsin x$), but it can fairly easily be confused with $\sin(x)^{-1}$. Why is it used, when $\arcsin x$ is easier to type and is ...
The Guy with The Hat's user avatar
14 votes
2 answers
5k views

Who invented the divisibility symbol and why is it backwards?

When we want to perform division, we write e.g. $8/2$ (this is what we already learn at school). But when we want to express that $2$ is a divisor of $8$, we write: $2\mid 8$. What the heck?? I do ...
SearchSpace's user avatar
14 votes
1 answer
15k views

Who invented short and long division?

I am curious who came up with algorithms that we use today to manually solve mathematical division problems, such as short or long division; how were they established or standardized that way and why?...
Rok's user avatar
  • 243
14 votes
2 answers
1k views

Where did John Wallis get the idea for $\infty$?

I read in an offhand comment in Amir Alexander's 2014 book Infinitesimal (p.280), that John Wallis introduced the symbol $\infty$ for infinity. Was there any logic, reason, or precedent for this ...
Joseph O'Rourke's user avatar
14 votes
1 answer
2k views

When was the vector notation in physics and other sciences first introduced?

The vector notation in physics is a very compact and easy way to write things down, and according to Feynman it also saves print. When exactly did scientists realize that they were summarizing things ...
Gonenc's user avatar
  • 775
13 votes
2 answers
22k views

How long has the order of priority of arithmetical operations been widely taught in high schools?

Browsing Facebook, I often come across posts like this, to test peoples' understanding of order of operations. This inevitably prompts a deluge of answers that either misunderstand the concept or ...
Joel Roberts's user avatar
13 votes
3 answers
639 views

Why did Cantor (and others) use $\mathfrak{c}$ for the continuum?

Kontinuum is German for continuum, but Cantor used $\mathfrak{c}$. Revision. J.W.Perry questions whether or not Cantor ever in fact used the symbol $\mathfrak{c}$. I must admit I just assumed that he ...
Joseph O'Rourke's user avatar
12 votes
5 answers
4k views

What is the origin of the $\hbar$ symbol?

Equations involving Planck's constant, $h ,$ are often simplified by instead writing them in terms of the reduced Planck's constant, $\hbar \equiv \frac{h}{2 \pi}.$ But where did the symbol for the ...
Nat's user avatar
  • 459
12 votes
2 answers
2k views

Is the symbol for set membership $\in$ derived from greek letter $\epsilon$?

Title self explains: Is the symbol for set membership $\in$ derived from greek letter $\epsilon$? What is their historical relationship? Obviously the letter must be older, since greek alphabet is ...
Santropedro's user avatar
12 votes
4 answers
2k views

Why is the letter $\vec{r}$ used for position?

I'm sorry if this is a dumb question but I've never heard a convincing explanation for why seemingly all of physics names the position vector "$\vec{r}$". I've tried translating it into just about ...
Jules Randolph's user avatar
12 votes
1 answer
1k views

What was the motivation for the choice of the subset symbol?

I gather that the symbols $\subset$ and $\supset$ were introduced by Ernst Schröder in his 1890 Vorlesungen über die Algebra der Logik. This account also appears—attributed to good old Cajori—in an ...
Paul Tanenbaum's user avatar
12 votes
3 answers
2k views

Was English mathematics behind Europe by many years because of Newton's notation?

Below are several quotes suggesting that Newton's notation had the effect of retarding English mathematics by 50 years, 100 years, or even centuries. Here is my simplistic two-sentence historical ...
user avatar
11 votes
3 answers
2k views

When was the function arrow notation $x \mapsto y$ first used?

The notation $x\mapsto \sin x $ and its meaning are well-known to most mathematicians. Less well-known seems to be the fact that $x \mapsto y$ means the same as Church's $\lambda x.y$ and Frege's $\...
Michael Bächtold's user avatar
11 votes
3 answers
908 views

$\frac{dy}{dx}$ versus $\frac{{\mathrm d}y}{{\mathrm d}x}$

When I first learned calculus a few decades ago, the books I read used italicized letter "d"s in derivatives (like this: $\frac{dy}{dx}$). But a few years ago, I started seeing upright "d"s (like ...
JRN's user avatar
  • 836
11 votes
5 answers
3k views

Who gets credit for the real numbers?

If Simon Stevin already pioneered the unending decimal representation for every number (rational, surd, etc.) at the end of the 16th century, why do Cantor and Dedekind (who certainly gave a more ...
Mikhail Katz's user avatar
  • 5,496
11 votes
2 answers
321 views

Who introduced the divisibility symbol $a\vert b$ ("$a$ divides $b$") and when?

I have just stumbled across this post and became curious about the same question, namely the part regarding the origin/history of the vertical bar symbol $a\vert b$ that we use to denote "a ...
BigbearZzz's user avatar
11 votes
1 answer
519 views

When and why did $\frac{dy}{dx}$ become $\frac{d}{dx}y$?

It's obvious for us, that $\frac{dy}{dx}$ can also be written as $\frac{d}{dx}y$, but skimming through Leibniz or Eulers writings I couldn't see them write the latter. I speculate that this change ...
Michael Bächtold's user avatar
11 votes
0 answers
264 views

Origin of the special Finnish notation for difference of antiderivative

Apologies for a question that is specific to one country (but perhaps others find it a curious example of how mathematical notation can vary between countries). In Finnish calculus texts, if $F$ is an ...
Jukka Kohonen's user avatar
10 votes
2 answers
10k views

Why was delta ($\Delta$) chosen to represent change of a quantity?

In many fields, it's common for $\Delta$ (the Greek letter delta) to represent a change or difference. Math uses it, physics uses it, engineering uses it, etc. Why was $\Delta$ chosen for this? I ...
Joe's user avatar
  • 687
10 votes
2 answers
3k views

Why do I , J and K in mechanics represent X , Y and Z in maths?

Why are letters $i$, $j$, and $k$ used for axes names in mechanics while letters $x$ , $y$ and $z$ are used in mathematics? Why these dimensions weren't called A, B and C or F, G and H?
Kiroloes Amir's user avatar
10 votes
4 answers
16k views

Old square bracket notation for units

As discussed in this answer https://physics.stackexchange.com/a/77691/667 there are several common conventions for the notation $[q]$ of a physical quantity $q$. However, I often see people to put ...
student's user avatar
  • 210
10 votes
2 answers
3k views

Why is the natural logarithm represented by $\ln$?

The natural logarithm is often represented by several different notations: $\log_e x$ $\log x$ (although this is also used for logarithms with a base of 10) $\ln x$ It is the third notation that has ...
HDE 226868's user avatar
  • 8,413
10 votes
1 answer
556 views

Notation from Weyl's algebraic number theory book

In Weyl's "Algebraic Theory of Numbers", which was written in 1940, there are many symbols that look handwritten, such as Fraktur (or Sütterlin, whatever you want to call it) letters for ideals. His ...
KCd's user avatar
  • 5,312
10 votes
1 answer
2k views

Origin of Q for the set of rational numbers?

It seems many sources$^1$ attribute the use of the letter "Q" to represent the rationals to the N. Bourbaki group (in the 1930's); however, the Wikipedia entry on rational numbers claims ...
Rax Adaam's user avatar
  • 474
10 votes
1 answer
4k views

How did the obelus ÷ come to stand for division?

The obelus ÷ represents division on calculator keyboards, and sometimes in elementary education. It has a long non-mathematical history starting before 200 BC. Its ...
Leopold says Reinstate Monica's user avatar
10 votes
1 answer
2k views

What is the origin of the "Japanese bracket"?

In discussions of Sobolev spaces one often sees the Japanese bracket, $$\langle x \rangle = (1+|x|^2)^{1/2},$$ as useful shorthand. I was not easily able to find information about this term. (1) What ...
mme's user avatar
  • 211
9 votes
2 answers
508 views

Who introduced the notation $y|_{x=a}$?

When a variable $y$ depends on other variables, say $y=c x^3$, one often writes $$y|_{x=2}$$ to say "$y$ when $x$ has value $2$". This might be more familiar in the context of derivatives where we ...
Michael Bächtold's user avatar
9 votes
1 answer
189 views

History of greater-than symbol used in reverse?

I was surprised to find that Oliver Byrne's 1847 marvelous The Elements of Euclid (color version)1 uses $\sqsubset$ to mean "greater than" and $\sqsupset$ to mean "less than," in ...
Joseph O'Rourke's user avatar
9 votes
1 answer
3k views

Who really invented the integration symbol?

Most of the sources online say that Leibniz invented the sign. There's also this answer on this site which says so. That is fine. But recently when I was watching Cosmos, I noticed this: See the ...
Yashbhatt's user avatar
  • 193

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