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Questions tagged [notation]

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

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21 votes
1 answer
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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
3 votes
2 answers
681 views

Introduction of $\imath$ and $\jmath$ notations for the imaginary unit

The imaginary unit is generally denoted $i$ or $\imath$. I have learned that the term imaginary ("imaginaires") was coined by R. Descartes in 1637, and the "i" notation was introduced by L. Euler (cf. ...
Laurent Duval's 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
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
9 votes
4 answers
4k views

Why is calculus missing from Newton's Principia?

I'm not suggesting that Newton did not discover calculus - the question is written this way to express my surprise that the Principia does not use the methods of calculus (or 'fluxions'). He instead ...
Mozibur Ullah's user avatar
4 votes
2 answers
876 views

Cartesian coordinate system in Newton's work

In the english translation of Newton's work "Enumeratio linearum tertii ordinis" by C.R.M. Talbot, we can see in a figure the depiction of a Cartesian coordinate system pretty much as we know it today:...
skol's user avatar
  • 103
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
16 votes
2 answers
14k 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
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
  • 785
13 votes
3 answers
649 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
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
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
  • 5,962
6 votes
1 answer
384 views

Who superseded Peano's dot notation in symbolic logic and when?

Bertrand Russel gave an exhaustive treatment of creating mathematics from logic in Principia Mathematica (1910-1913), using the logical notation created by Frege and Peano. As monumental as this is, I ...
Mitch's user avatar
  • 191
6 votes
6 answers
2k views

Has a digit ever been used to represent the number "10"?

Ten is special to humans, as there are 10 fingers on two hands, and fingers are still the basic counting medium for people. So, was there any digit representing the number "10" in a positional system ...
Barun's user avatar
  • 209
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
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
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
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
6 votes
3 answers
1k views

Was the concept of zero ever developed without relation to positional number systems?

Are there any ancient civilizations which had concept of zero but didn't not use positional numerals for any somewhat non-negligible (from historical point of view) amount of time? If there are such ...
Vlad's user avatar
  • 163
6 votes
4 answers
291 views

Use of $h$ in the Newton Quotient

Why do we typically use $h$ for $$\frac{\mathrm{d}f}{\mathrm{d}x}=\lim_{h\to0}\frac{f(x+h)-f(x)}{h}$$ A student asked me this the other day. My guess was that it was originally height, because ...
charlotte's user avatar
  • 181
4 votes
1 answer
446 views

Why do we use brackets for function parameters?

I know that a function is called "function" because it's an "execution" of operations. Abbreviated notation is f. But why do we write f(x) and not ...
SearchSpace's user avatar
4 votes
2 answers
687 views

What were the criticisms against the introduction of "vector analysis"?

Frequently, 19th century physicists—e.g., Helmholtz or Maxwell—did not use modern-day vector notation, which Gibbs contributed in large part to. For example, Helmholtz in his famous paper on the ...
Geremia's user avatar
  • 5,371
2 votes
2 answers
2k views

Who wrote down minus times minus is equal to plus? [duplicate]

I am not here to ask why "minus times minus is plus", this is a basic arithmetic fact. The related question most people ask is: why does $-\times-=+$. Of, course there may be several explanations for ...
Kushal Bhuyan's user avatar
58 votes
3 answers
14k views

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
33 votes
1 answer
5k 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
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
2 answers
846 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
15 votes
2 answers
6k 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
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
11 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
  • 484
10 votes
1 answer
208 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
976 views

Is the modern "is defined as" notation from computer programming?

To my knowledge, the symbol for "is defined as" historically has been the notation, $\equiv$ or $\triangleq$. More recently, however, the notation $:=$ seems to have overtaken the latter two notations ...
Matt Hancock's user avatar
9 votes
2 answers
515 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
8 votes
1 answer
351 views

Origin / first use of $\mathbb{Z}$ (blackboard bold Z)?

I'm aware that the choice of "Z" comes from German zahlen (for "numbers"); however, I was curious to know when the dedicated font, which I believe is called "blackboard bold&...
Rax Adaam's user avatar
  • 484
8 votes
1 answer
719 views

Why is the action from the principle of least action traditionally denoted $S$?

In theoretical physics, a Hamiltonian is traditionally denoted by some variant of $H$, a Lagrangian is some variant of $L$, but why is an action always some variant of S (as opposed to, say, $A$)? ...
user avatar
8 votes
3 answers
707 views

Where did Cartan introduce his notation for basis vectors and covectors?

There is a notation used in differential geometry and general relativity in which the partial derivative operators $\partial_\mu$ are used as the basis for the space of contravariant vectors, and ...
user avatar
8 votes
2 answers
787 views

Did Euler ever write $f(x)$, with parentheses?

Euler is often credited with introducing the notation $f(x)$, and people cite the example $f(\frac{x}{a}+c)$, where he had to use parentheses around the function argument. On the other hand, when the ...
Michael Bächtold's user avatar
8 votes
3 answers
533 views

When did physicists begin using the symbol $G$ for Newton's gravitational constant?

The Cavendish experiment was equivalent to measuring $G,$ Newton's gravitational constant. However, because physicists at the time did not write equations in the same way we do now, Cavendish didn't ...
Mark Eichenlaub's user avatar
6 votes
1 answer
163 views

Why are the symbols E, F, G, L, M, and N used for the coefficients of the fundamental forms?

In differential geometry, if $e_1$ and $e_2$ are bases for a tangent space $T_pM$, then the coefficients of the first fundamental form is: $$\begin{align}E&:=\left<e_1,e_1\right>\\F&:=\...
Slate's user avatar
  • 163
6 votes
1 answer
364 views

First appearance of tensor product symbol $\otimes$

I was asked recently if the tensor product symbol $\otimes$ had been used before Bourbaki's publication on multilinear algebra in 1948 (a draft of this document can be seen at http://sites.mathdoc.fr/...
KCd's user avatar
  • 5,647
5 votes
2 answers
344 views

First appearance of the product symbol ($\Pi$)

As far as I can tell, the first occurrence of the sum notation ($\Sigma$) was in Euler's book Institutiones calculi differentialis: Quemadmodum ad differentiam denotandam usi sumus signo $\Delta$, ...
user7118's user avatar
4 votes
2 answers
1k views

Why is electric potential denoted by $\phi$?

I haven't found any explanation for it, and I'm curious.
EB97's user avatar
  • 149
4 votes
1 answer
175 views

Who changed $i$ to $j$ in electronics?

In electronics, $j$ is used for a square root of $-1$, because $I$ is current. Who introduced this and when? And was it really necessary, given that (at least now) current's symbol is capitalised? ...
J.G.'s user avatar
  • 1,720
3 votes
1 answer
1k views

First use of curly braces to denote a set?

I was wondering who was the first person to Use curly braces to represent a finite set. Exempli gratia, $\{1,2,3\}$. Use set builder notation. Such as $\{2n:n \in \mathbb{Z}\}$ to represent the even ...
yberman's user avatar
  • 173
2 votes
2 answers
2k views

Dimension of the candela unit: What does J stand for?

The J symbol can represent the unit of energy but it's also the symbol for the dimension of the candela (or luminous intensity). For the energy unit, it clearly comes from the family name of the ...
DevonDahon's user avatar
1 vote
1 answer
1k views

Why was the reduced Planck constant introduced and when?

When was the reduced planck constant $\hbar= h/2\pi$ first introduced and what was the reason behind introducing such a constant? I know that $E=\hbar \omega$ and $p=\hbar k$ and writing again and ...
Gonenc's user avatar
  • 785
0 votes
1 answer
189 views

History of points with coordinates notation

In this MathEducator StackExchange article, "Notation of points with coordinates", it's posed the question about what is the best notation for geometrical points and their coordinates: $P(3, ...
Humberto José Bortolossi's user avatar