# What are some examples of notation that really improved mathematics? [closed]

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 conventions that simplify problem statements and ideas, (these are all almost ubiquitous today):

• $\binom{n}{k}$
• $\left \lfloor x \right \rfloor$ and $\left \lceil x \right \rceil$
• $\sum f(n)$
• $\int f(x) dx$
• $[P] = \begin{cases} 1 & \text{if } P \text{ is true;} \\ 0 & \text{otherwise} \end{cases}$

The last one being the Iverson Bracket. A motivating example for the use of this notation can be found here.

What are some other examples of notation that really improved mathematics over the years? Maybe also it is appropriate to ask what notational issues exist in mathematics today?

EDIT (11/7/13 4:35 PM): Just thought of this now, but the introduction of the Cartesian Coordinate System for plotting functions was a HUGE improvement! I don't think this is outside the bounds of my original question and note that I am considering the actual graphical object here and not the use of $(x,y)$ to denote a point in the plane.

• Positional notation to write numbers.
– OR.
Commented Nov 7, 2013 at 18:21
• As this doesn't seem like it would have a right answer, I am flagging to make the question community wiki Commented Nov 7, 2013 at 18:28
• $x^2+ax+b$ or, in general, using letters to denote numbers. Commented Nov 7, 2013 at 18:47
• @voromax Hmm, you're not from here, you come from Stack Overflow like me ;) I've noticed that here at Math.SE, these types of questions are encouraged. It's not the same as SO here Commented Nov 8, 2013 at 1:21
• I have to agree with ABC . . . positional notation is without a doubt the most significant advance in notation. Try simple addition with Roman numerals. Try ANYTHING with Roman Numerals, tally marks, or any other system of numeration.
– Marc
Commented Nov 8, 2013 at 2:17

Vector notation! The fact that you can write: $$\vec{a}\cdot\vec{b}$$ Instead of: $$a_1 b_1 + a_2 b_2 + \ldots$$ Or that you can use vector operators such as $\vec\nabla$, really helped develop linear algebra and its use in applied fields. To point out one famous example, Maxwell's original equations take up an entire page!

$\quad\quad\quad\quad\quad$

And here's the same using vector notation for comparison:

\boxed{\begin{align}\\\,\\\qquad&\qquad\nabla\cdot\mathbf D=\rho&&\text{(1)} \qquad\qquad\text{Gauss' law}&\\\,\\ \qquad&\qquad\nabla\cdot\mathbf B=0&&\text{(2)}\quad\text{Gauss' law for magnetism}&\\\,\\ \qquad&\,\,\,\nabla\times\mathbf E=-\dfrac{\partial\mathbf B}{\partial t}&&\text{(3)}\qquad\quad\,\text{Faraday's law}&\\\,\\ \qquad&\,\nabla\times\mathbf H=\dfrac{\partial\mathbf D}{\partial t}+\mathbf J&&\text{(4)}\qquad\text{Ampère-Maxwell law}\qquad\\\,\\ \end{align}}\\\,\\\textit{Maxwell's equations in vector form}

(Image from IEEE's history page)

• I would argue that "vector notation" isn't really special notation, just a useful definition of operators like the scalar product. In a programming language, we'd say it's just a custom-defined operator but all in the language's default syntax. Commented Nov 8, 2013 at 15:05
• vector notation also obviates the need to refer to spatial coordinates/parametrization. Commented Nov 9, 2013 at 2:55
• Maxwell actually wrote twenty equations, of which (4) is actually two equations. There are actually eight equations in vector form, (4) breaks into [math]\nabla\times H = C \quad C = J + \dot D[\math]. Maxwell actually invented something like Nabla, by writing these as the scalar and vector parts of a quarterion. Commented May 3, 2015 at 10:23
• @wendy.krieger We use a $stuff$ tag here, not a [math]stuff[\math] tag. That's $\nabla\times H = C \quad C = J + \dot D$. Commented May 3, 2015 at 12:49
• Late comment but lets not forget that when tensor notation was introduced Maxwell’s equations were simplified into just two! See here: en.m.wikipedia.org/wiki/… Commented Apr 2, 2018 at 17:18

Algebraic notation with letters instead of verbal descriptions for quantities that are not explicitly known. The transition from rhetorical mathematics to syncopated and then symbolic had a tremendous impact on mathematics, making it possible to state rules and algorithms in greater generality.

• I think that consistency in the choice of some letters for some mathematical objects has proven particularly helpful. I'm thinking about $n,\,m$ for integers, $i,\,j$ for indices, $z,\,x,\,y$ when talking about complex numbers, $p,\,q$ for prime numbers, $k$ for the base field, $\phi_i : U_i \rightarrow V_i$ for charts, and so on. Commented Nov 7, 2013 at 20:37
• For me this is by far the most important notational advance in mathematical history, whose impact on mathematics is so fundamental that it hard to express properly. Without it mathematics just couldn't have advanced much beyond its medieval state. I'll take the freedom to replace "numbers" in the first sentence though, as this is improper; apart from some rare cases like$~\pi$, letters are almost never used in place of (explicit) numbers. Commented Nov 8, 2013 at 8:42
• Agreed; this seems to me the single most important answer by far. It’s hard to conceive that modern mathematics could have developed if we were still writing out “that number, which when squared and added to five times itself yields 3” and the like. Commented Nov 8, 2013 at 15:58
• Just think how Indian mathematicians stated all of their theorems! Paragraphs! Implicit reference unrolled through winding sentences! Commented Nov 11, 2013 at 9:38

I would suggest Gauss's invention of the notation for congruences in modular arithmetic:

"The invention of [congruence notation] by Gauss affords a striking example of the advantage which may be derived from an appropriate notation, and marks an epoch in the development of the science of arithmetic."

G. B. Matthews (1861-1922)

Edit:

For anyone who is not sure what the notation actually is:

$$a \equiv b \pmod{k}$$ means that $a$ and $b$ give the same remainder on division by $k$, or equivalently, that $a-b$ is divisible by $k$.

• Absolutely. ${}{}{}{}$
– Pedro
Commented Nov 7, 2013 at 18:58
• It is hard to believe how they managed without it before then, I reckon. Commented Nov 7, 2013 at 18:59
• And yet I read somewhere that Gauss said something like, “We need new notions, not new notations.” Commented Nov 8, 2013 at 1:39
• @Lubin: "But in our opinion truths of this kind should be drawn from notions rather than from notations." - Gauss, Disquisitiones Arithmeticae, Article 76. (Of course, the original quote was in Latin.) This was a remark towards Edward Waring's Meditationes Algebraicae, another Latin work, wherein Waring states that a proof of Wilson's theorem (and other theorems of 'that sort') would be difficult because of a lack of notation to represent prime numbers.
– Reid
Commented Nov 8, 2013 at 7:03
• I agree fully that having a notation for congruences is invaluable, but I must admit that I've always found the $\equiv \pmod k$ notation itself to be cumbersome. I wish it were written $a=_k b$ or $a\equiv_k b$ or something like that. Commented Nov 10, 2013 at 1:43

Landau notation for describing the asymptotic behavior of a function:

$$O(n)\quad o(\log n)\quad \Omega(n!)\quad \Theta(2^n) \quad \dots$$

• I actually think the notation for this is not very good. For example, we say $2x^2 = O(x^2)$, but also that $2x^2 = O(x^3)$, even though $O(x^3) \neq O(x^2)$ (and whether we can say $O(x^2) = O(x^3)$ depends on what book you read). It would have been better to use =, <, and > to convey a partial-ordering, so that we could say $O(2x^2) = O(x^2) < O(x^3)$ Commented Nov 7, 2013 at 21:47
• @BlueRaja-DannyPflughoeft Maybe we should say, for instance, $2x^2 \in O(x^2)$ instead, interpreting $O(x^2)$ as a set of functions.
– lily
Commented Nov 7, 2013 at 21:50
• @BlueRaja-DannyPflughoeft: I think the most powerful way to use Landau notation is as shorthand for a generic function in the set, and combining it with other functions. E.g. $f(n) = n! + o(n^2)$. If you insist on treating $o(n^2)$ formally as a set, you'll have to replace this with something much more clunky. Commented Nov 7, 2013 at 21:58
• @BlueRaja-DannyPflughoeft I don't get what you mean. $2x^2 = O(x^2)$ is a notation abuse. The correct notation is $2x^2 \in O(x^2)$ since $O$ is a set. Hence there is absolutely nothing strange in having $2x^2 \in O(x^2) \wedge 2x^2 \in O(x^3) \wedge O(x^2) \neq O(x^3)$. Also you must use three different notations to describe the relationship between functions because we want to be able to express $f$ is at most as big as $g$, but also $f$ is at least as big as $g$ and even $f$ is "close" to $g$. Making these three cases distinct is clearer (explicit is better than implicit) Commented Nov 7, 2013 at 22:03
• @NateEldredge: $f(n) \in n! + o(n^2)$ isn't that clunky
– Max
Commented Nov 10, 2013 at 15:59

Commutative diagrams! Their use revolutionized whole areas of mathematics and probably paved the way to the discovery-invention of category theory.

Writing matrices with double subindices $$\begin{pmatrix}a_{11}&a_{12}&a_{13}&\dots\\a_{21}&a_{22}&a_{23}&\cdots\\\vdots&\vdots&\vdots&\ddots\end{pmatrix}$$

Here's a little information and images of how things were done before. It is thought Leibniz was the pioneer. In fact, you can find this in A Source Book in Mathematics:

• +1 for your invaluable link. I was just discussing the extraordinary creativity of Leibni(t)z with a friend this afternoon and that was before you taught me that the double index notation is also due to him. Voltaire was very, very wrong to mock Leibniz in Candide :-) [but the book is so well-written that one can't help forgiving him ...] Commented Nov 7, 2013 at 20:25
• Not to mention he originated the $\int$ notation for integrals ... Commented Nov 7, 2013 at 20:58
• @GeorgesElencwajg Thanks for the comment. Agreed! =)
– Pedro
Commented Nov 7, 2013 at 21:11

Einstein summation notation really helped simplify things in areas of linear algebra applied to physics and differential geometry.

For example:

$$y = \sum_{i=1}^3 c_i x^i = c_1 x^1 +c_2 x^2 + c_3 x^3$$

could be written as

$$y = c_i x^i$$

where the lower and upper indices imply "sum over $i$".

• "Really hate" would be a bit strong, but I seriously dislike it. Unlikely that I will ever win an Abel Prize though, and my chance for a Fields medal is gone, so take it just as my personal preference. Commented Nov 7, 2013 at 20:05
• A caveat: many mathematicians really hate this notation and it is actually not common at all in pure mathematics. Bourbaki, Serre, Henri Cartan certainly haven't used it and I would guess that very few Fields medalists and no Abel Prize winner ever used it. I don't know however about Henri Cartan's Daddy Elie ... (Just for clarification: I don't use that convention but it doesn't disturb me in the least to read documents which do use it) Commented Nov 7, 2013 at 20:11
• Pg 122 of "Michael Atiyah Collected Works Volume 5: Gauge Theories" The foot note reads "The Einstein summation convention is employed...". Sir Michael Atiyah (British) also collaborated with and inspired Edward Witten (American) who also wrote on Gauge theories and used the convention in countless papers. Michael Atiyah won the Abel prize in 2004, Edward Witten won the Fields medal in 1990. Commented Nov 9, 2013 at 17:59
• This is a great example of an advance in notation. Reducing clutter like the summation symbol is not trivial. A less cluttered notation allows one to think more clearly about mathematical ideas. It also saves chalk. Mathematicians avoid going to components but for some calculations it is preferable or even necessary. That is when Einstein's notation shows its power. (+1) Commented Nov 10, 2013 at 23:21
• @oen: You might find this answer interesting. Commented Jan 2, 2014 at 17:47

$$Y^X$$

This is some of the most suggestive notation I can think of (and the best notation for some object should be suggestive of the structure of that object). If $X$ and $Y$ are numbers, then we have the familiar rule $(Y^X)^Z = Y^{X\times Z}$, and this holds in more generality as well: if $X$, $Y$, and $Z$ are topological spaces, then $Y^X = \{f : X\to Y\mid f\textrm{ continuous}\}$, and $(Y^X)^Z \cong Y^{X\times Z}$. Moreover, if we are only thinking of $X$, $Y$, and $Z$ as sets, and of $Y^X$ as set-theoretic functions, we have $\left|Y^X\right| = \left|Y\right|^{\left|X\right|}$ (assuming the expressions make sense). It even works in some cases where one of the expressions doesn't make sense; we obtain the correct combinatorial interpretation of $0^0$ via this rule: $1 = \left|\emptyset^{\emptyset}\right| =" 0^0$ (one could take this as the reason that $0^0 = 1$ for discrete types of things, although there are others as well).

$$\frac{d}{dx}$$ Another piece of notation that is useful/suggestive is the Leibniz $\frac{d}{dx}$ notation. There's the chain rule $\frac{d f}{dx} = \frac{df}{dg}\cdot\frac{dg}{dx}$ (while we're not actually cancelling $dg$'s, it at least makes it easier to remember) and the rule $\frac{dy}{dx} = \frac{1}{dx/dy}$, for example. The $\frac{d}{dx}$ (and $\int$) notation can also be seen as an operator, and this also lends itself to some interesting formal calculations that turn out to give the right answer!

Of course, the suggestiveness of these notations is not their only benefit - they also are convenient/useful/space saving!

Another to add to this: functions as arrows from one space to another $$X\xrightarrow{f} Y$$ or $$f : X\to Y$$

• I think you could improve this by Commented Nov 7, 2013 at 20:25
• Another piece of notation that is useful/suggestive is the Leibniz $\frac{d}{dx}$ notation Leibniz! Him again! Commented Nov 7, 2013 at 20:29
• Haha the formal calculations document is pretty cool. Do you know if such techniques work in the general case? Commented Nov 8, 2013 at 1:22
• @fiftyeight: The arguments presented in the link actually can be made rigorous by interpreting the derivative and integral as operators and observing that they have operator norm less than $1$, so that it does make sense to use the infinite geometric series identity. However, I don't know all the details required to make the arguments precise. Commented Nov 8, 2013 at 4:13
• nice post and very cool article Commented Nov 9, 2013 at 2:44

Perhaps $\forall , \exists , \vee , \wedge , \neg , \Rightarrow , \Leftarrow , \Leftrightarrow , =$ belong in this list, together with $\in$.

• I'm not sure how to feel about sets $X$ with $\in \in X$. Commented Nov 7, 2013 at 19:05
• I've changed it; it bugged me :) Commented Nov 7, 2013 at 19:16
• That's why I write $\varepsilon \in X$ in my delta-epsilon arguments :) Commented Nov 7, 2013 at 19:30
• I enjoy the comment that these "belong in this list"... ∈
– Ray
Commented Nov 8, 2013 at 15:31
• @MikeF I think there aren't enough sets with ${\in}\in X$. :-P Commented Aug 18, 2015 at 1:06

The use of decimal notations for tenths, hundredths ...

When dealing with noninteger number early European mathematicians used sexagesimal fractions

for example Fibonacci gave the solution to the equation $x^3 + 2x^2 + 10x = 20$ as
$1^{\circ}22'07''42'''33^{IV}04^{V}40^{VI}$

From chapter 4.1 of The art of computer programming .

And in fact, this sexagesimal system is still used for minuts and seconds.

Try calculating eg a compound index using such a system!

• Fibonacci's notation nearly just gave me a seizure. Commented Nov 7, 2013 at 23:22
• Sexagesimal has the advantage of 60 being highly-composite, with 10 non-trivial divisors, compared to only 7 for 100 and only 2 for 10 itself. Fibonacci's notation is horrible, though. Something like 01;22:07:42:33:04:40 would give the same information with less clutter.
– Dan
Commented Nov 8, 2013 at 0:58
• @Dan: good points about divisors. However for your last point: Fibonnacci's notation straight-away tells it's a sexagesimal number [even many years later], whereas your alternative doesn't Commented Nov 8, 2013 at 11:18
• @OlivierDulac But if his had been used, we would know right away. I'd prefer (1.22 07 42 33 04 40)_{60}, though. (Don't know how you get spaces in mathjax, that last bit is meant to be a subscript) Commented Nov 8, 2013 at 17:42
• Stevins wrote decimal numbers using exactly the same form, but with a different second-series number: eg 1(0)6(1)1(2)8(3)3(5)4(6) for what we write 1.618034. Note he did not show 0(4). The units are primes, seconds, thirds, &c. A tenth metre is the unit 1(10) metre in this scale, we write it 1e-10 metre. Commented May 3, 2015 at 10:32

The equal sign.

The earliest known printed occurrence is Recorde's Whetstone of witte, England, 1557. It was a significant step in the direction of symbolic logic.

Recorde described it as two parallel lines of equal length; according to him nothing could be more equal than that.

• That and also $+$.
– John
Commented Nov 8, 2013 at 15:43
• How is equality expressed before that then? Commented Nov 9, 2013 at 3:55
• ...........Prose. Commented Nov 9, 2013 at 8:28

One huge innovation that flies so far under the radar that multiple other answers to this question (as well as the question itself!) have used it in one way or another without note: the notation $f(n)$ for (take your pick) expressing the value of the function $f$ at the parameter-value $n$, or applying the function/operator $f$ to the argument $n$. It's probably the single most-used notation in mathematics beyond the basic operations themselves, and I've always found it absolutely amazing that the calculus itself was invented before the standard notation for functions existed.

How about the basics: the notation of basic arithmetic expressions: $+$, $-$, $\times$, $\div$, and their precedence rules and parentheses. And of course "Arabic" numerals (including zero), as @ABC already noted in a comment.

Of course positional notation of numbers with zero.

How about Bra-Ket notation?

$\langle \phi | \psi \rangle$

$\langle \phi | A | \psi \rangle$

$| \phi \rangle \langle\psi |$

• Physicists may like this, but I'm not sure that most mathematicians find it an improvement over our usual notation for vectors and operators in Hilbert space. Commented Nov 7, 2013 at 19:59
• It's a matter of taste, of course. But for me, when I see $\sum \langle \phi | x \rangle |\phi\rangle$, my first thought is not "matrix multiplication" but rather "parse error, unbalanced delimiters". Commented Nov 7, 2013 at 20:28
• @NateEldredge Which version are you running? Commented Nov 8, 2013 at 1:58
• I love bra-ket notation. It really adds nothing conceptually to write it as $\langle \phi \mid \psi \rangle$ instead of $\langle \psi , \phi \rangle$, but dang, do I feel cool when I'm doing it. Commented Nov 10, 2013 at 1:48
• @swish, you forgot to mention the main reason to use bra-ket notation: all the great identities you get! For instance, $\langle \phi \mid (\mid \psi \rangle) = \langle \phi, \psi \rangle$, and $(\mid \phi \rangle \langle \psi \mid)^* = \mid \psi \rangle \langle \phi \mid$, and the fact that $\mid \phi \rangle \mapsto \langle \phi\mid$ is an isomorphism from $H$ to $H'$, etc. Commented Aug 7, 2014 at 3:12

The use of $\lambda$ for abstraction, as in Church's $\lambda$-calculus. It may not have improved mathematics generally speaking, but it beautifully complements the notation for application, and it simplified reasoning about currying. It certainly proved useful for computer science, at least.

• Lambda notation does make it clearer that functions are objects independent of their arguments. Syntactically $f=\lambda x. y$ is just solving for $f$ in $f(x)=y$. Commented Nov 8, 2013 at 3:14
• Related to this would the notation $\mapsto$. Commented Nov 8, 2013 at 6:43
• Actually the choice of using the symbol $\lambda$ for this is a pretty lousy one, since it is already used as a variable name (and often one of preference: eigenvalues, partitions,...) in many areas. I would argue that this choice, while harmless in the subfield where the notation emerged, has greatly hampered the spread of anonymous function notation throughout mathematics. For what it's worth, I use a notation like $v\mapsto(\alpha\mapsto \alpha(v))$ for the map $V\to V^{**}$ in linear algebra, without feeling much need to explain the notation. Commented Nov 8, 2013 at 8:33
• Originally it was not supposed to be $\lambda x. x$, but $\hat{x}.x$, as in Russell and Whitehead's Principia Mathematica. Typewriters printed this as $\hat{} x. x$, and $\hat{}$ was mistaken as a $\Lambda$, which then became $\lambda$. Commented Nov 8, 2013 at 9:20

The only one that comes to mind is the representation of zero.

Going from having to write ML or 1    5      and hoping the omission would allow the number to be understood, to 1050 as an unambiguous representation

Leibniz notation has been immensely useful in the improvement of applied mathematics. The notion of $dx$ denoting infinitesimal increments of a quantity $x$ (while not strictly consistent Newton's limit interpretation) has helped many a scientist arrive at a better intuitive understanding of the relationship between various physical quantities and facilitated modelling of physical processes and objects.

I find Euler and Bernoulli's work in applied math (which prefers Leibniz's notation) speaks for how intuitive Leibniz calculus is compared to the alternative.

How about complex and imaginary number notation: $$a + bi$$

Includes the phase plane representation of vectors $$z = a + bi = |z|\space(\cos( \theta) + i\space \sin (\theta)) = |z| \space e^{i \space \theta}$$

based on one of the more elegant statements in all mathematics, Euler's formula:

$$e^{ix} = \cos(x) + i \space \sin(x)$$

or in a more limited application, a beautiful equation relating the fundamental constants: $\pi,i,e,1,$ and $0$.

$$e^{i\pi} + 1 = 0$$

• If we are talking about improvement, we must replace $2\pi \rightarrow \tau$. It is outrageous that we are stuck with $2\pi$, especially in the context of notation-induced improvements.
– Val
Commented Nov 8, 2013 at 13:07
• Tau is used for a LOT of symbols, it's already stretched extremely thin even in the context of frequency analysis. They should just come up with a new symbol altogether. Commented Dec 29, 2013 at 5:57
• @Val I'm not an enthusiast of changing $\pi$ to anything else, the pros of switching to a new symbol aren't that great to be honest. Commented Aug 25, 2016 at 16:32
• @DanielV 1. Pi also has a lot of other meanings. 2. Only a few of the other meanings of $\tau$ don't have any other notations and would be used in the same context as $\tau=2\pi$. 3. $\pi$ is actually sometimes used to mean different things at the same time., and so is $e$. Commented Sep 24, 2019 at 12:58
• Commented Sep 24, 2019 at 12:59

So - what about fraction notation? Using this:

$$\frac{a+b}{c+d}$$

$$(a+b)\div(c+d)$$

And to some extent anything else that trades vertical space for horizontal compactness, e.g.:

$$\sum^{10}_{x=1}x^2$$

instead of (for example) $\mathrm{sum}(x,1,10,x\uparrow 2)$

The arrow notation invented by Knuth for representing large numbers (i.e. so large that scientific notation isn't practical).

• When'd one use those, though? The universe is estimated to contain only 1e82 particles. Commented Nov 12, 2013 at 14:11
• In pure mathematics, analyses and proofs will often require far larger numbers than are tied to any physical reality. See this article about Graham's Number.
– JDM
Commented Jun 17, 2015 at 18:41

The mutli-index notation is used to write tuples and express sums, powers and products of these tuples. The $n$ dimensional multi-index is a $n$-tuple $$\alpha = (\alpha_1, \alpha_2, \alpha_3, \ldots, \alpha_n)$$ with each $\alpha_i > 0$

For multi-indices $\alpha, \beta$ we have

• Component wise summation $$\alpha \pm \beta = (\alpha_1 \pm \beta_1, \alpha_2 \pm \beta_2, \ldots, \alpha_n \pm \beta_n)$$
• For $x = (x_1, x_2, \ldots, x_n) \in \mathbb{R}$, we have $$x^\alpha = x_1^{\alpha_1}x_2^{\alpha_2}\ldots x_n^{\alpha_n}$$
• Partial derivatives $$\partial^\alpha = \partial_1^{\alpha_1}\partial_2^{\alpha_2}\ldots \partial_n^{\alpha_n}$$ where each $\partial_i^{\alpha_i} = \partial^{\alpha_i}/\partial x_i ^{\alpha_i}$.

This notation helps in topics such as functional analysis and pseudo differential operators. Look at the Wikipedia article for more examples

• I've edited my answer. Commented Nov 8, 2013 at 6:38
• Actually, the idea of using indices for a whole bunch of different values instead of letters of some alphabet is a groundbreaking revolution in my opinion. Commented Nov 9, 2013 at 0:02

The switch from Roman to Arabic numerals has to be a fundamental improvement.

Try to calculate IL divided by VII !!

I love the exclamation mark for factorial

x! = 1 * 2 * .. * x

because of the way that bang indicates intensity in natural language. Seems quite fitting to its fast growth.

I like the French invention ∃! to designate unicity. It is nicely useful in mathematics but is not universally used. There are a lot of illustrations of this ("∃x such that f(x) is something and this x is unique" changes by "∃!x such that f(x) is something" or "Prove that ∃!x such that something". A professor of University of Toledo (Ohio) adopted this notation after I mentioned it).

Matrix notation: Ax=mx, A*=(P^-1)BP etc.., block matrices. Without it it will be very hard to express these operations.

Coxeter co-joined Stott's operator and Schläfli's description of regular polytopes into a combined notation that is used to this day.

Stott wrote the truncated icosahedron as e_1 I. Wythoff wrote it as t_0,1 I. Coxeter writes it as t_0,1 {3,5}. Krieger writes it as x3x5o. If one wants to write the figure where the inter-hexagon edge is phi, then only Krieger's notation works: f3x5o.

I greatly revamped the notation in terms of vectors and phase-space the effect is similar from going from roman notation to digital forms, and with the new notation, we describe figures well past the scope of Coxeter's Stott-Schläfli construction.

Conspicuous by its absence among the above answers is any celebration of the usual notation for multiplication.