What are some examples of notation that really improved mathematics?

Question

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.

Answer 1

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 $i$".

Answer 2

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

Answer 3

$$ Y^X $$

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 $$ f : X\to Y $$

Answer 4

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:

Answer 5

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 $k$.

Answer 6

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)

Answer 7

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.

Answer 8

How about Bra-Ket notation?

$\langle \phi | \psi \rangle$

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

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

Answer 9

Landau notation for describing the asymptotic behavior of a function:

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

Answer 10

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.

Answer 11

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.

Answer 12

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!

Answer 13

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

See this discussion on MO.

Answer 14

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 $0$.

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

Answer 15

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.

Answer 16

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.

Answer 17

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

Answer 18

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.

Answer 19

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

Answer 20

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

Answer 21

Of course positional notation of numbers with zero.

Answer 22

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.

Answer 23

So - what about fraction notation? Using this:

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

Instead of this:

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

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) $\mathrm{sum}(x,1,10,x\uparrow 2)$

Answer 24

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