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.

Answer

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\quadMaxwell's equations

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)

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