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Sometimes, coming up with good mathematical notation is key to understanding parts of mathematics. For example, consider the quadratic formula.

Brahmagupta formulated a version of the quadratic formula in 628 A.D., which goes as follows:

To the absolute number multiplied by four times the [coefficient of the] square, add the square of the [coefficient of the] middle term; the square root of the same, less the [coefficient of the] middle term, being divided by twice the [coefficient of the] square is the value.

Compare this to our modern notation:

$$ \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $$

Not only is the modern notation more succinct, it also makes certain properties of quadratic equations more apparent. For example, it's easy to determine the number of real roots of a quadratic equation using the modern formula. But Brahmagupta's formula is more opaque, and it's not immediately obvious how to find the number of real roots.

A second example is the introduction of the digit 0 and the place value system. When doing complicated arithmetic, Roman numerals are difficult to manage. But number systems with place value systems can handle addition much more intuitively.

What I am looking for is specific examples of mathematical notation, where its introduction accelerated progress in that particular field. I'm especially interested in more elementary mathematics--i.e. everything before the 1700s--but examples in more modern math are welcome too.

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    $\begingroup$ Descartes with analityc geometry and Leibniz with calculus. $\endgroup$
    – Mauro ALLEGRANZA
    Sep 27, 2017 at 9:56
  • $\begingroup$ 0 to denote blank digit $\endgroup$
    – timur
    Sep 28, 2017 at 2:35
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    $\begingroup$ Before Descartes and Leibniz, Vieta introduced proto-symbolic notation that was exploited in particular by Fermat and led to explosive development of mathematics. There are so many examples of this as to make the question too broad. $\endgroup$
    – Mikhail Katz
    Sep 28, 2017 at 8:05

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The most fundamental example the OP has already mentioned themselves: they called it 'place value system'. This is indeed a very important example. It seems to have taken humanity many millenia to realize that that makes it possible to harness the magic of the logarithm function to speed up specified tasks. Most of what you see around you is ultimately based on the 'place value system' (to use you words, which conveniently relieves me of having to pick one of the synonyms).

Another, slightly more 'theoretical' example are:

exponents in the form of Hindu-Arabic numerals

I am sure I once saw an original copy of a book of Euler in which he still writes something like $xxx$ for $x^3$. It is not that long ago that the latter was an innovation. The 'acceleration' (to use the OP's word) roughly corresponds to the advantages of Positional notation over unary number systems.

I also have to say that your question is rather broad and somehow too unfocused.

A very good advanced theoretical example, quite surprisingly not mentioned so far, is

Einstein summation convention.

for which I refer you to the following recommendable summary:

Willie Wong: Why use Einstein Summation Notation?, URL (version: 2017-04-13): https://math.stackexchange.com/q/1926173

Note that Einstein Summation Notation

  • does not only have an 'acceleration' effect

but also

  • has one of those 'limit the language' effects.
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Variables; function notation; partial derivatives; vector calculus; exterior calculus; matrices; tensors; operational calculus; delta function; set operations; equivalence relations; mapping arrows; exact sequences; commutative diagrams; Feynman diagrams; abstract index notation; $\infty$; $i$; ...too broad!

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  • $\begingroup$ Einstein notation. $\endgroup$
    – Keith McClary
    Nov 6, 2017 at 4:55
  • $\begingroup$ @KeithMcClary If you mean the summation convention, that was already mentioned in P. Heinig’s answer. $\endgroup$
    – Francois Ziegler
    Mar 16, 2018 at 17:47
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The classic example is Diophantus of Alexandria (ca. 250 AD), the first mathematician to use something that can reasonably be called algebraic notation. His work famously influenced Fermat, a millennium and a half later.

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The introduction (in Europe) of decimal fractions by Simon Stevin in 1585 and the simplification in its notation introduced by John Neper a few decades later were very important, because sudenly a much larger number of people were able to do numerical computations. This had a large impact in scientific research.

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