While there is no universal agreement, the word notation can be used in several senses:

1. In its most common usage, notation refers to the typographical conventions or rules of symbol usage that are followed, e.g., within a book or article. Examples of this usage include:

• • Infix notation for representing calculations in an almost linguistic format, such as a+b-c
  • Prefix notation or Polish notation, which places the operator before the operands, such as + a b
  • Postfix notation or reverse Polish notation, which places the operator after the operands, such as a b +
  • Numeral systems, notation for writing numbers, including
    • Positional notation, which made writing numbers more systematic than a tally method like Roman numerals
    • binary notation, a positional notation in base two
    • decimal notation, a positional notation in base ten
    • scientific notation for expressing large and small numbers

1. In another sense, notation is short for notational system, meaning an interpreted system of tokens having a syntax and a semantics. The medium used by a notational system (e.g., paper, clay tablets, digital memory) defines and delimits the possible characteristics of its tokens. The tokens of a notational system designate (refer to) abstractions that are reified by the notational system. The following is a small set of examples of notational systems used in various disciplines:

• • In musical composition, the music is represented by musical notation
  • In chemistry, the Lewis notation denotes chemical bonds
  • In quantum mechanics, Dirac's Bra-ket notation is another representation of probability amplititudes
  • In relativity, for example, Tensor notation is a general way to represent a gravitational field
  • In characterising dance various forms of dance notation are used to document movement such as Labanotation, Benesh Notation , and Eshkol-Wachman notation
  • In mathematics, mathematical notation is used to represent mathematical ideas, such as:
    • Conway's chained arrow, Knuth's up-arrow notation and Steinhaus polygon notation can represent much larger quantities than scientific notation
    • Cartesian Coordinate System, for representing position and other spatial concepts in analytic geometry
    • Leibnitz notation is the standard representation for Calculus
    • Big O notation, used for example in analysis to represent less significant elements of an expression, to indicate that they will be neglected
    • Z notation - a formal notation for specifying objects using Zermelo-Fraenkel set theory and first-order predicate logic

Under its broader definition, notational systems would include speech (a way of inscribing semantically meaningful vibrations upon the medium of air); money (a combined token-and-medium that attempts to represent value); logical notation; cartography; and of course writing. While the alphabet is the most famous revolutionary notational system, there have been many notational revolutions in the development of homo sapiens sapiens.

While notational systems have had a great impact on civilization, they are not themselves studied comparatively and longitudinally as a coherent subject, but rather are considered bit layers within the disciplines they support and enable. -