Notation

3 Key excerpts on "Notation"

• 

  eBook - ePub

  Mathematical Reasoning

  Patterns, Problems, Conjectures, and Proofs

  • Raymond Nickerson(Author)
  • 2011(Publication Date)
  • Psychology Press

    (Publisher)

  The introduction of new Notational conventions has provided significant economies of expression and greatly facilitated the performance of mathematical operations. And the Notational systems invented to represent new mathematical ideas have stimulated and made possible further advances in mathematical thinking. So central are representations to mathematics that, according to one view, “mathematics can be said to be about levels of representation, which build on one another as the mathematical ideas become more abstract” (Kilpatrick, Swafford, & Findell, 2001, p. 19). Mathematicians who have developed new areas of mathematics have often found it essential to invent new Notational schemes in order to make progress. Diophantus, Descartes, Euler, and Leibniz are all remembered for their original contributions to mathematics; each of them also introduced new Notational conventions and did so because the existing ones were not adequate to represent the thinking they wished to do. Jourdain (1913/1956) claims that Leibniz, who is remembered for numerous contributions to philosophy, science, and mathematics, attributed all his mathematical discoveries to his improvements in Notation. As discussed in Chapter 4, mathematical ideas have progressed from the more concrete to the more abstract. The emergence of new Notational conventions often has been forced by the need to represent a new level of abstraction. This progression is illustrated by the symbols 3, x, and f(x), which represent the increasingly abstract ideas of number, variable, and function. The concept three, as distinct from three stones or three sheep, is an abstraction; threeness is the property that three sheep and three stones have in common. The concept number is a further abstraction; numberness is what 3, 17, and 64 have in common

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  eBook - ePub

  Introduction to Actuarial and Financial Mathematical Methods

  • Stephen Garrett(Author)
  • 2015(Publication Date)
  • Academic Press

    (Publisher)

  Chapter 1

  Mathematical Language

  Abstract

  In this chapter, we state and illustrate the use of common mathematical Notation that will be used without further comment throughout this book. It is assumed that much of this section will have been familiar to you at some point of your education and is included as an aide-mémoire . Of course, given that the book will explore many areas of the application of mathematics, the material presented here may well prove to be incomplete. It should therefore be considered as an illustration of the level of mathematics that will be assumed as prerequisite, rather than a definitive list.

  Keywords Number systems Mathematical symbols Set Notation Interval Notation Quantifiers Equations Identities Inequalities

  Contents

  1.1  Common Mathematical Notation   3

  1.1.1  Number systems   3

  1.1.2  Mathematical symbols   6

  1.2  More Advanced Notation   8

  1.2.1  Set Notation   8

  1.2.2  Interval Notation   12

  1.2.3  Quantifiers and statements   13

  1.3  Algebraic Expressions   14

  1.3.1  Equations and identities   14

  1.3.2  An introduction to mathematics on your computer   17

  1.3.3  Inequalities   18

  1.4  Questions   20

  Prerequisite knowledge	Learning objectives
  •  “School” mathematics •  use of a calculator •  algebraic manipulation •  analytical solution of simple polynomial expressions •  Familiarity with basic use of Excel	•  Define, recognize, and use •  number systems •  mathematical Notation including set Notation •  bracket Notation •  quantifiers •  equations, identities, and inequalities

  In this chapter, we state and illustrate the use of common mathematical Notation that will be used without further comment throughout this book. It is assumed that much of this section will have been familiar to you at some point of your education and is included as an aide-mémoire

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  eBook - PDF

  Rethinking Writing

  • Roy Harris(Author)
  • 2005(Publication Date)
  • Continuum

    (Publisher)

  This is done by means of the following ten letters or figures: i [sic], 2, 3, 4, 5, 6, 7, 8, 9, 0? 2 D.E. Smith (ed.), A Source Book in Mathematics, New York, McGraw-Hill, 1929,pp.2-3. Notes on Notation 95 This 'first operation' is, in effect, the choice of a Notation, and the Notation here presented has become one of the most familiar in modern Western culture: the so-called 'Arabic numerals'. From this 'first operation' there will follow a whole range of decisions that need to be taken concerning the system(s) of mathematical writing to be based upon it. For example, if it is desired to express more than ten numbers, some convention will have to be determined for combining the individual Notational marks (i, 2, 3, etc.) into complex mathematical signs. Any such convention, however, will belong to that particular system of mathematical writing, not to the Notation as such. Many such conventions might be devised, all based on the same original Notation, and irrespective of the numerical values assigned to the basic Notational marks. For non-mathematicians, however, it is very easy to fall into confusion about what belongs to the Notation and what does not. For example, there may be a temptation to suppose that 5 always designates the number five, this value being somehow built in to the definition of that Notational unit. That temptation arises, if it does, simply because of the arithmetic value most commonly assigned to the figure 5 in the calculations of daily life. Worse still, it is not uncommon to speak of 'the number five' to refer either to the number or to the figure, as if the two were indissolubly asso-ciated. Thus National Lottery results are announced as winning 'numbers'. This and similar usages may encourage people to sup-pose that the figure 5 always has a numerical value, even when it occurs in post-codes or telephone 'numbers'.

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