Mathematical notation Chris Sangwin School of Mathematics University of Birmingham January 2013 Chris Sangwin (Birmingham) Mathematical notation January 2013 1 / 46 This talk is based on Chapter 5 of Sangwin, C. J., Computer Aided Assessment of Mathematics, Oxford University Press, 2013, ISBN: 978-0-19-966035-3. Chris Sangwin (Birmingham) Mathematical notation January 2013 2 / 46 Introduction This talk is about mathematical notation. Joseph Phillip’s copybook 1858 (age 10) Chris Sangwin (Birmingham) Mathematical notation January 2013 3 / 46 Background: assessing answers My motivation is automatic online assessment. In order to assess a student’s answer we need them to enter it. Mathematical notation, Meaning of expressions, Establishing properties. Chris Sangwin (Birmingham) Mathematical notation January 2013 4 / 46 Background: assessing answers My motivation is automatic online assessment. In order to assess a student’s answer we need them to enter it. Mathematical notation, Meaning of expressions, Establishing properties. Chris Sangwin (Birmingham) Mathematical notation January 2013 4 / 46 Chris Sangwin (Birmingham) Mathematical notation January 2013 5 / 46 Importance of notation Examples of the power of a well contrived notation to condense into small space, a meaning which would in ordinary language require several lines or even pages, can hardly have escaped the notice of most of my readers. The advantage of selecting in our signs, those which have some resemblance to, or which from some circumstance are associated in the mind with the thing signified, has scarcely been stated with sufficient force... C. Babbage, (1827) Chris Sangwin (Birmingham) Mathematical notation January 2013 6 / 46 Importance of notation Examples of the power of a well contrived notation to condense into small space, a meaning which would in ordinary language require several lines or even pages, can hardly have escaped the notice of most of my readers. The advantage of selecting in our signs, those which have some resemblance to, or which from some circumstance are associated in the mind with the thing signified, has scarcely been stated with sufficient force... C. Babbage, (1827) Chris Sangwin (Birmingham) Mathematical notation January 2013 7 / 46 Themes This quotation points to 1 communicative power of a good notation 2 the ability to aid strict formal calculation 3 recognition, meaning, analogy and intuition. Chris Sangwin (Birmingham) Mathematical notation January 2013 8 / 46 Themes This quotation points to 1 communicative power of a good notation 2 the ability to aid strict formal calculation 3 recognition, meaning, analogy and intuition. Chris Sangwin (Birmingham) Mathematical notation January 2013 8 / 46 Themes This quotation points to 1 communicative power of a good notation 2 the ability to aid strict formal calculation 3 recognition, meaning, analogy and intuition. Chris Sangwin (Birmingham) Mathematical notation January 2013 8 / 46 Conventions and human recognition Writing mathematical expressions in standard forms enables recognition. x2 + 3x + 1 not x + x2 + x + 1 + x. sin(t)2 − 1 as “the difference of two squares”. Unification... (Computers are not very good at recognition...) Chris Sangwin (Birmingham) Mathematical notation January 2013 9 / 46 Conventions and human recognition Writing mathematical expressions in standard forms enables recognition. x2 + 3x + 1 not x + x2 + x + 1 + x. sin(t)2 − 1 as “the difference of two squares”. Unification... (Computers are not very good at recognition...) Chris Sangwin (Birmingham) Mathematical notation January 2013 9 / 46 Conventions and human recognition Writing mathematical expressions in standard forms enables recognition. x2 + 3x + 1 not x + x2 + x + 1 + x. sin(t)2 − 1 as “the difference of two squares”. Unification... (Computers are not very good at recognition...) Chris Sangwin (Birmingham) Mathematical notation January 2013 9 / 46 Conventions and human recognition Writing mathematical expressions in standard forms enables recognition. x2 + 3x + 1 not x + x2 + x + 1 + x. sin(t)2 − 1 as “the difference of two squares”. Unification... (Computers are not very good at recognition...) Chris Sangwin (Birmingham) Mathematical notation January 2013 9 / 46 Conventions and human recognition Writing mathematical expressions in standard forms enables recognition. x2 + 3x + 1 not x + x2 + x + 1 + x. sin(t)2 − 1 as “the difference of two squares”. Unification... (Computers are not very good at recognition...) Chris Sangwin (Birmingham) Mathematical notation January 2013 9 / 46 Human recognition... E.g. the complex logarithm is a multivalued inverse of w ! ew : Lambert W function is the multivalued inverse of the function w ! wew : Chris Sangwin (Birmingham) Mathematical notation January 2013 10 / 46 Human recognition... E.g. the complex logarithm is a multivalued inverse of w ! ew : Lambert W function is the multivalued inverse of the function w ! wew : Chris Sangwin (Birmingham) Mathematical notation January 2013 10 / 46 Human recognition... E.g. the complex logarithm is a multivalued inverse of w ! ew : Lambert W function is the multivalued inverse of the function w ! wew : Chris Sangwin (Birmingham) Mathematical notation January 2013 10 / 46 Johann Heinrich Lambert (1728–1777) Corless 1996 concludes Names are important. The Lambert W function has been widely used in many fields, but because of differing notation and the absence of a standard name, awareness of the function was not as high as it should have been. Chris Sangwin (Birmingham) Mathematical notation January 2013 11 / 46 Johann Heinrich Lambert (1728–1777) Corless 1996 concludes Names are important. The Lambert W function has been widely used in many fields, but because of differing notation and the absence of a standard name, awareness of the function was not as high as it should have been. Chris Sangwin (Birmingham) Mathematical notation January 2013 11 / 46 However.... Symboles are poor unhandsome (though necessary) scaffolds of demonstration; and ought no more to appear in publique, then the most deformed necessary business which you do in your chambers. T. Hobbes, (1656) Chris Sangwin (Birmingham) Mathematical notation January 2013 12 / 46 Early history One motivations for algebraic symbolism is abbreviation. R. Recorde, (1557), The Whetstone of Witte ...to avoid the tedious repetition of these words: “is equal to”, I will set (as I do often in work use) a pair of parallels of one length (thus =), because no two things can be more equal. Chris Sangwin (Birmingham) Mathematical notation January 2013 13 / 46 Early history One motivations for algebraic symbolism is abbreviation. R. Recorde, (1557), The Whetstone of Witte ...to avoid the tedious repetition of these words: “is equal to”, I will set (as I do often in work use) a pair of parallels of one length (thus =), because no two things can be more equal. Chris Sangwin (Birmingham) Mathematical notation January 2013 13 / 46 Cossic arts The initial letters of words were used as abbreviations. “unknown” in Latin causa, in Italian cosa, in German coss. co as the “unknown”. Chris Sangwin (Birmingham) Mathematical notation January 2013 14 / 46 William Oughtred (1573–1660) Clavis Mathematicae (Key to Mathematics) Written 1628, Published London in 1631, English edition 1647. Last edition 1693. Chris Sangwin (Birmingham) Mathematical notation January 2013 15 / 46 Oughtred’s notation Introduced the × symbol for multiplication. (The symbol actually appears in 1618 in an anonymous appendix to Edward Wright’s translation of John Napier’s Mirifici Logarithmorum Canonis Descriptio (Cajori vol. 1, page 197). However, this appendix is believed to have been written by Oughtred.) Oughtred wrote N to denote the unknown, and Q to denote the square of the unknown, and C to denote the cube of the unknown. Imagine expanding out a product! N × Q = C: How does this generalize? What about structure and patterns? Chris Sangwin (Birmingham) Mathematical notation January 2013 16 / 46 Oughtred’s notation Introduced the × symbol for multiplication. (The symbol actually appears in 1618 in an anonymous appendix to Edward Wright’s translation of John Napier’s Mirifici Logarithmorum Canonis Descriptio (Cajori vol. 1, page 197). However, this appendix is believed to have been written by Oughtred.) Oughtred wrote N to denote the unknown, and Q to denote the square of the unknown, and C to denote the cube of the unknown. Imagine expanding out a product! N × Q = C: How does this generalize? What about structure and patterns? Chris Sangwin (Birmingham) Mathematical notation January 2013 16 / 46 Oughtred’s notation Introduced the × symbol for multiplication. (The symbol actually appears in 1618 in an anonymous appendix to Edward Wright’s translation of John Napier’s Mirifici Logarithmorum Canonis Descriptio (Cajori vol. 1, page 197). However, this appendix is believed to have been written by Oughtred.) Oughtred wrote N to denote the unknown, and Q to denote the square of the unknown, and C to denote the cube of the unknown. Imagine expanding out a product! N × Q = C: How does this generalize? What about structure and patterns? Chris Sangwin (Birmingham) Mathematical notation January 2013 16 / 46 Oughtred’s notation Introduced the × symbol for multiplication. (The symbol actually appears in 1618 in an anonymous appendix to Edward Wright’s translation of John Napier’s Mirifici Logarithmorum Canonis Descriptio (Cajori vol. 1, page 197). However, this appendix is believed to have been written by Oughtred.) Oughtred wrote N to denote the