Mathematical Notation

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

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

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)

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 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 Isaac Newton

..... learned algebra from Clavis Mathematicæ. His annotated copy survives. Asked for a recommendation on a proposed mathematics course:

And now I have told you my opinion in these things, I will give you Mr Oughtred’s, a Man whose judgment (if any man’s) may be safely relied upon. Isaac Newton, 25th May 1694.

Chris Sangwin (Birmingham) Mathematical notation January 2013 17 / 46 Isaac Newton

..... learned algebra from Clavis Mathematicæ. His annotated copy survives. Asked for a recommendation on a proposed mathematics course:

And now I have told you my opinion in these things, I will give you Mr Oughtred’s, a Man whose judgment (if any man’s) may be safely relied upon. Isaac Newton, 25th May 1694.

Chris Sangwin (Birmingham) Mathematical notation January 2013 17 / 46 Isaac Newton

..... learned algebra from Clavis Mathematicæ. His annotated copy survives. Asked for a recommendation on a proposed mathematics course:

And now I have told you my opinion in these things, I will give you Mr Oughtred’s, a Man whose judgment (if any man’s) may be safely relied upon. Isaac Newton, 25th May 1694.

Chris Sangwin (Birmingham) Mathematical notation January 2013 17 / 46 One episode from history

xn The n in xn indicates the number of terms in the product

xn := x × x × · · · × x.

The law xn × xm = xn+m. (1)

Repetition was common, e.g. Descartes preferred aa to a2.

Chris Sangwin (Birmingham) Mathematical notation January 2013 18 / 46 One episode from history

xn The n in xn indicates the number of terms in the product

xn := x × x × · · · × x.

The law xn × xm = xn+m. (1)

Repetition was common, e.g. Descartes preferred aa to a2.

Chris Sangwin (Birmingham) Mathematical notation January 2013 18 / 46 One episode from history

xn The n in xn indicates the number of terms in the product

xn := x × x × · · · × x.

The law xn × xm = xn+m. (1)

Repetition was common, e.g. Descartes preferred aa to a2.

Chris Sangwin (Birmingham) Mathematical notation January 2013 18 / 46 One episode from history

xn The n in xn indicates the number of terms in the product

xn := x × x × · · · × x.

The law xn × xm = xn+m. (1)

Repetition was common, e.g. Descartes preferred aa to a2.

Chris Sangwin (Birmingham) Mathematical notation January 2013 18 / 46 1634, Pierre Hérigone wrote 5a4. 1636, James Hume published Viete’s work using 5aiv . 1637, René Descartes’s Géométry 5a4.

Chris Sangwin (Birmingham) Mathematical notation January 2013 19 / 46 Comparison...

x3 − 6x2 + 11x = 6

Author Example Pacioli 1cu. m. 6ce. p. 11 co. equale 6ni Bombelli 1 ^3 .m. 6 ^2 p. 11 ^1 equale 6 Stevinus 1z−6y+11x: eguale 6 Viete 1C − 6Q + 11N egal 6 Harriot 1.aaa − 6.aa + 11.a = 6 Modern x3 − 6x2 + 11x = 6

The comparison of mathematical notation by Babbage (1830).

Chris Sangwin (Birmingham) Mathematical notation January 2013 20 / 46 Isaac Newton (1642–1726)

The ﬁrst conﬁdent statement on this subject.

2 3 4 Since algebraists write a , a √, a , etc.,√ for aa, aaa, aaaa, etc., 1 3 5 √ −1 −2 so I write a 2 , a 2 , a 2 , for a, a3, a5; and I write a , a , −3 1 1 1 a , etc., for a , aa , aaa , etc. m He then uses this notation in his binomial formula for (x + y) n .

Chris Sangwin (Birmingham) Mathematical notation January 2013 21 / 46 Isaac Newton (1642–1726)

The ﬁrst conﬁdent statement on this subject.

Chris Sangwin (Birmingham) Mathematical notation January 2013 21 / 46 Leonhard Euler (1707–1783)

Imaginary powers,

From these equations we understand how complex exponentials can be expressed by real sines and cosines, √ √ since e+v −1 = cos .v + − 1 sin .v and √ √ e−v −1 = cos .v − − 1 sin .v. L. Euler (1740) §138 Introduction to Analysis of the Inﬁnite

Chris Sangwin (Birmingham) Mathematical notation January 2013 22 / 46 Euler and analogy...

We may even apply this method to equations which go on to inﬁnity. The following will furnish an example:

x∞ = x∞−1 + x∞−2 + x∞−3 + x∞−4+, &c.

§799, Elements of algebra.

Chris Sangwin (Birmingham) Mathematical notation January 2013 23 / 46 Notation is powerful....

In analogy with exponentiation as ab.

The logarithm loga(b) is written ab.

E.g. natural logarithm ex .

ax Formally at least, a = aax = x. To solve ax = b, we “operate with a ↓” to obtain

x a = b, ⇒ aax = ab, ⇒ x = ab.

Which of the following are true?

ab×c = ab +ac, abc = (ab)×c, (ab)×(ba) = 1, (ab) = (ac)×(cb).

M. Brown (1974)

Chris Sangwin (Birmingham) Mathematical notation January 2013 24 / 46 Notation is powerful....

In analogy with exponentiation as ab.

The logarithm loga(b) is written ab.

E.g. natural logarithm ex .

ax Formally at least, a = aax = x. To solve ax = b, we “operate with a ↓” to obtain

x a = b, ⇒ aax = ab, ⇒ x = ab.

Which of the following are true?

ab×c = ab +ac, abc = (ab)×c, (ab)×(ba) = 1, (ab) = (ac)×(cb).

M. Brown (1974)

Chris Sangwin (Birmingham) Mathematical notation January 2013 24 / 46 Notation is powerful....

In analogy with exponentiation as ab.

The logarithm loga(b) is written ab.

E.g. natural logarithm ex .

ax Formally at least, a = aax = x. To solve ax = b, we “operate with a ↓” to obtain

x a = b, ⇒ aax = ab, ⇒ x = ab.

Which of the following are true?

ab×c = ab +ac, abc = (ab)×c, (ab)×(ba) = 1, (ab) = (ac)×(cb).

M. Brown (1974)

Chris Sangwin (Birmingham) Mathematical notation January 2013 24 / 46 Notation is powerful....

In analogy with exponentiation as ab.

The logarithm loga(b) is written ab.

E.g. natural logarithm ex .

ax Formally at least, a = aax = x. To solve ax = b, we “operate with a ↓” to obtain

x a = b, ⇒ aax = ab, ⇒ x = ab.

Which of the following are true?

ab×c = ab +ac, abc = (ab)×c, (ab)×(ba) = 1, (ab) = (ac)×(cb).

M. Brown (1974)

Chris Sangwin (Birmingham) Mathematical notation January 2013 24 / 46 Notation is powerful....

In analogy with exponentiation as ab.

The logarithm loga(b) is written ab.

E.g. natural logarithm ex .

ax Formally at least, a = aax = x. To solve ax = b, we “operate with a ↓” to obtain

x a = b, ⇒ aax = ab, ⇒ x = ab.

Which of the following are true?

ab×c = ab +ac, abc = (ab)×c, (ab)×(ba) = 1, (ab) = (ac)×(cb).

M. Brown (1974)

Chris Sangwin (Birmingham) Mathematical notation January 2013 24 / 46 Notation is powerful....

In analogy with exponentiation as ab.

The logarithm loga(b) is written ab.

E.g. natural logarithm ex .

ax Formally at least, a = aax = x. To solve ax = b, we “operate with a ↓” to obtain

x a = b, ⇒ aax = ab, ⇒ x = ab.

Which of the following are true?

ab×c = ab +ac, abc = (ab)×c, (ab)×(ba) = 1, (ab) = (ac)×(cb).

M. Brown (1974)

Chris Sangwin (Birmingham) Mathematical notation January 2013 24 / 46 Notation is powerful....

In analogy with exponentiation as ab.

The logarithm loga(b) is written ab.

E.g. natural logarithm ex .

ax Formally at least, a = aax = x. To solve ax = b, we “operate with a ↓” to obtain

x a = b, ⇒ aax = ab, ⇒ x = ab.

Which of the following are true?

ab×c = ab +ac, abc = (ab)×c, (ab)×(ba) = 1, (ab) = (ac)×(cb).

M. Brown (1974)

Chris Sangwin (Birmingham) Mathematical notation January 2013 24 / 46 Visual salience

for some students the surface features of ordinary notation provide a necessary cue to successful syntax decisions Kirshner (1989)

The quality of visual salience is easy to recognize but difﬁcult to deﬁne. Visually salient rules have a visual coherence that makes the left- and right-hand sides of the equations appear naturally related to each other. Kirshner (2004)

Some students operate only at the surface level of visual pattern matching.

Chris Sangwin (Birmingham) Mathematical notation January 2013 25 / 46 Visual salience

for some students the surface features of ordinary notation provide a necessary cue to successful syntax decisions Kirshner (1989)

Some students operate only at the surface level of visual pattern matching.

Chris Sangwin (Birmingham) Mathematical notation January 2013 25 / 46 Visual salience

for some students the surface features of ordinary notation provide a necessary cue to successful syntax decisions Kirshner (1989)

Some students operate only at the surface level of visual pattern matching.

Chris Sangwin (Birmingham) Mathematical notation January 2013 25 / 46 Common “errors"

Evaluation as juxtaposition: If x = 6, 4x → 46. Minus sign difficulties: x = −3, y = −5, xy → −8. Incorrect parsing: 2(−3) → −1, (−1)3 → −3. Identity difficulties: a 1 → 0, 0 × a → a. a √ √ √ Distribution/linearity: (a + b)n → an + bn, a + b → a + b, sin(a + b) → sin(a) + sin(b). Partial distribution: 2(x + 3) → 2x + 3, −(3a + b) → −3a + b Excessive distribution: a × (bc) → ab × ac. a a a a+b a b Fraction difficulties: b+c ↔ b + c , c+d ↔ c + d . Laws of indices: 2a+b → 2a + 2b, 2ab → 2a2b.

Chris Sangwin (Birmingham) Mathematical notation January 2013 26 / 46 Slippy cases

16 19 , 64 94

1 7 1 + 7 + → 5 4 5 + 4 deﬁnes the mediant of two fractions.

Chris Sangwin (Birmingham) Mathematical notation January 2013 27 / 46 Slippy cases

16 19 , 64 94

1 7 1 + 7 + → 5 4 5 + 4 deﬁnes the mediant of two fractions.

Chris Sangwin (Birmingham) Mathematical notation January 2013 27 / 46 Mindless symbol pushing

√ √ √ √ §148. Moreover,√ as a multiplied by√ b makes ab we shall have 6 for the value of −2 multiplied by −3; [...] √ √ §149. It is the same with regard to division; for a divided by b √ √ √ q a making b , it is evident that −4 divided by −1 will make +4 or 2; [...] √ q √ and that 1 divided by −1 gives +1 , or −1; because 1 is equal to √ −1 +1. L. Euler, (1770), Elements of algebra.

Chris Sangwin (Birmingham) Mathematical notation January 2013 28 / 46 Mindless symbol pushing

Chris Sangwin (Birmingham) Mathematical notation January 2013 28 / 46 Notation and meaning

The object of mathematical rigour is to sanction and legitimate the conquests of intuition, and there never was any other object for it. Jacques Hadamard (1865–1963)

Chris Sangwin (Birmingham) Mathematical notation January 2013 29 / 46 Notation and meaning

The object of mathematical rigour is to sanction and legitimate the conquests of intuition, and there never was any other object for it. Jacques Hadamard (1865–1963)

Chris Sangwin (Birmingham) Mathematical notation January 2013 29 / 46 Meaning in a CAS

Maxima is a system for working with expressions, such as x + y, sin(a + bπ), and u · v − v · u. Maxima is not much worried about the meaning of an expression. Whether an expression is meaningful is for the user to decide. R. Dodier (2005), Minimal Maxima ...but what about type and domain checking?

Chris Sangwin (Birmingham) Mathematical notation January 2013 30 / 46 Meaning in a CAS

Chris Sangwin (Birmingham) Mathematical notation January 2013 30 / 46 Notation as a design problem

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 31 / 46 Examples

f is a function, t is time n, m are integers

Conventions

a, b, c ... parameters ... x, y, z unknowns

Chris Sangwin (Birmingham) Mathematical notation January 2013 32 / 46 Shape

In choosing inﬁx symbols, there is a simple principle that really helps our ability to calculate: we should choose symmetric symbols for symmetric operators, and asymmetric symbols for asymmetric operators, and choose the reverse of an asymmetric symbol for the reverse operator. The beneﬁt is that a lot of laws become visual: we can write an expression backwards and get an equivalent expression. For example, x + y < z is equivalent to z > y + x. By this principle, the arithmetic symbols + × < > = are well chosen but − and 6= are not. Hehner (2004)

Chris Sangwin (Birmingham) Mathematical notation January 2013 33 / 46 Babbage’s principles

(B1) All notation should be as simple as the nature of the operations to be indicated will admit. (B2) We must adhere to one notation for one thing. (B3) Not to multiply the number of signs without necessity. (B4) When it is required to express new relations that are analogous to others for which signs are already contrived, we should employ a notation as nearly allied to those signs as we conveniently can. (B5) Whenever we wish to denote the inverse of any operation, we must use the same characteristic with the index −1.

Chris Sangwin (Birmingham) Mathematical notation January 2013 34 / 46 Babbage’s principles

(B6) Every operation ought to be capable of indicating a law. (B7) It is better to make any expression an apparent function of n, than let it consist of operations n times repeated. (B8) All notations should be so contrived as to have parts being capable of being employed separately. (B9) All letters that denote quantity should be printed in Italics, but all those which indicate operations, should be printed in a Roman Character. (B10) Every functional characteristic affects all symbols which follow it, just as if they constituted one letter. (B11) Parentheses may be omitted, if it can be done without introducing ambiguity.

Chris Sangwin (Birmingham) Mathematical notation January 2013 35 / 46 sinn(x)

Hence, the meanings of

sin(xn), sin(x)n, sinn(x).

sinn(x) must be repeated composition. sin−1(x) is the inverse.

1 sin 2 (x) is what?

Chris Sangwin (Birmingham) Mathematical notation January 2013 36 / 46 sinn(x)

Hence, the meanings of

sin(xn), sin(x)n, sinn(x).

sinn(x) must be repeated composition. sin−1(x) is the inverse.

1 sin 2 (x) is what?

Chris Sangwin (Birmingham) Mathematical notation January 2013 36 / 46 sinn(x)

Hence, the meanings of

sin(xn), sin(x)n, sinn(x).

sinn(x) must be repeated composition. sin−1(x) is the inverse.

1 sin 2 (x) is what?

Chris Sangwin (Birmingham) Mathematical notation January 2013 36 / 46 sinn(x)

Hence, the meanings of

sin(xn), sin(x)n, sinn(x).

sinn(x) must be repeated composition. sin−1(x) is the inverse.

1 sin 2 (x) is what?

Chris Sangwin (Birmingham) Mathematical notation January 2013 36 / 46 Contemporary notation

Ambiguities 2 3 , 3(2 + 5) and x(t + 1). 5 Inconsistencies sin2, sin−1

Chris Sangwin (Birmingham) Mathematical notation January 2013 37 / 46 Contemporary notation

Ambiguities 2 3 , 3(2 + 5) and x(t + 1). 5 Inconsistencies sin2, sin−1

Chris Sangwin (Birmingham) Mathematical notation January 2013 37 / 46 Overloading

1 Assignment of a value to a variable: let n = 1. 2 An equation yet to be solved x2 = 1 3 Deﬁnition of a function f (x) = x2 4 A function =: C → {true, false}.

if n = 1 then ··· else ···

Chris Sangwin (Birmingham) Mathematical notation January 2013 38 / 46 CAS syntax

CAS Assignment Equation Function Inﬁx

Axiom := = == = Derive := = := = Maple := = := = Mathematica = (or :=) == := (or =) == Maxima : = := =

Chris Sangwin (Birmingham) Mathematical notation January 2013 39 / 46 Procept

Elementary operations are a calculation “to do".

By using the notation ambiguously to represent either process or product, whichever is convenient at the time, the mathematician manages to encompass both — neatly side-stepping a possible object/process dichotomy. E. Gray & D. Tall (1994)

Need noun as well as verb forms. Need ﬁne control over properties of the operations.

Chris Sangwin (Birmingham) Mathematical notation January 2013 40 / 46 Procept

Elementary operations are a calculation “to do".

Need noun as well as verb forms. Need ﬁne control over properties of the operations.

Chris Sangwin (Birmingham) Mathematical notation January 2013 40 / 46 Procept

Elementary operations are a calculation “to do".

Need noun as well as verb forms. Need ﬁne control over properties of the operations.

Chris Sangwin (Birmingham) Mathematical notation January 2013 40 / 46 Cultural differences

Most obvious: decimal and digit separators. British Standard BS 6727:1987. Representation of numerical values in character strings for information interchange. (19 pages)

Chris Sangwin (Birmingham) Mathematical notation January 2013 41 / 46 Cultural differences

Most obvious: decimal and digit separators. British Standard BS 6727:1987. Representation of numerical values in character strings for information interchange. (19 pages)

Chris Sangwin (Birmingham) Mathematical notation January 2013 41 / 46 Syntax

Typed on a keyboard.

1 One dimensional, 2 Limited set of symbols.

Some surprising decisions, e.g.−42

... a profusion of notations (when we regard the whole science) which threaten, if not duly corrected, to multiply our difﬁculties instead of promoting our progress. C. Babbage (1827)

Chris Sangwin (Birmingham) Mathematical notation January 2013 42 / 46 Syntax

Typed on a keyboard.

1 One dimensional, 2 Limited set of symbols.

Some surprising decisions, e.g.−42

... a profusion of notations (when we regard the whole science) which threaten, if not duly corrected, to multiply our difﬁculties instead of promoting our progress. C. Babbage (1827)

Chris Sangwin (Birmingham) Mathematical notation January 2013 42 / 46 Syntax

Typed on a keyboard.

1 One dimensional, 2 Limited set of symbols.

Some surprising decisions, e.g.−42

... a profusion of notations (when we regard the whole science) which threaten, if not duly corrected, to multiply our difﬁculties instead of promoting our progress. C. Babbage (1827)

Chris Sangwin (Birmingham) Mathematical notation January 2013 42 / 46 Dragmath

Drag-and-drop interface to the computer aided assessment system STACK.

Chris Sangwin (Birmingham) Mathematical notation January 2013 43 / 46 STACK input syntax

A consistent syntax is effectively impossible in assessment. External pressure under assessment conditions is too strong. Options set by the teacher

“Insert *". i, j, vectors. (n+1)! Assumptions, e.g. n! .

Chris Sangwin (Birmingham) Mathematical notation January 2013 44 / 46 STACK input syntax

A consistent syntax is effectively impossible in assessment. External pressure under assessment conditions is too strong. Options set by the teacher

“Insert *". i, j, vectors. (n+1)! Assumptions, e.g. n! .

Chris Sangwin (Birmingham) Mathematical notation January 2013 44 / 46 STACK input syntax

A consistent syntax is effectively impossible in assessment. External pressure under assessment conditions is too strong. Options set by the teacher

“Insert *". i, j, vectors. (n+1)! Assumptions, e.g. n! .

Chris Sangwin (Birmingham) Mathematical notation January 2013 44 / 46 Students should learn syntax!

Chris Sangwin (Birmingham) Mathematical notation January 2013 45 / 46 Conclusion

Elementary algebraic notation is

more complex than we might suppose; almost impossible to “improve” or change; difﬁcult to learn.

Is it any wonder students get confused? Automatic computer aided assessment

reveals these problems.

Chris Sangwin (Birmingham) Mathematical notation January 2013 46 / 46 Conclusion

Elementary algebraic notation is

more complex than we might suppose; almost impossible to “improve” or change; difﬁcult to learn.

Is it any wonder students get confused? Automatic computer aided assessment

reveals these problems.

Chris Sangwin (Birmingham) Mathematical notation January 2013 46 / 46 Conclusion

Elementary algebraic notation is

more complex than we might suppose; almost impossible to “improve” or change; difﬁcult to learn.

Is it any wonder students get confused? Automatic computer aided assessment

reveals these problems.

Chris Sangwin (Birmingham) Mathematical notation January 2013 46 / 46