Anecdotes From the History of Mathematics

1. Numbers

Anecdotes from the history of mathematics : Ways of selling

mathematics

Dennis Almeida, University of Exeter

To start off with – a puzzle:What has this 1835 painting by Turner got to do

with progress in mathematics?

Number sense

Cardinal number sense

Number words

Counting (influenced by anatomy)

Discovery of zero

Development of arithmetic

Number sense – critical for survival of the species

The ability to recognize whether a small collection of objects has increased or decreased

Have we lost someone whilst out hunting?

Is our group size sufficient to defend against or attack the opposing tribe?

Early cardinal number sense – giving prototypical structure to number sense

• The size of the community/group compared with a fixed collection of objects or marks- pebbles, notches on a stick, or fingers on the hand.

The development of number words

• The abstraction of number words to abstract symbols came much later. As Bertrand Russell stated "It must have required many ages to discover that a brace of pheasants and a couple of days were both instances of the number 2”.

Counting

• To be able to count one has to place numbers/number words in order or succession.

• This is ordinal number sense: one, two, three, .........

Words to symbols - symbolic ordinal number systems

Babylonian (present day Iraq – c 3000 BC): Base 60

Egyptian (c 300 BC): Base 10

Indian ( c 11th century AD): Base 10

Need for the development of arithmetic

Organising military affairs

Calculations in trade, taxation, and the recording of time

The need to record calculations on paper led to widespread adoption of different arithmetic by the 15th century.

Early Arithmetic

37 × 11 and you don’t know place value arithmetic?

40737742921137

1372378371137

292837

146437

74237

:1137

=++=×

×+×+×=×⎪⎭

⎪⎬

⎫

=×=×=×

×

Early Arithmetic

23÷ 4 and you don’t know place value arithmetic?

4

35

4

1

2

1144231241623

14

1

22

1

164

82

41

:423 =+++=÷⇒+++=

⎪⎪⎪⎪⎪

⎭

⎪⎪⎪⎪⎪

⎬

⎫

→

→

→→→

÷

Indo-Arabic Arithmetic37 × 11 and you know

place value arithmetic?

23÷ 4 and you know

place value arithmetic?

In the first example notice the use of 0 as place value:

the 0 in 407 signifies zero=no tens. Also multiplication

proceeds Right to Left. Division in reverse.

7407

1111

737377

×←×

3

2020

2342345423

55

43 ←←=÷

A feature of the Hindu-Arabic Numerals

• There is direct evidence that the original Hindu numerals were meant to be used in two ways.

• Either in the Left-Right orientation: Increase in place value L to R

213 = 2 + 10 + 300• Or the Right-left orientation: Increase in place value R to L• 213 = 200+ 10 + 3

See A.K. Bag: Mathematics in Ancient and Medieval India, Chaukhamba

Orientalia, 1976, Delhi

•

A feature of the Indo-Arabic Numerals• Islamic mathematicians adopted the Indian system and

transmitted it Westwards. • Arabic being written in the right-left orientation is probably

the reason why the right-left orientation is universally used.

•

880508

444

72172172122121

×←×←×

The 1835 painting by Turner depicts .........the houses of parliament burning in 1834

• Resistance to the new arithmetic … tally sticks were in use until the 19th century …. The fire indirectly due to the enormous tally sticks kept in the houses.

The 1835 painting by TurnerCharles Dickens commented at the time: "... it took until 1826 to get these sticks abolished. ….In 1834 there was a considerable accumulation of them.

The sticks were housed in Westminster…… and so the order went out that they should be privately and confidentially burned.

It came to pass that they were burned in a stove in the House of Lords. The stove, overgorged with these preposterous sticks, set fire to the panelling; the panelling set fire to the House of Commons; the two houses were reduced to ashes"

2. Algebra


mathematics

First - a puzzle:

What has bone setting got to do with algebra?

Key developments in ancient and medieval algebra

Extraction of square roots

Method for solution of practical problems

Method for approximate solutions of these problems

Some problems of ancient and medieval times that required algebra.

•Right angled triangles. •Length of the hypotenuse.•Implied the need to extract square roots.

Extracting square roots - The Babylonian method

Step 1 Given a non-square number N find a number a such that a2 is near N.Step 2 Then set b = |N – a2|and c = b/2a Step 3 N a + c if a2 < N; N a – c otherwise

Example N = 2 Step 1. Choose a =17/12 Step 2. Then b = 289/144 –2 = 1/144 and c = 1/144 34/12 = 1/(1234)Step 3 2 = 17/12 - 1/(1234) = 1.414215686…

Solving simple equations – Early generalisations

The rule of three. To find the cost multiply the fruit by the requisition, and divide the resulting product by the argument.

Example 1. If A = 6 [the argument] books cost F = 12 units [the fruit], what will R = 10 [requisition] books cost? Rule of 3 Cost = F × R = 12 × 10 = 20 units A 6

The Fourth rule: x2 + 10x = 50

Make a square with x and half the number of things.

(x+5)2 = 25 + 50

x = √75 - 5

Solving quadratic equations: Al Khwarizmi (820) and Pedro Nunes (1567)

Half the number of things

5 xnumber of things

255 5x

5x x2

(x + half the number of things) squared = square of (half the number of things) placed next to the number.

number

To find x subtract from the root half the number of things

Picture (x+5)2 -25 = 50

x

Cubic equations – Jamshid al Kashi (15th century AD)

Problem from antiquity: Find sin 10.

Al Kashi knew sin 30 ≈ 0.0523359562429448 and that sin 3 = 3sin – 4sin3 . sin 30 = 3sin 10 – 4sin3 10 If x = sin 10 then 3x – 4x3 = 0.0523359562429448 Re-arranging gives x = (0.0523359562429448 + 4x3)/31st approximation x0 = 0.016 2nd approximation x1 = (0.0523359562429448 + 4x0 3)/3 =

0.0174507800809816 3rd approximation x2 = (0.0523359562429448 + 4x1 3)/3 =

0.0174524044560038

...017452406.01sin 0 =

al-Kashi‘s fixed point iteration

This is exactly the fixed-point iteration used in post 16mathematics.

y = x

y = g(x)

Location of exact root

x1 x2 x3

x = g(x) In the example g(x) = (0.0523359562429448 + 4x3)/3


• Al-Khwarizmi wrote the first treatise on algebra: Hisab al-jabr w’al-muqabala in 820 AD. The word algebra is a corruption of al-jabr which means restoration.

• In Spain, where the Arabs held sway for a long period, there arose a profession of ‘algebrista’s’ who dealt in bone setting.


• álgebra. Del lat. tardío algebra, y este abrev. del ár. clás. algabru walmuqabalah, reducción y cotejo.

• 1. f. Parte de las matemáticas en la cual las operaciones aritméticas son generalizadas empleando números, letras y signos.

• 2. f. desus. Arte de restituir a su lugar los huesos dislocados

Translation: the art of restoring broken bones to their correct positions

3. Geometry: the mother of algebra


mathematics

How do these paintings show how geometry influenced art?

Melchior Broederlam (c1394) Pietro Perugino fresco at the Sistine

Chapel (1481)

Some features in the development of Geometry

Practical knowledge for construction of buildings Practical knowledge for patterning and art Generalisation of geometry Axiomatic deductive geometry

Practical geometry in real life

The 3, 4, 5 rope for ensuring a right angle in building

construction – ropes.

Artisans in ancient and medieval times used a loop of

rope of length 12 units knotted at 3 and 4 units toensure a right angle was formed.

5

4

3

Practical calculation of areas – the quadrilateral

The surveyors rule - first evidenced in Babylonian mathematics (c 2000 BC) – for calculating the area of a quadrilateral. Walk along the 4 sides a, b, c, and d – measure – substitute into the formula.

The formula gives exact area only in the case of a rectangle. In

all other cases it is an overestimate.

2

)(

2

)( dbcaA

+×

+=

a

db

c

Greek Geometry - Euclid

Euclid (c. 300 BC) theorised geometry deriving results using axioms and deductive logic in a series of 13 books called the Elements. One such axiom is that an isosceles triangles has equal angles opposite the equal sides.

A long line of non-Greek, mainly Islamic, scholars called Euclidisi’s kept the Elements alive by manually producing editions of the work after Greek culture fell in decay.

The importance of Euclid and Greek geometry

Greek geometry was constructed in a culture of democracy where all issues were subject to debate.

Greek geometry naturally followed this tradition of having to argue the case against all sceptics.

It could be argued that this democratic, intellectual feature enabled Euclidean geometry to plant itself in foreign soil and, therefore, survive long after the decline of Greek culture.

Geometry of plane patterns - tessellations

Just how does a builder make a pattern that repeats in order to tile a floor or a wall?

North African geometers between the 8th and 16th centuries worked out that there were just 17 different types of tessellations

A result mathematically proved only in 1935. Four of the 17 possibilities are depicted in these pictures of tilings from the Alhambra in Granada, Spain (all 17 are to be found there).

P4 P3P6M P4G

Geometry the mother of algebra

•There are just 7 types of frieze patterns•The realisation that Islamic geometers had given structure to patterns in the plane motivated 19th and 20th century mathematicians algebriasing geometry.•The study of geometric symmetry directly leads to methods for the solutions of polynomials – Galois Theory.

1. ⎣ ⎣ ⎣ ⎣ ⎣ ⎣ ⎣ ⎣ ⎣ ⎣ 2. ⎣ ⎡ ⎣ ⎡ ⎣ ⎡ ⎣ ⎡ ⎣ ⎡ 3. →→→→→→ 4. ⎣⎤ ⎣⎤ ⎣⎤ ⎣⎤ ⎣⎤ ⎣⎤ 5. ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ 6. ↑ ↓ ↑ ↓ ↑ ↓ ↑ ↓ ↑ 7. ⇔ ⇔ ⇔ ⇔ ⇔ ⇔

Geometrical perspective – how geometry influenced art

Filippo Brunelleschi (1377 –1446 ) discovered theory of perspective.

Essentially in parallel lines on a horizontal plane depicted in the vertical plane meet – at the vanishing point. Only objects in perspective look realistic.

Cuboid with 1 vanishing point

Pietro Perugino’s fresco clearly shows perspective.

While Broederlam’s painting does not look natural … parallel lines in the painting meet at different points.

Melchior Broederlam (c1394) Pietro Perugino fresco (1481)

4. Who said calculus was hard?


mathematics

What has a piece of string go to do with calculus?

Some key points in the history of calculus

Early work on integration; calculation of areas and volumes

The realisation that integration means sum of power series

The conquest of infinity: summation of infinite terms

Calculation of lengths of curved lines

Integration: the determination of lengths, areas and volumes.

Early Integration.

Tsu Ch’ung Chi c.430 - c.501) did the same thing reputedly using a polygon of 24,576 sides thereby computing the value of π correct to 6 d.p.

Archimedes (c 225 BC) approximated the length of a circle and, hence, of π by approximating a circle by inscribed and circumscribed regular polygons. Using one of 96 sides he found π is between 223⁄71 and 22⁄7. So π ~ 3.1419.

Early Integration of area under a curve – the technical problem

The area A under the curve y=xk between 0 and n is approximated by the areas of the rectangles, each of width 1 and height given by xk

A ≈ 1k + 2k + 3k + ……(n-1)k + nk

Need to be able to sum powers of integers.

Archimedes and Ibn al Haytam (965-1039) were able to do this for some values of n. Later (12th -14th centuries) al Samawal (Iraq), Zhu Shijie (China), and Narayana Pandit (India) for general values of n.

y= xk

Early Integration of area under a curve – Better approximations

y=xk

The area A under the curve y=xk between 0 and 1 is approximated by the areas of the rectangles, each of width 1/n and height given by xk

A ≈ 1k + 2k + 3k + ……(n-1)k + nk

nk+1

As n →∞ the sum on the left becomes the exact area .The first appearance of a solution (A = 1/(k+1) ) was in 1530 – in the Yuktibhasa of Jyesthadeva. Later tackled in the 17th century by Fermat, Pascal, Wallis, etc.

Infinity conquered – the calculation of the derivative

Derivative at P =gradient of tangent at P

0)()(

→−−

hash

hxfxf

Newton and Leibniz independently discovered the generalised method late 17th century

xx - h

Derivative =f(x-h)

f(x)

P

Historical problems that gave rise to the calculus.• Arc length calculation:

• Approximate small sections of arc by straight lines. • What happens as the sections get smaller and

smaller?

http://upload.wikimedia.org/wikipedia/en/1/11/Arclength.png

Arc length calculation using the calculus• Each arc segment ≈ (dx2 + dy2)1/2 = (1 + [dy/dx]2)1/2 × dx

• So the total arc length ≈ Sum of all (1 + [dy/dx]2)1/2 × dx’ s

= ∫ (1 + [dy/dx]2)1/2 dxy

xx1 x2 x4xnx3

A1

A2 A3 A4

An

dx

dy =y2-y1

dxdx

dyAAlengthArc

a

bn

2

12

11∫ ⎥

⎦

⎤⎢⎣

⎡⎟⎠

⎞⎜⎝

⎛+=

• In the primary classroom one may see curved length calculation as follows: lay a piece of string along the curve, mark the ends of the curve along it, straighten the string, and then measure the marked length.


Lay a piece of string along the curve, mark the ends of the curve along it, straighten the string, and then measure the marked length. This is essentially the principle employed in the deriving the arc length formula

This was also a principle used in ancient mathematics. Good mathematics is when you first simplify the problem to easily deduce the solution and then develop the solution for the complex case.

dxdx

dylengthArc

a

b

2

12

1∫⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠

⎞⎜⎝

⎛+=


5. Using one’s imagination


mathematics

What has special effects in the cinema got to do with mathematics?

Source of fractal pictures: www.comp.dit.ie/

Using imagination - i the complex square root of -1

What kind of pictures would arise from

repeatedly applying a function of the complex

numbers?

These imaginings were that of Gaston Julia in 1915 and the resulting pictures were called Julia sets. Julia sets had no conceived applications at the Time and these later gave rise to Fractal Geometry.

The picture from repeatedly applying z z2 + i.

Fractal Geometry in the classroom: The van Koch snowflake

The mapping to be applied repeatedly: Rotate every equilateral triangle by 600 about its centre.

Fractal Geometry: The van Koch snowflake at stage 2


Fractal Geometry: The van Koch snowflake at stage 3

v


Development of a van Koch snowflake fractal

Observe: Each stellation is congruent to the original equilateral triangle

An application of fractal geometry

The van Koch snowflake fractal has

the amazing property that its

perimeter tends to infinity while its

area is finite [certainly less than the

area of the bounding rectangle

containing it].

This is the perfect design for

antennae for mobile phone and

microwave communications.

Source of fractal antenna picture: Wikipedia

Picture of a fern leaf computer generated using Fractals

Benoit Mandelbrot, the mathematician who gave fractal geometry impetus by using computers, said: “Clouds are not spheres, mountains are not cones, coastlines are not circles and bark is not smooth, nor does lightning travel in a straight line.”

Fractal imagery using computers

These pictures show the use of fractals in computer generated imagery in the cinema.

Source of fractal pictures: www.comp.dit.ie/

What has special effects in the cinema got to do with mathematics?

6. Using one’s imagination 2


mathematics

What has the auto-focus in your camera got to do with mathematics?

TWO VALUED LOGIC

• At the turn of the last century mathematics was defined by the 23 problems posed by the German mathematician David Hilbert.

• Hilbert’s problems were preponderantly about proving conjectures. That is, they were entirely to do with pure mathematics where 2 valued logic reigns: either a statement is true (1) or it is false (0).

FUZZY LOGIC: The rise of the imaginative maverick

In 1965 a computer scientist by the name of Lofti Zadeh proposed an infinite valued logic.

The logic would take any value x in the range 0 ≤ x ≤ 1

This was called FUZZY LOGIC.

FUZZY LOGIC• Fuzzy logic was not an abstract phenomenon. Zadeh

knew it could be applied from the outset.

• “Well, I knew it was going to be important. That much I knew. In fact, I had thought about sealing it in a dated envelope with my predictions and then opening it 20-30 years later to see if my intuitions were right. I used to think about it this way: that one day Fuzzy Logic would turn out to be one of the most important things to come out of our Electrical Engineering Computer Systems Division at Berkeley.”

APPLICATIONS OF FUZZY LOGIC

CLIMATE CONTROL: To keep the temperature in the operating theatre constant the control device has to direct the heating or cooling to come on when the temperature changes. The question is: how much does the room have to cool down (or heat up) before the heating (or cooling) comes on? What should the device do if it is ‘warm’?

To enable this the temperature has 3 truth values: 0.8 = a bit cold; 0.2 = a little warm; and 0 = hot. Other temperatures will give different values to the 3 functions. Depending on the (infinite) triplets of values the control device can activate heating or cooling or neither.

1 0

Cool Warm Hot

The success of Fuzzy Logic.Amongst hundreds of industrial applications

of Fuzzy Logic are the following: • Handwriting recognition by computers (Sony) • Medicine technology: cancer diagnosis (Kawasaki Medical

School) • Back light control for camcorders (Sanyo) • Single button control for washing-machines (Matsushita) • Voice Recognition (CSK, Hitachi, Ricoh)• Improved fuel-consumption for automobiles (Nippon Tools)

Source: http://www.esru.strath.ac.uk/Reference/concepts/fuzzy/fuzzy_appl.10.htm

Most people put their digital cameras on auto focus mode. But how does the camera knows what to focus on? Is it the necessarily the object you are trying to photograph? Is this object the nearest in the field of vision? Etc?

The camera uses Fuzzy logic to make assumptions on behalf of the owner. Occasionally the choice is to focus on the object closest to the centre of the viewer. On other occasions it focuses on the object closest to the camera. The margins of error are acceptable for the non-expert camera user whose concern is album pictures.

Fuzzy logic enables a digital camera to focus on the right object more often than not

What has digital camera auto-focus got to do with mathematics?

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Anecdotes From the History of Mathematics