IST 4Information and Logic

Lectures are at:paradise caltech edu/ist4/lectures htmlparadise.caltech.edu/ist4/lectures.html

Homeworks are at:paradise caltech edu/ist4/homeworks htmlparadise.caltech.edu/ist4/homeworks.html

mon tue wed thr fri31 M1 1= todayT

7 M1

14 1 2M2x= hw#x out

= todayT

T oh oh

oh oh M21 228 M2

x= hw#x due

idt

oh oh

h h28 M2

5 312 3 4Mx= MQx out

midtermsoh

oh oh

oh = office hours oh

12 3 419 4 526

Mx= MQx out

Mx= MQx dueoh oh

ohoh

26

2 5oh

oh

oh

oh

MQ1E h ift!Everyone has a gift!

Due Thursday 4/10/2012 by 10pmDue Thursday 4/10/2012 by 10pm

Please email PDFlastname-firstname.pdfpto ta4@paradise.caltech.edu

Languages help in reasoning beyond our natural sensebeyond our natural sense

What is our number sense?

Need a language for quantities!g g q

What is this item?

Lebombo bone ~40 000 years agoLebombo bone 40,000 years ago

hint?

A very early device for di titi b m ki recording quantities by making

marks on a bone of a monkey

Early Counting Devices ~40,000 ago

lebombo bone

1970's excavations in Lebombo Mountains, a piece of the fibula of a baboon was found marked with 29 clearly fibula of a baboon was found marked with 29 clearly defined notches ??????

Lunar cycle (month): ~29.5 days Lu y (m ) 9. y

Source: Wikipedia

5 5

Day and night: 210 10

Moon – Earth (month): ~29 days

Earth – Sun (year): ~365 days

Igno-Info break

12 is a 12 is a ‘special’ numberspecial number..E l ?Examples?

150 000 d h H i~150,000 years ago: modern humans - Homo sapiens

~50,000 years ago: emergence of languages50,000 years ago emergence of languages

~5,000 years ago: emergence of written languages

~500 years ago: emergence of printing

~50 years ago: emergence of electronic forms

Wh t s in nt d fi st?What was invented first?a. Written text

1. Numbers

a and 1 together

3200 2700 BC: writing was used for accounting

Denise Schmandt-Besserat1933 -

3200 – 2700 BC: writing was used for accounting...

Source: Her 2009 paper,1933

“Tokens and Writing: the Cognitive Development,”

Will be posted on the class web page

Tokens 8,000 ya, yPhysical Symbols:

shape = meaningshape = meaning

Source: DSB 2009 paper

one jar of oil one garment Idea: Represent goods with tokens

one sheep one jar of oil one garment

one ingot of metal

Token are physical symbols of real items – a language!

h b

Source: DSB 2009 paper

one honeycomb one garment

Secure Transactions:E l (B ll ) Envelopes (Bulla) and Seals, 3,700BCand Seals, 3,700BC

cylinder Seals

New Idea: Mark on the outside what is inside...

Writing: h i l b l t i i t d b l physical symbols to imprinted symbols

Envelope showing the imprint of three ovoid tokens with an incised line

3,300 BC

tokens with an incised line representing jars of oilSource: DSB 2009 paper

A crazy idea: No need for an envelope! A crazy idea: p

The tablet is born!

Tablet showing the impression of spheres and cones representing measures of grain3 100BC3,100BC

Source: DSB 2009 paper

One more crazy idea:Separate between the quantity and the item....

Too many jars of oils m y j ffor markings on the envelope / tablet

Source: DSB 2009 paper

Separate between the quantity and the itemSeparate between the quantity and the item...

The NUMBER is born!!!

10 1

The NUMBER is born!!!

110

Twelve jars

LLanguage for quantities!!!

Source: DSB 2009 paper

tablet featuring an account of 33 f il 3 100 BC il

oneten

33 measures of oil, 3,100 BC oil

Source: DSB 2009 paper

Language for quantities!!!Language for quantities!!!

The item (jar of oil) is not restricted The item (jar of oil) is not restricted to accounting... Not restricted to envelopes

G l iti i b !!!General writing is born!!!

A great story... Not covered in IST4g y

Where did it start?Where did it start?

Babylonians and Egyptians5000 ~5000 years ago

The Egyptians Preferred 10 The Egyptians Preferred 0

How will you representHow will you representa trillion1,000,000,000,000?

2 100 7 10 6 1 2762x100 + 7x10 + 6x1 = 276

Babylonians

The first Positional number system

11

10

The Babylonian preferred 60

Babylonians

The first Positional number systems

4x60 + 36x1 = 276

Babylonians were Masters of Abstractions

Why 60?

y

Why 60?Good for computation and representing fractions:

1/2 = 30/601/2 30/601/3 = 20/601/4 = 15/60 1/5 = 12/601/5 = 12/601/6 = 10/60

Babylonian Number Systemsy y

What is the number?

31

What is the number?

31 60 31 189131x60 31+ = 1891

The Babyloniansknew everything!knew everything!

How did we learn about the Babylonians?Babylonians?Expeditions (Ninveh, Nippur...):

-Claudius Rich – 1811-Paul-Émile Botta - 1842-Austen Layard - 1851

-Peters and HilprechtPeters and Hilprecht,U Penn. 1889-1900 (4)

Chi U P -Chicago, U. Penn, 1948-1990 (19)

Hilprecht (Ed), Babylonian Expeditions, U. of Pennsylvania, 1906

Deciphering the ‘code’Ignace Jay Gelb

1907-1985

B i P l d

1952

Born in Poland

PhD in Italy - 1929y

Professor in Chicago

A tablet called: Plimpton 322, from 1800 BC, at Columbia U

9x13 cm

Otto NeugebauerAustria (BS) –> Germany (BS) –> Denmark (PHD) –> US (Brown U)

Otto Neugebauer1899-1990

Mathematics –> History of Exact Sciences

His son:Gerry Neugebauer (PhD ‘60)(PhD 60)Millikan Professor of Physics, Emeritus

Asger AaboeMathematics –> Teaching Math ->History of Exact Sciences

Denmark –> US (PhD 1957, Brown U) -> 1961,Yale U 1922–2007

Otto Neugebauer1899 19901899-1990

Otto Neugebauer1899-1990

Asger Aaboe1922–2007

Today you will get to be Otto and Asger!Today, you will get to be Otto and Asger!

1 159 2,15

Multiplication table for 9123

5161718

91827

, 52,242,332,422 514

567

1819?20

40

364554

63 1 3

2,513,04,306,0

27030789

10

4050??

631,121,211 30

1,3 6,07,3072 450

811011121

1,301,391,481 5713

14

1,572,6

Multiplication instead of division

????

fraction

7 and 11 and other numbers are missing?7 and 11 and other numbers are missing?

fraction

Regular numbers = divide 60n

A number that its prime factors are at most 5: 5-smooth

Is 24 a regular number? 2x2x2x3

Is 896 a regular number? 2x2x2x2x2x2x2x7

Is 900 a regular number? 2x2x3x3x5x5

Generate all the regular numbers??Generate all the regular numbers??

Regular numbers = divide 60n

Also: 5-smooth, Hamming numbers

1 – 81

Source: Wikipedia

1 – 81Appear in the Babylonian tablet

-The Babylonians had multiplicationand reciprocals tables for the regular p gnumbers + the number 7

-Also they had tables of squaresAlso they had tables of squares

-And many tricks... We call it algorithms...

Question: Suppose you can add, subtract, have small multiplication tables, large tables for n2,can you use it to compute arbitrary multiplications?

The Babyloniansknew everything!knew everything!

even Geometryeven Geometry...

3

945

The Babyloniansknew everything!knew everything!

even Geometryeven Geometry...

3

945

The Babyloniansknew everything!knew everything!

even Geometryeven Geometry...

3

945

The Babyloniansknew everything!knew everything!

even Geometryeven Geometry...

12 again!!12

12 again!!

2,24?12

Which Number Does not Belong?g

is a space between two digits

Which Number Does not Belong?g

139 25512763139 25512763

64-1 128-1 256-1

Positional number systems

10

2

6060

B se b P siti n l S stemsBase-10 is embedded in our language and thought

Base-b Positional Systems

B se b C nversi n t B se BTranslation between languages!

Base-b Conversion to Base-B

B se b C nversi n t B se BTranslation between languages!

Base-b Conversion to Base-B

Base b to base 10 Base 10 to base B b B10

Successive division by B using base-10 arithmetic

Sum the corresponding weights using base-10 arithmetic