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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 [email protected]
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