1 - Caltech

IST 4
Information and Logic
Lectures are at:
paradise caltech edu/ist4/lectures html
paradise.caltech.edu/ist4/lectures.html
Homeworks are at:
paradise caltech edu/ist4/homeworks html
paradise.caltech.edu/ist4/homeworks.html
T
= today
x= hw#x out
x= hw#x due
oh
= office hours
mon tue wed thr
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2M
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midterms
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MQ1
E
Everyone
has
h a gift!
ift!
Due Thursday 4/10/2012 by 10pm
Please email PDF
p
lastname-firstname.pdf
to ta4@paradise.caltech.edu
Languages help in reasoning
beyond our natural sense
What is our number sense?
Need a language
g g for q
quantities!
What is this item?
Lebombo bone ~40
40,000
000 years ago
hint?
A very early device for
recording
di quantities
titi b
by m
making
ki
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
defined notches
???
29 clearly
Lunar cycle
Lu
y
(month):
(m
) ~29.5
9. days
y
Source: Wikipedia
5
5
10
Day and night: 2
10
Moon – Earth (month): ~29 days
Earth – Sun (year): ~365 days
Igno-Info break
12 is a
‘special’
special number
number..
E
Examples?
l ?
~150,000
150 000 years ago: modern
d
h
humans - Homo
H
sapiens
i
~50,000
50,000 years ago
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 wass invented
What
in nt d fi
first?
st?
a. Written text
1. Numbers
a and 1 together
3200 – 2700 BC: writing was used for accounting...
accounting
Denise Schmandt-Besserat
1933 -
Source: Her 2009 paper,
“Tokens and Writing: the
Cognitive Development,”
Will be posted on the class web page
Tokens 8,000
,
ya
y
Physical Symbols:
shape = meaning
Source: DSB 2009 paper
Idea: Represent goods with tokens
one garment
one jar of oil
one ingot of metal
Token are physical
one honeycomb
h
b
Source: DSB 2009 paper
one sheep
symbols of real items – a language!
one garment
Secure Transactions:
E
Envelopes
l
(Bulla)
(B ll )
and Seals, 3,700BC
cylinder Seals
New Idea:
Mark on the outside what is inside...
Writing:
physical
h i l symbols
b l tto imprinted
i
i t d symbols
b l
Envelope showing the
imprint of three ovoid
tokens with an incised line
representing jars of oil
Source: DSB 2009 paper
3,300 BC
A crazy idea:
No need for an envelope!
p
The tablet is born!
Tablet showing the
impression of spheres and
cones representing
measures of grain
3 100BC
3,100BC
Source: DSB 2009 paper
One more crazy idea:
Separate between the quantity and the item....
Too m
many
y jars
j
of
f oils
for markings on
the envelope / tablet
Source: DSB 2009 paper
Separate between the
The
quantity
NUMBER
10
1
and the
is born!!!
1
Twelve jars
Language for quantities!!!
Source: DSB 2009 paper
item...
tablet featuring an account of
33 measures of
f oil,
il 3
3,100
100 BC
Source: DSB 2009 paper
ten
one
oil
il
Language for quantities!!!
The item (jar of oil) is not restricted
to accounting... Not restricted to envelopes
G
General
l writing
iti is
i born!!!
b
!!!
A g
great story...
y
Not covered in IST4
Where did it start?
Babylonians and Egyptians
~5000
5000 years ago
The Egyptians Preferred
10
0
How will you represent
a trillion
1,000,000,000,000?
2 100 + 7
2x100
7x10
10 + 6
6x11 = 276
Babylonians
The first
Positional
number system
1
10
The Babylonian preferred
60
Babylonians
The first
Positional
number systems
4x60 + 36x1 = 276
Babylonians
y
were Masters of Abstractions
Why 60?
Good for computation and
representing fractions:
1/2
1/3
1/4
1/5
1/6
=
=
=
=
=
30/60
20/60
15/60
12/60
10/60
Babylonian
y
Number Systems
y
What is the number?
31
What is the number?
31 60
31x60
+
31
=
1891
The Babylonians
knew everything!
How did we learn about the
Babylonians?
Expeditions (Ninveh, Nippur...):
-Claudius Rich – 1811
-Paul-Émile Botta - 1842
-Austen Layard - 1851
Peters and Hilprecht,
Hilprecht
-Peters
U Penn. 1889-1900 (4)
-Chicago,
Chi
U.
U Penn,
P
1948-1990 (19)
Hilprecht (Ed), Babylonian Expeditions, U. of Pennsylvania, 1906
Deciphering the ‘code’
Ignace Jay Gelb
1907-1985
B
Born
in
i P
Poland
l d
PhD in Italy
y - 1929
Professor in Chicago
1952
A tablet called: Plimpton 322, from 1800 BC, at Columbia U
9x13 cm
Austria (BS) –> Germany (BS) –> Denmark (PHD) –> US (Brown U)
Otto Neugebauer
1899-1990
Mathematics –> History of Exact Sciences
His son:
Gerry Neugebauer
(PhD ‘60)
60)
Millikan Professor of
Physics, Emeritus
Mathematics –> Teaching Math ->History of Exact Sciences
Asger Aaboe
1922–2007
Denmark –> US (PhD 1957, Brown U) -> 1961,Yale U
Otto Neugebauer
1899 1990
1899-1990
Asger Aaboe
1922–2007
Otto Neugebauer
1899-1990
Today you will get to be Otto and Asger!
Today,
Multiplication table for 9
1
9
2
3
4
5
6
7
8
9
10
18
11
12
1
13
14
63
72
81
27
36
45
54
1 3
1,3
1,12
1,21
1 30
1,30
1,39
1,48
1 57
1,57
2,6
15
5
16
17
18
19?
20
30 270
40
450
50
??
2,15
, 5
2,24
2,33
2,42
2 51
2,51
3,0
4,30
6,0
7,30
Multiplication
instead of division
??
fraction
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??
Regular numbers = divide 60n
Also: 5-smooth, Hamming numbers
1 – 81
Appear in the
Babylonian tablet
Source: Wikipedia
-The Babylonians had multiplication
and reciprocals
p
tables for the regular
g
numbers + the number 7
-Also
Also 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 Babylonians
knew everything!
even Geometry...
Geometry
3
45
9
The Babylonians
knew everything!
even Geometry...
Geometry
3
45
9
The Babylonians
knew everything!
even Geometry...
Geometry
3
45
9
The Babylonians
knew everything!
even Geometry...
Geometry
12 again!!
12
12
?
2,24
Which Number Does not Belong?
g
is a space between two digits
Which Number Does not Belong?
g
139
63
127
255
64-1
128-1
256-1
Positional number systems
10
2
60
Base-10
is embedded in our language and thought
B se b P
Base-b
Positional
siti n l S
Systems
stems
Translation between languages!
B se b C
Base-b
Conversion
nversi n tto B
Base-B
se B
Translation between languages!
B se b C
Base-b
Conversion
nversi n tto B
Base-B
se B
b
Base b to base 10
Sum the corresponding weights
using base-10 arithmetic
10
Base 10 to base B
Successive division by B
using base-10 arithmetic
B