Post on 18-Mar-2020
IST 4Information and Logic
MQ1 d il dMQ1 grades were emailedplease let the TAs know if you submitted MQ1
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HW1 will be returned todayAverage is 44/48 ~= 92Average is 44/48 ~= 92
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MemoryMQ2• You are invited to write short essay on the topic of theM t Q ti
QDeadline Tuesday 4/28/2015 at 10pm
Magenta Question.• Recommended length is 3 pages (not more)• Submit the essay in PDF format to ta4@paradise.caltech.eduSubmit the essay in PDF format to ta4@paradise.caltech.edu
file name lastname-firstname.pdf• No collaboration. No extensionsGrading of MQ:3 points (out of 103)
50% for content quality, 50% for writing quality
Some students will be given an opportunityto give a short presentation for up to 3 additional points
Last time...Building blocks
geometricgeometricnumeric
b
a b m ?
symbolica b
0 0
10
1
1 M y1 0
11
1
0 m
separationBuilding blocks
The path from geometricpnumeric to
symbolic and beyond?
geometricnumeric
symbolic and beyond?
b
a b m ?a b
0 0
10
1
1 M symbolic1 0
11
1
0 m
y
The geography The geography of numbers representations
MCMXXX = 1930
The Roman Numeral Puzzle: The survival of the unfitThe survival of the unfit...
Babylonia 5,000 yafirst positional number system
Base 60
India, China 2,500 yaRoman EmpireMM = 2000 ya Base 10
Central Asia, Persia, Arabs 1,200 yaRoman Empire
B 10
Europe 800 yapositional number system in Europe
Base 10
system in Europe
Babylonians and Egyptians5000 ~5000 years ago
Merv, 2014 Merv, 2014
Merv, 2014
source: wikipedia
Greek - Alexander~2500 years ago 2500 years ago
source: wikipedia
Roman Empire~2000 years ago 2000 years ago
source: wikipedia
Arabs~1200 years ago
Muhammad ibn Mūsā al-Khwārizmī
~1200 years ago
Roman numerals and the abacus
A Refresher on the Roman Numeral SystemRoman Numeral System
Roman Numeral Number Large
Roman NumberI 1V 5
NumeralsV 5,000
X 10L 50
X 10,000L 50,000C 100 000C 100
D 500M 1000
C 100,000D 500,000M 1 000 000M 1000 M 1,000,000
LCD Monitor
The Abacus: It’s all About Syntax2 1
5 1
It s all About Syntax
VLDVIIIII V
2 1
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The Abacus: It’s all About Syntax
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It s all About SyntaxWhat is the number?
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The Abacus: It’s all About Syntax
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MDCI 601
The Abacus: It’s all About Syntax
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Touch the middle: Yes or NoVLDVIIIII
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Lit is a binary mechanism
CCCCC D
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MDCI 601
The AbacusCalculating Machine are Based on Syntax
The first actual calculating mechanism known to us is th b s hi h is th ht t h b n in nt d
Calculating Machine are Based on Syntax
the abacus, which is thought to have been invented by the Babylonians sometime between 1,000 BC and 500 BC
The original concept referred to a flat stone covered with sand or dust with pebbles being placed on lines with sand or dust, with pebbles being placed on lines drawn in the sand
Source: Wikipedia
The AbacusCalculating Machine are Based on Syntax
The original concept referred to a flat stone covered
Calculating Machine are Based on Syntax
The original concept referred to a flat stone covered with sand or dust, with pebbles being placed on lines drawn in the sand ?
?
In Phoenician the word abak means sand
In Hebrew the word abhaq ָאָבק means dust
Calculus is Latin for pebble
Source: Wikipedia
The Abacus: It’s all About Syntax
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It s all About Syntax
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V
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The Abacus: It’s all About Syntax
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IXCM
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MCCCCLXXXVIIII 489
The Abacus: It’s all About Syntax2 1
5 1
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It s all About Syntax2 1
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The Abacus: It’s all About Syntax2 1
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It s all About Syntax2 1
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CCCCLXXXVIIII 489DCI 601 +
The Abacus: It’s all About Syntax2 1
5 1
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It s all About Syntax2 1
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IXCM
CCCCLXXXVIIII 489DCI 601 +
The Abacus: It’s all About Syntax2 1
5 1
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It s all About Syntax2 1
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IXCM
CCCCLXXXVIIII 489DCI 601 +
The Abacus: It’s all About Syntax2 1
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It s all About Syntax2 1
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IXCM
CCCCLXXXVIIII 489DCI 601 +
The Abacus: It’s all About Syntax2 1
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It s all About Syntax2 1
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The Abacus: It’s all About Syntax2 1
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The Abacus: It’s all About Syntax2 1
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CCCCLXXXVIIII 489DCI 601 +
The Abacus: It’s all About Syntax2 1
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It s all About Syntax2 1
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The Abacus: It’s all About Syntax2 1
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The Abacus: It’s all About Syntax
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MLXXXX ??
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The Abacus: It’s all About Syntax
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What is the decimal VLDVWhat is the decimal representation?
MLXXXX 0901
IXCMWhat’s wrongwith this picture?
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Roman Numerals and Base 10 Systemsy
VLDVMLXXXX VLDVMLXXXXRoman numerals
The representation in the abacusis a positional base 10 representation
0901m um
used for numberRepresentation
is a positional base 10 representation
For calculation:we used the abacus
IXCMwe used the abacus
F Ph l ( b ) From Physical (abacus) to Symbolsy
Algorizmsg
Algorizmig
A positional number system Operations are done on syntaxp yis a key enabler for efficient arithmetic operations
Operations are done on syntax
Muhammad ibn Mūsā al-Khwārizmī
الخوارزمي موسى بن محمد 780 850ADي رز و ى و 850AD-780 بن
A Persian mathematician, who wrote on Hindu-Arabic numerals and was among the first to use zero as a place numerals and was among the first to use zero as a place holder in positional base notation. The word algorithm derives from his name. His book Kitab al-jabr w'al-muqabala gives us the word algebra
Source: Wikipedia
The Beginning of the “Algebra Book ” by “Algorizmi ”
Everything requires computation...
The Beginning of the “Algebra Book ” by “Algorizmi ”Positional: order is important; the base – multiplication by 10;
from 1 to infinity...
Example from the “Algebra Book ” by “Algorizmi ”computation = single digit syntax manipulation
????
It is rhetorical (words) no symbols
Algorithms and Algebra in Europeg g pLeonardo Fibonacci1170-1250AD Leonardo was born in Pisa his father directed Leonardo was born in Pisa, his father directed
a trading post in Bugia, a port east of Algiers in North Africa, as a young boy Leonardo traveled there to help him This is where he learned about there to help him. This is where he learned about the Arabic numeral system
Perceiving that arithmetic with Arabic numerals is simpler and more efficient than with Roman numerals, Fibonacci traveled ,throughout the Mediterranean world to study under the leading Arab mathematicians of the time, returning around 1200. t me, return ng around 00. In 1202, at age 32, he published what he had learned in Liber Abaci, or Book of Calculation
Source: Wikipedia
Liber Abaci – First ChapterL F p
Introduction of the syntax; from 1 to infinity...
Liber Abaci – First ChapterL F p
Positional: order is importantPositional: order is important
Algorizmi Algorizmi andand
“brain surgery” brain surgery
0 1 2 3 4 5 6 7 8 9
0 0 1 2 3 4 5 6 7 8 9
1 1 2 3 4 5 6 7 8 9 102 2 3 4 5 6 7 8 9 10 112 2 3 4 5 6 7 8 9 10 113 3 4 5 6 7 8 9 10 11 124 4 5 6 7 8 9 10 11 12 135 5 6 7 8 9 10 11 12 13 145 5 6 7 8 9 10 11 12 13 146 6 7 8 9 10 11 12 13 14 157 7 8 9 10 11 12 13 14 15 168 8 9 10 11 12 13 14 15 16 178 8 9 10 11 12 13 14 15 16 179 9 10 11 12 13 14 15 16 17 18
In first grade:We use our BRAIN for remembering We use our BRAIN for remembering
Algorizmi’s syntax
Arithmetic boxesArithmetic boxesdecimal decimal
Algorithms – Syntax Manipulation
digit 1 digit 2digit 1 digit 2
2 symbol addercarry carry
sum
Algorithms – Syntax Manipulation
digit 1 digit 28 6
digit 1 digit 2
2 symbol addercarry carry0
sum
Algorithms – Syntax Manipulation
digit 1 digit 28 6
digit 1 digit 2
2 symbol addercarry 1 carry0
44
sum
d1 d2 d1 d2 d1 d2 d1 d2
2 symbol adderc
c
c 2 symbol adderc
c
c 2 symbol adderc
c
c 2 symbol adderc
c
c
1891 + 8709 = ??
1 d2 8 d2 9 d2 1 d2
2 symbol adderc
c
c 2 symbol adderc
c
c 2 symbol adderc
c
c 2 symbol adderc
c
0
1891 + 8709 = ??
1 8 8 7 9 0 1 9
2 symbol adderc
c
c 2 symbol adderc
c
c 2 symbol adderc
c
c 2 symbol adderc
c
0
1891 + 8709 = ??
1 8 8 7 9 0 1 9
2 symbol adderc
c
c 2 symbol adderc
c
c 2 symbol adderc
c
c 2 symbol adder1
0
0
1891 + 8709 = ??
1 8 8 7 1 99 0
2 symbol adderc
c
c 2 symbol adderc
c
c 2 symbol adder1
0
02 symbol adder1
0
1
1891 + 8709 = ??
1 8 8 7 9 0 1 9
2 symbol adderc
c
c 2 symbol adder1
6
1 2 symbol adder1
0
1 2 symbol adder1
0
0
1891 + 8709 = ??
1 8 8 7 9 0 1 9
2 symbol adder1
0
1 2 symbol adder1
6
1 2 symbol adder1
0
1 2 symbol adder1
0
0
1891 + 8709 = ??
1 8 8 7 9 0 1 9
2 symbol adder1
0
1 2 symbol adder1
6
1 2 symbol adder1
0
1 2 symbol adder1
0
0
1891 + 8709 = 10,600
How big are the tables h 2 b l dd in the 2 symbol adder?
1 8 8 7 9 0 1 9
Two cases: carry = 0 or carry =1
2 symbol adder1
0
1 2 symbol adder1
6
1 2 symbol adder1
0
1 2 symbol adder1
0
0
0 0 1 2 3 4 5 6 7 8 90 0 1 2 3 4 5 6 7 8 9
1 9 0 1 2 3 4 5 6 7 80 0 1 2 3 4 5 6 7 8 90 0 0 0 0 0 0 0 0 0 0 0 0 1 2 3 4 5 6 7 8 9
1 1 2 3 4 5 6 7 8 9 0
2 2 3 4 5 6 7 8 9 0 1
0 0 0 0 0 0 0 0 0 0 01 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 1 13 0 0 0 0 0 0 0 1 1 1 3 3 4 5 6 7 8 9 0 1 2
4 4 5 6 7 8 9 0 1 2 35 5 6 7 8 9 0 1 2 3 4
3 0 0 0 0 0 0 0 1 1 14 0 0 0 0 0 0 1 1 1 15 0 0 0 0 0 1 1 1 1 16 0 0 0 0 1 1 1 1 1 1 6 6 7 8 9 0 1 2 3 4 5
7 7 8 9 0 1 2 3 4 5 68 8 9 0 1 2 3 4 5 6 7
6 0 0 0 0 1 1 1 1 1 17 0 0 0 1 1 1 1 1 1 18 0 0 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 9 9 0 1 2 3 4 5 6 7 89 0 1 1 1 1 1 1 1 1 1
Algorithm = Algorithm = a procedure for syntax manipulation
Dear Algo and Fibo, we get confused with those glarge tables... can we use a
smaller syntax?m y
The BinaryThe Binary
When was the binary When was the binary system “invented”?system invented ?
(11010100111)(11010100111)2
(1 03)(1703)10
1024 512 128 32 4 2 1 1 03 1024+512+128+32+4+2+1 = 1703
Leibniz Binar S stemGottfried Leibniz
1646-1716
Leibniz – Binary System
Gottfried Leibniz1646-1716
Gottfried Leibniz1646-1716
Gottfried Leibniz1646-1716
Use the smallest syntax possible Binary – 0 and 1Binary 0 and 1
Gottfried Leibniz1646-1716
??
8
??
8
Binary AdditionGottfried Leibniz
1646-1716 y
carry
Binary MultiplicationGottfried Leibniz
1646-1716
Binary Multiplication
3 5 5206 2010
Addition of shifted versions
Leibniz: No Need for Flash Cards!No Need for Flash Cards!
I t d th Bi S t ?Gottfried Leibniz
1646-1716
Invented the Binary System?
Leibniz Binar S stemGottfried Leibniz
1646-1716
Leibniz – Binary SystemWen Wang (who flourished in about 1150 BC) g ( f )is traditionally thought to have been author of the present hexagrams
63
Leibniz Binar S stemGottfried Leibniz
1646-1716
Leibniz – Binary System
Arithmetic BoxesArithmetic Boxesbinarybinary
Adding Bits
digit 1 digit 2digit 1 digit 2
3 bits to 2 bits
2 symbol addercarry carry
sum
Adding Bits
digit 1 digit 20 0
digit 1 digit 2
3 bits to 2 bits
2 symbol addercarry 0 carry0
0sum
Adding Bits
digit 1 digit 21 0
digit 1 digit 2
3 bits to 2 bits
2 symbol addercarry 0 carry0
1sum
Adding Bits
digit 1 digit 21 1
digit 1 digit 2
3 bits to 2 bits
2 symbol addercarry 1 carry0
0sum
Adding Bits
digit 1 digit 21 0
digit 1 digit 2
3 bits to 2 bits
2 symbol addercarry 1 carry1
0sum
Adding Bits
digit 1 digit 20 1
digit 1 digit 2
3 bits to 2 bits
2 symbol addercarry 1 carry1
0sum
Adding Bits
digit 1 digit 21 1
digit 1 digit 2
3 bits to 2 bits
2 symbol addercarry 1 carry1
1sum
Algorithms – Syntax Manipulation
0
0 1 2 3 4 5 6 7 8 90 0 1 2 3 4 5 6 7 8 9
0
0 0 1 2 3 4 5 6 7 8 91 1 2 3 4 5 6 7 8 92 2 3 4 5 6 7 8 93 3 4 5 6 7 8 9
2 symbol adderc
d1 d2
c3 3 4 5 6 7 8 94 4 5 6 7 8 95 5 6 7 8 96 6 7 8 9
y
s
6 6 7 8 97 7 8 98 8 99 9
Algorithms – Syntax Manipulation0 1
0 0 11 1 2
0 0 1
0 1 21 2 3
1
0 1
sum
1 1 2 1 2 3
2 symbol adderc
d1 d2
c
0 1
0 0 11 1 0
0 1
0 1 01 0 1
sum
y
s
1 1 0 1 0 1
0 1
0 1
0
0 1
1
carry0 0 01 0 1
0 0 11 1 1
Algorithms – Syntax Manipulation
0 1
0 1
0
0 1
1
2 symbol adder1
1 1
0
0 0 11 1 0
0 1 01 0 1
y
0 0 1
0 1
0 0 01 0 1
0 1
0 0 11 1 11 0 1 1 1 1
Algorithms – Syntax Manipulation
0 1
0 1
0
0 1
1
2 symbol adder1
1 0
1
0 0 11 1 0
0 1 01 0 1
y
0 0 1
0 1
0 0 01 0 1
0 1
0 0 11 1 11 0 1 1 1 1
1 1 1 0 0 0 1 1
2 symbol adderc
c
c 2 symbol adderc
c
c 2 symbol adderc
c
c 2 symbol adderc
c
0
0 11101 + 1001 = ??
0 1
0 0 11 1 0
0 1
0 1 01 0 11 1 0 1 0 1
0 1
0 1
0
0 1
1
0 0 01 0 1
0 0 11 1 1
1 1 1 0 0 0 1 1
2 symbol adderc
c
c 2 symbol adderc
c
c 2 symbol adderc
c
c 2 symbol adder1
0
0
0 11101 + 1001 = ??
0 1
0 0 11 1 0
0 1
0 1 01 0 11 1 0 1 0 1
0 1
0 1
0
0 1
1
0 0 01 0 1
0 0 11 1 1
1 1 1 0 0 0 1 1
2 symbol adderc
c
c 2 symbol adderc
c
c 2 symbol adder0
1
1 2 symbol adder1
0
0
0 11101 + 1001 = ??
0 1
0 0 11 1 0
0 1
0 1 01 0 11 1 0 1 0 1
0 1
0 1
0
0 1
1
0 0 01 0 1
0 0 11 1 1
1 1 1 0 0 0 1 1
2 symbol adderc
c
c 2 symbol adder0
1
0 2 symbol adder0
1
1 2 symbol adder1
0
0
0 11101 + 1001 = ??
0 1
0 0 11 1 0
0 1
0 1 01 0 11 1 0 1 0 1
0 1
0 1
0
0 1
1
0 0 01 0 1
0 0 11 1 1
1 1 1 0 0 0 1 1
2 symbol adder1
0
0 2 symbol adder0
1
0 2 symbol adder0
1
1 2 symbol adder1
0
0
0 11101 + 1001 = ??
0 1
0 0 11 1 0
0 1
0 1 01 0 11 1 0 1 0 1
0 1
0 1
0
0 1
1
0 0 01 0 1
0 0 11 1 1
1 1 1 0 0 0 1 1
2 symbol adder1
0
0 2 symbol adder0
1
0 2 symbol adder0
1
1 2 symbol adder1
0
0
0 1
0 1
0 0 11 1 0
0 1
0 1 01 0 11 1 0 1 0 1
0 1
1101 + 1001 = 10110
0 1
0
0 1
113 + 9 = 22
0 0 01 0 1
0 0 11 1 1
1 1 1 0 0 0 1 1
2 symbol adder1
0
0 2 symbol adder0
1
0 2 symbol adder0
1
1 2 symbol adder1
0
0
0 1
M i b d t k ith ti !!0 1
0 0 11 1 0
0 1
0 1 01 0 1
Magic boxes do not know arithmetic!!However...
1 1 0 1 0 1
0 1
1101 + 1001 = 10110
0 1
0
0 1
113 + 9 = 22
0 0 01 0 1
0 0 11 1 1
Magic Syntax Boxes
There is a finite universal set of building blocks
C st t ‘everything’Can construct everythinga b
a b m
0 0 110
1 011
m11 0
Magic boxes can compute the sum
0 1d1 d2
0 1
compute the sum
0 1
0 0 11 1 0
0 1
0 1 01 0 1
2 symbol adderc
s
c
a bd1 d2 s d1 d2 sd1 d2 s
0 010
01
d1 d2
0 010
s
1010
1 011
110
101 0
11
00111 0
m11 1
Magic boxes can compute the carry
0 1d1 d2
0 1
compute the carry
0 1
0 0 01 0 1
0 10 0 11 1 1
2 symbol adderc
s
c
a bccd1 d2 d1 d2 c
0 010
01
0 010
c
00
d1 d2 d1 d2
101 0
11
111
101 0
11
00111 1
m111
1 1 1 0 0 0 1 1
2 symbol adder1
0
0 2 symbol adder0
1
0 2 symbol adder0
1
1 2 symbol adder1
0
0
0 1Magic boxes do not know arithmetic,however they can add large numbers!
0 1
0 0 11 1 0
0 1
0 1 01 0 1
however, they can add large numbers!
1 1 0 1 0 1
0 1
1101 + 1001 = 10110
0 1
0
0 1
113 + 9 = 22
0 0 01 0 1
0 0 11 1 1
Game timeGame time
Game TimeGame Time
Who can WIN WIN the game
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Who can WIN
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Who can WIN
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can take as many sticks
y uin removing sticks
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the player that removesp ythe last stick wins
Game TimeGame Time
Who can WIN
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Game TimeGame Time
Who can WIN
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Game TimeGame Time
Who can WIN
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the game
Game TimeGame Time
Who can WIN
WIN
the game
Game TimeGame Time
Who can WIN
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the game
Game TimeGame Time
Who can WIN
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the game
Game TimeGame Time
Who can WIN
WIN
the game
Game TimeGame Time
Who can WIN
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the game
Game TimeGame Time
Who can WIN
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the game
Game TimeGame Time
Who can WIN
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the game
Game TimeGame Time
Who can WIN
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the game
Game TimeGame Time
Who can WIN
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the game
Game TimeGame Time
Who can WIN
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Game TimeGame Time
Who can WIN
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I CANI CAN
Game TimeGame Time
Who can WIN WIN the game
VI=110
Game TimeAn even number of 1s in every Game Timeposition
Who can WIN
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011011
100
110
001
Game TimeAn even number of 1s in every Game Timeposition
Who can WIN
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the game
011011
100
110
001
Game TimeAn even number of 1s in every Game Timeposition
Who can WIN
WIN
the game
010010
100
110
000
Game TimeAn even number of 1s in every Game Timeposition
Who can WIN
WIN
the game
010010
100
110
000
Game TimeAn even number of 1s in every Game Timeposition
Who can WIN
WIN
the game
010010
100
001
111
Game TimeAn even number of 1s in every Game Timeposition
Who can WIN
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the game
010010
100
001
111
Game TimeAn even number of 1s in every Game Timeposition
Who can WIN
WIN
the game
010010
011
001
000
Game TimeAn even number of 1s in every Game Timeposition
Who can WIN
WIN
the game
010010
011
001
000
Game TimeAn even number of 1s in every Game Timeposition
Who can WIN
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the game
010010
011
001
000
Game TimeAn even number of 1s in every Game Timeposition
Who can WIN
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the game
010010
000
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Game TimeAn even number of 1s in every Game Timeposition
Who can WIN
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the game
010010
000
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Game TimeAn even number of 1s in every Game Timeposition
Who can WIN
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the game
001001
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Game TimeAn even number of 1s in every Game Timeposition
Who can WIN
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001001
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Game TimeAn even number of 1s in every Game Timeposition
Who can WIN
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000
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position
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