IST 4 Information and Logic MQ1 grades were emaile d please let the TAs know if you submitted MQ1 and di d not get t he email
HW1 will be returned today Average is 44/48 ~= 92 mon tue wed thr fri
30 M1 T = today 1 6 oh M1 oh x= hw#x out 13 oh oh 1 2M2 x= hw#x due 20 oh T oh 2
27 oh M2 oh midterms oh = office hours 4 3 oh oh Mx= MQx out 11 oh 3 4 oh 18 Mx= MQx due oh oh 4 5
25 oh oh
1 oh 5 oh MQ2 Memory Deadline Tuesday 4/28/2015 at 10pm • You are invited to write short essay on the topic of the MtMagenta QtiQuestion. • Recommended length is 3 pages (not more) • Submit the essay in PDF format to [email protected] file name lastname-firstname.pdf • No collaboration. No extensions Grading of MQ: 3 points (out of 103)
50% for content quality, 50% for writing quality
Some students will be given an opportunity to give a short presentation for up to 3 additional points Last time... Building blocks
geometric numeric a b m ?
a b 0 0 1 0 1 1 M symbolic 1 0 1
1 1 0 m separation Building blocks
The path from geometric numeric to numeric symbolic and beyond? a b m ?
a b 0 0 1 0 1 1 M symbolic 1 0 1
1 1 0 m The geography of numbers representations MCMXXX = 1930 The Roman Numeral Puzzle: The survival of the unfit... first positional Babylonia Base 60 5,000 ya number system
MM = 2000 ya India, China Base 10 2,500 ya Roman Empire Central Asia, Persia, Arabs 1,200 ya Base 10 positional number Europe system in Europe 800 ya Babylonians and Egyptians ~5000 years ago Merv, 2014 Merv, 2014
Merv, 2014
source: wikipedia Greek - Alexander ~2500 years ago
source: wikipedia Roman Empire ~2000 years ago
source: wikipedia Muhammad ibn Mūsā Arabs al-Khwārizmī ~1200 years ago Roman numerals and the abacus A Refresher on the Roman Numeral System
Roman Number Large Numeral Roman Number I1 Numerals V5 V 5,000 X10 X 10,000 L50 L 50,000 C 100 C 100, 000 D 500 D 500,000 M 1000 M 1, 000, 000 LCD Monitor 5 1 The Abacus: 2 1 It’ s all About Syntax
IIIII V V D L V
XXXXX L
CCCCC D
MMMMM V
VV X M C X I LL C
DD M The Abacus: It’ s all About Syntax What is the number? IIIII V V D L V
XXXXX L
CCCCC D
MMMMM V
VV X M C X I LL C
DD M The Abacus: It’ s all About Syntax What is the number? IIIII V V D L V
XXXXX L
CCCCC D
MMMMM V
VV X M C X I LL C DCI 601 DD M The Abacus: It’ s all About Syntax Touch the middle: Yes or No IIIII V V D L V it is a binary mechanism XXXXX L
CCCCC D
MMMMM V
VV X M C X I LL C DCI 601 DD M The Abacus Calculating Machine are Based on Syntax
The first actual calculating mechanism known to us is the ab acus , whi ch is thought to have been invent ed 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 drawn in the sand
Source: Wikipedia The Abacus 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 means dust אָבָק In Hebrew the word abhaq
Calculus is Latin for pebble
Source: Wikipedia The Abacus: It’ s all About Syntax
IIIII V V D L V
XXXXX L
CCCCC D
MMMMM V
VV X M C X I LL C DCI 601 DD M The Abacus: It’ s all About Syntax What is the number? IIIII V V D L V
XXXXX L
CCCCC D
MMMMM V
VV X M C X I LL C CCCCLXXXVIIII DD M 489 5 1 The Abacus: 2 1 It’ s all About Syntax
V D L V
M C X I
DCI 601 + CCCCLXXXVIIII 489 5 1 The Abacus: 2 1 It’ s all About Syntax
V D L V
M C X I
DCI 601 + CCCCLXXXVIIII 489 5 1 The Abacus: 2 1 It’ s all About Syntax
V D L V
M C X I
DCI 601 + CCCCLXXXVIIII 489 5 1 The Abacus: 2 1 It’ s all About Syntax
V D L V
M C X I
DCI 601 + CCCCLXXXVIIII 489 5 1 The Abacus: 2 1 It’ s all About Syntax
V D L V
M C X I
DCI 601 + CCCCLXXXVIIII 489 5 1 The Abacus: 2 1 It’ s all About Syntax
V D L V
M C X I
DCI 601 + CCCCLXXXVIIII 489 5 1 The Abacus: 2 1 It’ s all About Syntax
V D L V
M C X I
DCI 601 + CCCCLXXXVIIII 489 5 1 The Abacus: 2 1 It’ s all About Syntax
V D L V
M C X I
DCI 601 + CCCCLXXXVIIII 489 5 1 The Abacus: 2 1 It’ s all About Syntax
V D L V
M C X I
DCI 601 + CCCCLXXXVIIII 489 5 1 The Abacus: 2 1 It’ s all About Syntax
V D L V
M C X I
DCI 601 + CCCCLXXXVIIII 489 The Abacus: It’ s all About Syntax
What is the decimal V D L V representation?
MLXXXX ??
M C X I
DCI 601 + CCCCLXXXVIIII 489 The Abacus: It’ s all About Syntax
What is the decimal V D L V representation?
MLXXXX 1 0 9 0
What’s wrong M C X I with this picture?
DCI 601 + CCCCLXXXVIIII 489 Roman Numerals and Base 10 Systems
MLXXXX V D L V The representation in the abacus Rommuman numerals is a positional base 10 representation used for number Representation 1 0 9 0
For calculation: we used the abacus M C X I From PhPhlysical ((b)abacus) to Symbols Algorizms Algorizmi
A positional number system Operations are done on syntax is a key enabler for efficient arithmetic operations
Muhammad ibn Mūsā al-Khwārizmī ﻣﺤﻤﺪ ﺑﻦ ﻣﻮﺳﻮﻰﻰ اﻟﺨﻮارزﻣﻮرزﻲﻲ 780-850AD A Persian mathematician, who wrote on Hindu-Arabic 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 Algggporithms and Algebra in Europe
Leonardo Fibonacci 1170-1250AD 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 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 ti me, returni ng around 12 00. In 1202, at age 32, he published what he had learned in Liber Abaci, or Book of Calculation Source: Wikipedia Liber Abaci – FFpirst Chapter
Introduction of the syntax; from 1 to infinity... Liber Abaci – FFpirst Chapter
Positional: order is important Algorizmi and “brain surgery” 0123456789
00123456789 1 12345678910 2 2 3 4 5 6 7 8 9 10 11 33456789101112 4 4 5 6 7 8 9 10 11 12 13 5 5 6 7 8 9 10 11 12 13 14 6 6 7 8 9 10 11 12 13 14 15 7 7 8 9 10111213141516 8 8 9 10 11 12 13 14 15 16 17 9 9 10 11 12 13 14 15 16 17 18 In first grade: We use our BRAIN for remembering Algorizmi’s syntax Arithmetic boxes decimal Algorithms – Syntax Manipulation
digit 1 digit 2
carry 2 symbol adder carry
sum Algorithms – Syntax Manipulation
digit 1 digit 2 8 6
carry 2 symbol adder 0 carry
sum Algorithms – Syntax Manipulation
digit 1 digit 2 8 6
carry 1 2 symbol adder 0 carry
4 sum d1 d2 d1 d2 d1 d2 d1 d2 c 2 symbol adder c c 2 symbol adder c c 2 symbol adder c c 2 symbol adder c
c c c c
1891 + 8709 = ?? 1 d2 8 d2 9 d2 1 d2 c 2 symbol adder c c 2 symbol adder c c 2 symbol adder c c 2 symbol adder 0
c c c c
1891 + 8709 = ?? 1 8 8 7 9 0 1 9 c 2 symbol adder c c 2 symbol adder c c 2 symbol adder c c 2 symbol adder 0
c c c c
1891 + 8709 = ?? 1 8 8 7 9 0 1 9 c 2 symbol adder c c 2 symbol adder c c 2 symbol adder c 1 2 symbol adder 0
c c c 0
1891 + 8709 = ?? 1 8 8 7 9 0 1 9 c 2 symbol adder c c 2 symbol adder c 1 2 symbol adder 1 1 2 symbol adder 0
c c 0 0
1891 + 8709 = ?? 1 8 8 7 9 0 1 9 c 2 symbol adder c 1 2 symbol adder 1 1 2 symbol adder 1 1 2 symbol adder 0
c 6 0 0
1891 + 8709 = ?? 1 8 8 7 9 0 1 9
1 2 symbol adder 1 1 2 symbol adder 1 1 2 symbol adder 1 1 2 symbol adder 0
0 6 0 0
1891 + 8709 = ?? 1 8 8 7 9 0 1 9
1 2 symbol adder 1 1 2 symbol adder 1 1 2 symbol adder 1 1 2 symbol adder 0
0 6 0 0
1891 + 8709 = 10,600
How big are the tables in the 2 symb blol adder? Two cases: carry = 0 or carry =1
1 8 8 7 9 0 1 9
1 2 symbol adder 1 1 2 symbol adder 1 1 2 symbol adder 1 1 2 symbol adder 0
0 6 0 0
19012345678 00123456789 00123456789 0 0 0 0 0 0 0 0 0 0 0 0 0 1 2 3 4 5 6 7 8 9 10000000001 1 1234567890 2000000001 1 22345678901 3 0 0 0 0 0 0 0 1 1 1 33456789012 40000001 1 1 1 44567890123 5000001 1 1 1 1 55678901234 6 0 0 0 0 1 1 1 1 1 1 66789012345 70001 1 1 1 1 1 1 77890123456 80011111111 88901234567 901 1 1 1 1 1 1 1 1 99012345678 Algorithm = a procedure for syntax manipulation
Dear Algo and Fibo, we get confused with those large tables... can we use a smmyaller syntax? The Binary When was the binary system “invented ”?
(11010100111)2
(17 03)10 1024+ 512+ 128+ 32+ 4+ 2+ 1 = 17 03 Gottfried Leibniz 1646-1716 Leibniz – Binary Sy stem Gottfried Leibniz 1646-1716 Gottfried Leibniz 1646-1716 Gottfried Leibniz 1646-1716
Use the smallest syntax possible Binary – 0 and 1 Gottfried Leibniz 1646-1716
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8 Gottfried Leibniz 1646-1716 Binary Addition
carry Gottfried Leibniz 1646-1716 Binary Multiplication
3 5 5 6 10 20
Addition of shifted versions Leibniz: No Need for Flash Cards! Gottfried Leibniz 1646-1716 ItdInvented the Binary St?System? Gottfried Leibniz 1646-1716 Leibniz – Binary Sy stem Wen Wangg( (who flourished in about 1150 BC) is traditionally thought to have been author of the present hexagrams
63 Gottfried Leibniz 1646-1716 Leibniz – Binary Sy stem Arithmetic Boxes binary Adding Bits
digit 1 digit 2
3 bits to 2 bits
carry 2 symbol adder carry
sum Adding Bits
digit 1 digit 2 0 0 3 bits to 2 bits
carry 0 2 symbol adder 0 carry
0 sum Adding Bits
digit 1 digit 2 1 0 3 bits to 2 bits
carry 0 2 symbol adder 0 carry
1 sum Adding Bits
digit 1 digit 2 1 1 3 bits to 2 bits
carry 1 2 symbol adder 0 carry
0 sum Adding Bits
digit 1 digit 2 1 0 3 bits to 2 bits
carry 1 2 symbol adder 1 carry
0 sum Adding Bits
digit 1 digit 2 0 1 3 bits to 2 bits
carry 1 2 symbol adder 1 carry
0 sum Adding Bits
digit 1 digit 2 1 1 3 bits to 2 bits
carry 1 2 symbol adder 1 carry
1 sum Algorithms – Syntax Manipulation
0
0123456789 0 0 1 2 3 4 5 6 7 8 9 d1 d2 1 123456789 223456789 c 2 symbol adder c 3 3 4 5 6 7 8 9
s 4456789 556789 6 6 7 8 9 7789 889 99 Algorithms – Syntax Manipulation
0 01 1 01 001 012 1 1 2 1 2 3
0 1 sum 01 01 d1 d2 001 010 c 2 symbol adder c 1 1 0 1 0 1
s
0 1
carry 01 01 000 001 101 111 Algorithms – Syntax Manipulation
0 1
01 01 001 010 1 1 110 101
1 2 symbol adder 0
0 0 1
01 01 000 001 1 0 1 1 1 1 Algorithms – Syntax Manipulation
0 1
01 01 001 010 1 0 110 101
1 2 symbol adder 1
0 0 1
01 01 000 001 1 0 1 1 1 1 1 1 1 0 0 0 1 1
c 2 symbol adder c c 2 symbol adder c c 2 symbol adder c c 2 symbol adder 0
c c c c
0 1 1101 + 1001 = ?? 01 01 001 010 1 1 0 1 0 1
0 1
01 01 000 001 101 111 1 1 1 0 0 0 1 1
c 2 symbol adder c c 2 symbol adder c c 2 symbol adder c 1 2 symbol adder 0
c c c 0
0 1 1101 + 1001 = ?? 01 01 001 010 1 1 0 1 0 1
0 1
01 01 000 001 101 111 1 1 1 0 0 0 1 1
c 2 symbol adder c c 2 symbol adder c 0 2 symbol adder 1 1 2 symbol adder 0
c c 1 0
0 1 1101 + 1001 = ?? 01 01 001 010 1 1 0 1 0 1
0 1
01 01 000 001 101 111 1 1 1 0 0 0 1 1
c 2 symbol adder c 0 2 symbol adder 0 0 2 symbol adder 1 1 2 symbol adder 0
c 1 1 0
0 1 1101 + 1001 = ?? 01 01 001 010 1 1 0 1 0 1
0 1
01 01 000 001 101 111 1 1 1 0 0 0 1 1
1 2 symbol adder 0 0 2 symbol adder 0 0 2 symbol adder 1 1 2 symbol adder 0
0 1 1 0
0 1 1101 + 1001 = ?? 01 01 001 010 1 1 0 1 0 1
0 1
01 01 000 001 101 111 1 1 1 0 0 0 1 1
1 2 symbol adder 0 0 2 symbol adder 0 0 2 symbol adder 1 1 2 symbol adder 0
0 1 1 0
0 1
01 01 001 010 1 1 0 1 0 1
1101 + 1001 = 10110 0 1 13 + 9 = 22
01 01 000 001 101 111 1 1 1 0 0 0 1 1
1 2 symbol adder 0 0 2 symbol adder 0 0 2 symbol adder 1 1 2 symbol adder 0
0 1 1 0
0 1
MiMagic boxes d01o not know01 arith ithti!!metic!! 001However...010 1 1 0 1 0 1
1101 + 1001 = 10110 0 1 13 + 9 = 22
01 01 000 001 101 111 Magic Syntax Boxes
There is a finite universal set of building blocks Can conststtruct ‘everything’
a b a b m 0 0 1 0 1 1 1 0 1 1 1 0 m Magic boxes can compute the sum 0 1 d1 d2 0 1 01 c 2 symbol adder c 001 010 110 s 101
a b d1 d2 s d1 d2 s 0 0 0 0 0 1 0 1 1 0 1 0 1 0 1 1 0 0 1 1 0 1 1 1 m Magic boxes can compute the carry 0 1 d1 d2 0 1 01 c 2 symbol adder c 000 001
101 s 111
a b d1 d2 c d1 d2 c 0 0 0 0 0 0 0 1 0 0 1 1 1 0 0 1 0 1 1 1 1 1 1 1 m 1 1 1 0 0 0 1 1
1 2 symbol adder 0 0 2 symbol adder 0 0 2 symbol adder 1 1 2 symbol adder 0
0 1 1 0
Magic boxes do0 not know1 arithmetic, however, they can01 add large01 numbers! 001 010 1 1 0 1 0 1
1101 + 1001 = 10110 0 1 13 + 9 = 22
01 01 000 001 101 111 Game time Game Time
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