IST 4 Information and Logic
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1 IST 4 Information and Logic
2 Quizzes grade (6): average of top n-2 T = today x= hw#x out x= hw#x due mon tue wed thr fri 1 M1 oh 1 8 oh M1 15 oh 1 T 2 oh M2 22 oh PCP oh 2 oh sun oh 29 oh M2 oh = office hours oh 6 3 oh Midterms oh oh Mx= MQx out 13 oh 3 4 oh Mx= MQx due 20 oh PCP oh 4 5 PCP= Programming Challenge 27 3 oh 5 oh oh oh oh
3 information systems (limited) memory and innovation process (artificial languages) - Lecture 5: Babylonian mathematics vs Greek mathematics - Lecture 1: Life DNA sequences and evolution - Lecture 2: The human brain natural languages - Lecture 3: Artificial languages - numbers and writing - Lecture 4: Languages for quantities: Babylonians
4 Babylonian Clay Tablets X 9 Π Asger Aaboe Otto Neugebauer Memory of mathematical knowledge
5 Questions on languages Q (feasibility): Can we compute / represent everything with a language? Q (complexity): How complex is the computation / representation?
6 The language of numbers Q: Can we represent everything with integers? Π 3 Approximations
7 The Babylonians knew everything! Yale U., March 2018 YBC 7289 ~1700BC
8 The Babylonians knew everything! YBC 7289 ~1700BC
9 30 (1,24, 51,10) (42,2 5, 35)
10 30 (42,25,35) x (2)???? (1,24, 51,10) (42,2 5, 35)
11 30 (1,24,51,10) x (0;30) (42,25,35) (42,25,35) x (2) (1,10) + (50,0) (1,24,0,0) (1,24,51,10) (2) x (0;30) = (1) (1,24, 51,10) (42,2 5, 35)
12 The Babylonians loved reciprocals!!! (1;24,51,10) x (0;30) 30 So what?? (0;42,25,35) (1;24,51,10) x (0;42,25,35)???? ~(1) Assume it is exactly 1... (1,24, 51,10) (42,2 5, 35)??
13 30 So what?? The Babylonians knew (i) Pythagoras Theorem (ii) how to approximate square roots... (1,24, 51,10) (42,2 5, 35)
14 30 (1,24, 51,10) (42,2 5, 35) The Babylonians knew Pythagoras Theorem and how to approximate square roots...
15 approximating a
16 ideas for approximations? searching and correcting simple and slow
17 simple and slow
18 The Babylonian method How to improve the estimate? Imagine!!! 2 1 and?? squaring the rectangle Idea: Imagine a square of area 2 Approximate the square with a rectangle of identical area!
19 The Babylonian method How to improve the estimate? (2 + 1)/2 = 1.5 2/1.51 and?? squaring the rectangle Idea: Average of the two sides of the rectangle Approximate the square with a rectangle of identical area!
20 The Babylonian method 1.5 (2/ )/2 = /1.5 Idea: Average of the two sides of the rectangle Approximate the square with a rectangle of identical area!
21 The Babylonian method (2/ )/2 = /1.416 Idea: Average of the two sides of the rectangles Approximate the square with a rectangle of identical area!
22 The Babylonian method 2/estimate + estimate 2
23 Why does it work? The Babylonian method 2/long + long new long = 2? estimate is strictly decreasing, converges from above short new long long
24 In general: The Babylonian method??
25 In general: The Babylonian method??? arithmetic mean geometric mean
26 In general: The Babylonian method?? Proof: arithmetic mean geometric mean
27 The Babylonians knew Pythagoras Theorem and how to approximate the square root... The Babylonians knew everything...
28 The Babylonians knew Pythagoras Theorem and how to approximate the square root...
29 A tablet called: Plimpton 322, from 1800 BC, at Columbia U Why? Trigonometric tables 9x13 cm source: wikipedia
30 The Babylonians knew Pythagoras Theorem and how to approximate the square root... and compute Pythagorean triples??
31 The Babylonians knew Pythagoras Theorem and how to approximate the square root... and compute Pythagorean triples??
32 The Babylonians knew everything... They created a highly advanced civilization: music, literature, law, medicine, science, engineering, mathematics... Code of Hammurabi, 1754 BC so that the strong should not harm the weak Louvre Museum, France 2.25 m (7.4 ft)
33 The Babylonians knew everything... However,... They created a highly advanced civilization: music, literature, law, medicine, science, engineering, mathematics... They had (schools) a formal education system! Schools had both boys and girls! Edubba house of tablets
34 The Babylonians knew everything... However,... For 1,000 years they made very little progress in mathematics... My Conjecture: They taught only the how and did not teach the why... a utilitarian approach NO (documented...) proofs... Why are proofs important?
35 The Babylonians knew everything... However,... For 1,000 years they made very little progress in mathematics... To make progress: We need to impart the sensation of ideas as they are conceived and not only as they are known NO (documented...) proofs... Why are proofs important?
36 The Babylonians NO proofs... knew everything... They taught the how and not the why The solution came with the Greeks
37 Alexander the Great, BC Captured Egypt, Babylonia 331BC Died in June, 323 BC, age 32 in Babylonia Recorded in the Babylonian astronomical diaries 800 years of records!! source: wikipedia
38 Babylonians vs Greeks The language of proofs rational numbers
39 Pythagoras BC Proofs Euclid,300BC The solution came with the Greeks
40 Pythagoras BC Proof: Euclid,300BC proof - contradiction, parity... Assume that p and q are relatively prime (simplified) Reach a contradiction!! divisible by 4 = not divisible by 4 p odd? p odd, q even: NO p odd, q odd: NO which had to be demonstrated p even? p even, q even: NO p even, q odd: NO Quod Erat Demonstrandum QED
41 The language of proofs Babylonians vs Greeks Pythagoras Theorem
42 Proof?? Pythagoras BC Euclid,300BC Thm: Given a right triangle with sides a, b and c, where a and b are the legs, then:
43 China ~400BC Book named 'Chou pei Suan Ching'
44 Idea: Compute the area QED
45 The language of proofs Babylonian / Pythagorean Triples
46
47 Primitive triples smaller than 100: (3, 4, 5) (5, 12, 13) (8, 15, 17) (7, 24, 25) (20, 21, 29) (12, 35, 37) (9, 40, 41) (28, 45, 53) (11, 60, 61) (16, 63, 65) (33, 56, 65) (48, 55, 73) (13, 84, 85) (36, 77, 85) (39, 80, 89) (65, 72, 97) Question: Is the number of primitive triples infinite? Question: Closed form solution for all the primitive triples? Euclid,300BC
48 Primitive triples smaller than 100: (3, 4, 5) (5, 12, 13) (8, 15, 17) (7, 24, 25) (20, 21, 29) (12, 35, 37) (9, 40, 41) (28, 45, 53) (11, 60, 61) (16, 63, 65) (33, 56, 65) (48, 55, 73) (13, 84, 85) (36, 77, 85) (39, 80, 89) (65, 72, 97) Question: Is the number of primitive triples infinite? Idea: every odd number can be expressed as a difference between two (consecutive) squares 5 = = 16-9 Hence, every odd square is part of a Pythagorean triple 9 = Euclid,300BC
49 Primitive triples smaller than 100: (3, 4, 5) (5, 12, 13) (8, 15, 17) (7, 24, 25) (20, 21, 29) (12, 35, 37) (9, 40, 41) (28, 45, 53) (11, 60, 61) (16, 63, 65) (33, 56, 65) (48, 55, 73) (13, 84, 85) (36, 77, 85) (39, 80, 89) (65, 72, 97) 2m+1 = 9 m = 4 m+1 = 5 2m+1 = 25 m = 12 m+1=13 2m+1 = 121 m = 60 m+1 = 61 2m+1 = 169 m = 84 m+1 = 85 Euclid,300BC
50 Primitive triples smaller than 100: (3, 4, 5) (5, 12, 13) (8, 15, 17) (7, 24, 25) (20, 21, 29) (12, 35, 37) (9, 40, 41) (28, 45, 53) (11, 60, 61) (16, 63, 65) (33, 56, 65) (48, 55, 73) (13, 84, 85) (36, 77, 85) (39, 80, 89) (65, 72, 97) m=7, n =2 Question: Closed from solution for all the triples? 1. m and n are relatively prime 2. one of m and n must be odd and the other even a = 49-4= 45 b = 2x7x2 = 28 c = = 53 Euclid,300BC
51 m=125, n =54
52 The Babylonians NO proofs... knew everything... They taught the how and not the why The Greeks documented the why via proofs
53 Babylonian tablets HOW Greek proofs WHY
54 Quiz #3
55 Quiz #3 10min Prove that is not rational. Namely, it cannot be expressed as, with p and q integers.
56 Prove that is not rational. Namely, it cannot be expressed as, with p and q integers. Proof for sqrt(2) p is not divisible by 3 p is divisible by 3 Assume that p and q are relatively prime (simplified) Reach a contradiction!! divisible by 4 = not divisible by 4 p odd? p odd, q even: NO p odd, q odd: NO p even? p even, q even: NO p even, q odd: NO QED
57 Due next week Thursday 4/25 Start early!
58
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60
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63 The range of digits is increasing
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