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(1)

IST 4

Information and Logic

(2)

mon tue wed thr fri

2 M1

1

9 M1

16

1 2

M2

23

PCP

30

2

M2

7

3

14

21

3 4

28

4 5

4

5

x= hw#x out x= hw#x due

Mx= MQx out Mx= MQx due

classno

oh oh

oh oh

oh oh

oh

oh oh oh

oh

oh oh oh

oh oh

oh = office hours oh

T

= today

T oh

oh

oh

sun

oh

revised 4/19/2018

PCP = Programing Challenge

midterms

oh

oh oh

(3)

- Lecture 1: Life – DNA sequences and evolution - Lecture 2: The human brain – natural languages

- Lecture 3: Artificial languages - numbers and writing

(limited) memory and innovation process (artificial languages)

information systems

- Lecture 4: Languages for quantities: Babylonians

- Lecture 5: Babylonian mathematics vs Greek mathematics - Lecture 6: Flow of ideas over time and space

Babylonians to Euclid to Leibniz

(4)

Babylonian tablets

‘HOW’ Greek proofs

‘WHY’

Euclid, 300BC 2000BC

(5)

Algorithms Greek proofs

‘WHY’

Euclid, 300BC

How are algorithms

similar to proofs?

(6)

Algorithms

How are algorithms similar to proofs?

premise / axioms / input

‘Legal’ steps

Theorem-QED

correct output

(7)

Squaring the Rectangle

Algorithms

(8)

Squaring the Rectangle

Tiling the rectangle with squares

Smallest number of squares

(9)

Can we find a smaller number of identical squares?

(10)

Can we find a smaller number of identical squares?

How?

(11)

Idea: Be greedy

How?

(12)

We are almost done!

Now what?

From Geometry to Numbers?

(13)

Euclidean algorithm for finding the GCD Greatest Common Divisor

15 40

GCD = 5

Euclid,300BC 10

15 5

10

(14)

??

(15)

The flow of ideas

2000BC Euclid, 300BC

(16)

Pythagoras 570-495 BC

(17)

Will be posted

on the class website

“In November 26, 1949, Albert Einstein published an essay in the Saturday Review of Literature in which he described two pivotal moments in his childhood.”

“The first involved a compass that his father showed him when he was four or five. Einstein recalled his sense of wonderment that the needle always pointed north, even

though nothing appeared to be pulling it in that direction. He came to a conclusion, then and there, about the structure of the physical world:

“Something deeply hidden had to be behind things.””

(18)

Will be posted

on the class website

“The second moment occurred soon after he turned twelve,

when he was given “a little book dealing with Euclidean plane geometry.””

The book’s “lucidity,” he wrote—the idea that a mathematical assertion could

“be proved with such certainty that any doubt appeared to be out of the question”— provoked

“wonder of a totally different nature.”

“Pure thought could be just as powerful as geomagnetism.”

(19)

How does pure

thought travel?

Pure thought could be just as

powerful as geomagnetism” S.

Strogatz

(20)

Communications: books, interaction

Trade and imperialism

How does pure

thought travel?

(21)

Babylonians and Egyptians

~5000 years ago

(22)

source: wikipedia

Greek - Alexander

~2500 years ago

(23)

Roman Empire

~2000 years ago

source: wikipedia

(24)

Arabs

~1200 years ago

Muhammad ibn Mūsā al-Khwārizmī

House of Wisdom, Bagdad

source: wikipedia

(25)

source: wikipedia Merv, 2014

Merv, 2014

Dunhuang

Trade:

Silk Road

(26)

The geography

and flow of ideas

number system

(27)

Roman numerals and the abacus

(28)

Roman

Numeral Number

I 1

V 5

X 10

L 50

C 100

D 500

M 1000

A Refresher on the Roman Numeral System

Large Roman

Numerals Number

V 5,000

X 10,000

L 50,000

C 100,000

D 500,000

M 1,000,000 LCD Monitor

(29)

I V

X L

C D V

M LL

DD

C M

VV X

IIIII XXXXX

V

CCCCC MMMMM

L D V

The Abacus:

It’s all About Syntax

What is the number?

(30)

I V

X L

C D V

M LL

DD

C M

VV X

IIIII XXXXX

V

CCCCC MMMMM

L D V

The Abacus:

It’s all About Syntax

What is the number?

DCI 601

(31)

I V

X L

C D V

M LL

DD

C M

VV X

IIIII XXXXX

V

CCCCC MMMMM

L D V

DCI 601

The Abacus:

It’s all About Syntax

Touch the middle: Yes or No

it is a binary mechanism

(32)

The first actual calculating mechanism known to us is the abacus, which is thought to have been invented

by the

Babylonians

sometime between 1,000 BC and 500 BC

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

Source: Wikipedia

(33)

The original concept referred to a flat stone covered with sand or dust, with pebbles being placed on lines drawn in the sand

The Abacus

Calculating Machine are Based on Syntax

In Phoenician the word abak means sand

In Hebrew the word abhaq

ק בָ אָ

means dust

Calculus is Latin for

pebble

? ?

Source: Wikipedia

(34)

The Abacus:

It’s all About Syntax

I V

X L

C D V

M LL

DD

C M

VV X

IIIII XXXXX

V

CCCCC MMMMM

L D V

2 1

5 1

(35)

I V

X L

C D V

M LL

DD

C M

VV X

IIIII XXXXX

V

CCCCC MMMMM

L D V

DCI 601

The Abacus:

It’s all About Syntax

(36)

I V

X L

C D V

M LL

DD

C M

VV X

IIIII XXXXX

V

CCCCC MMMMM

L D V

CCCCLXXXVIIII 489

The Abacus:

It’s all About Syntax

What is the number?

(37)

I V

X L

C D V

M

CCCCLXXXVIIII 489 DCI 601 +

The Abacus:

It’s all About Syntax

2 1

5 1

(38)

I V

X L

C D V

M

CCCCLXXXVIIII 489 DCI 601 +

The Abacus:

It’s all About Syntax

2 1

5 1

(39)

I V

X L

C D V

M

CCCCLXXXVIIII 489 DCI 601 +

The Abacus:

It’s all About Syntax

2 1

5 1

(40)

I V

X L

C D V

M

CCCCLXXXVIIII 489 DCI 601 +

The Abacus:

It’s all About Syntax

2 1

5 1

(41)

I V

X L

C D V

M

CCCCLXXXVIIII 489 DCI 601 +

The Abacus:

It’s all About Syntax

2 1

5 1

(42)

I V

X L

C D V

M

CCCCLXXXVIIII 489 DCI 601 +

The Abacus:

It’s all About Syntax

2 1

5 1

(43)

I V

X L

C D V

M

CCCCLXXXVIIII 489 DCI 601 +

The Abacus:

It’s all About Syntax

2 1

5 1

(44)

I V

X L

C D V

M

CCCCLXXXVIIII 489 DCI 601 +

The Abacus:

It’s all About Syntax

2 1

5 1

(45)

I V

X L

C D V

M

CCCCLXXXVIIII 489 DCI 601 +

The Abacus:

It’s all About Syntax

2 1

5 1

(46)

I V

X L

C D V

M

CCCCLXXXVIIII 489 DCI 601 +

The Abacus:

It’s all About Syntax

2 1

5 1

(47)

I V

X L

C D V

M

MLXXXX

CCCCLXXXVIIII 489 DCI 601 +

The Abacus:

It’s all About Syntax

What is the decimal representation?

??

(48)

I V

X L

C D V

M

MLXXXX 1 0 9 0

What’s surprising in this picture?

CCCCLXXXVIIII 489 DCI 601 +

The Abacus:

It’s all About Syntax

What is the decimal representation?

(49)

Roman Numerals and Base 10 Systems

I V

X L

C D V

M

MLXXXX

0 9

0 1

Roman numerals used for number

representation

For

calculation

: we used the abacus

The

representation in the abacus

is a

positional base 10

representation

(50)

From Physical (abacus) to Symbols

Algorizms

(51)

Algorizmi

A positional number system is a key enabler for efficient arithmetic operations

Operations are done on syntax

Muhammad ibn Mūsā al-Khwārizmī

ﺣ ﻣ د ﺑ ن ﻣ و ﺳ ﻰ اﻟ ﺧ و ا ر ز ﻣ

ﻲ 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

(52)

The Beginning of the “Algebra Book ” by “Algorizmi ” Everything requires computation...

(53)

The Beginning of the “Algebra Book ” by “Algorizmi ” Positional: order is important; the base – multiplication by 10;

from 1 to infinity...

(54)

Example from the “Algebra Book ” by “Algorizmi

It is rhetorical (words) no symbols

??

computation = single digit syntax manipulation

Left to right…

(55)

Algorithms and Algebra in Europe

Leonardo Fibonacci 1170-1250AD

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.

In 1202, at age 32, he published what he had learned in Liber Abaci, or Book of Calculation

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

Source: Wikipedia

(56)

Liber Abaci – First Chapter

Introduction of the syntax; from 1 to infinity...

(57)

Liber Abaci – First Chapter

Positional: order is important

(58)

Algorizmi

“brain surgery” and

(59)

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 10

2 2 3 4 5 6 7 8 9 10 11

3 3 4 5 6 7 8 9 10 11 12

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 10 11 12 13 14 15 16 8 8 9 10 11 12 13 14 15 16 17 9 9 10 11 12 13 14 15 16 17 18

(60)

In first grade:

We use our BRAIN for remembering

Algorizmi’s syntax

(61)

Arithmetic boxes

decimal

(62)

Algorithms – Syntax Manipulation

2 symbol adder digit 1 digit 2

carry

sum

carry

(63)

Algorithms – Syntax Manipulation

2 symbol adder

8 6

digit 1 digit 2

carry

sum

carry 0

(64)

Algorithms – Syntax Manipulation

2 symbol adder

8 6

4

digit 1 digit 2

carry

sum

1 0 carry

(65)

2 symbol adder c

c

d1 d2

c

1891 + 8709 = ??

2 symbol adder c

c

d1 d2

c c 2 symbol adder c

d1 d2

c c 2 symbol adder c

d1 d2

c

(66)

2 symbol adder c

c

1 d2

c c 2 symbol adder c

8 d2

c c 2 symbol adder c

9 d2

c c 2 symbol adder c

1 d2

0

1891 + 8709 = ??

(67)

2 symbol adder c

c

1 8

c c 2 symbol adder c

8 7

c c 2 symbol adder c

9 0

c c 2 symbol adder c

1 9

0

1891 + 8709 = ??

(68)

2 symbol adder c

c

1 8

c c 2 symbol adder c

8 7

c c 2 symbol adder c

9 0

c 1 2 symbol adder 0

1 9

0

1891 + 8709 = ??

(69)

2 symbol adder c

c

1 8

c c 2 symbol adder c

8 7

c 1 2 symbol adder

0

1 9

0 2 symbol adder

1

0

9 0

1

1891 + 8709 = ??

(70)

2 symbol adder c

c

1 8

c 1 2 symbol adder 6

8 7

1 1 2 symbol adder 0

9 0

1 1 2 symbol adder 0

1 9

0

1891 + 8709 = ??

(71)

2 symbol adder 1

0

1 8

1 1 2 symbol adder 6

8 7

1 1 2 symbol adder 0

9 0

1 1 2 symbol adder 0

1 9

0

1891 + 8709 = ??

(72)

2 symbol adder 1

0

1 8

1 1 2 symbol adder 6

8 7

1 1 2 symbol adder 0

9 0

1 1 2 symbol adder 0

1 9

0

1891 + 8709 = 10,600

(73)

Algorithm =

a procedure for syntax manipulation

Dear Algo and Fibo,

we get confused with those large tables... can we use a

smaller syntax?

(74)

The Binary

(75)

When was the binary system “invented”?

(11010100111) 2 (1703) 10

1024+512+128+32+4+2+1 = 1703

(76)

Leibniz – Binary System

Gottfried Leibniz 1646-1716

(77)

Gottfried Leibniz 1646-1716

(78)

Gottfried Leibniz 1646-1716

(79)

Gottfried Leibniz 1646-1716

8

??

Use the smallest syntax possible Binary – 0 and 1

(80)

Binary Addition

carry

Gottfried Leibniz 1646-1716

(81)

Binary Multiplication

Addition of

shifted versions

3 6

5 5

10 20

Gottfried Leibniz 1646-1716

(82)

Leibniz:

No Need for Flash Cards!

(83)

Invented the Binary System?

Gottfried Leibniz 1646-1716

(84)

Leibniz – Binary System

Gottfried Leibniz 1646-1716

Wen Wang (who flourished in about 1150 BC) is traditionally thought to have been author of the present hexagrams

63

(85)

Leibniz – Binary System

Gottfried Leibniz 1646-1716

(86)

We are learning about people with

good ideas and strong initiative…

(87)

• You are invited to write short essay on the topic of the

Magenta Question. Extra Credit of 3 points Entrepreneurship:

Good ideas and strong initiative!

MQ2

Due Tuesday 5/1/2018 at 10pm

Your story – ideas, dreams

Someone you know

Someone known

Anything…

(88)

• You are invited to write short essay on the topic of the

Magenta Question.

• Recommended length is 3 pages (not more)

Submit the essay in PDF format to istta4@paradise.caltech.edu file name lastname-firstname.pdf

No collaboration. No extensions

MQ2

Grading of MQ:

3 points (out of 105)

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 Due Tuesday 5/1/2018 at 10pm

(89)

Quiz time

(90)

Quiz #4 – 10min

Compute the multiplication base-2 (in base-2) Show your work!

1010 x 1000

?

1011 x 1011

?

100111 x 101

?

Addition of

shifted versions

References

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