IST 4 Information and Logic

IST 4

Information and Logic

mon tue wed thr fri

3 _M1

1

10 _M1

17

1 2

24

2

1

8

3

15

3 4

22 4 5

29

5

5

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

Mx= MQx out Mx= MQx due

midterms

oh 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

oh

sun

Babylonian tablets

‘HOW’ Greek proofs

‘WHY’

Euclid, 300BC 2000BC

A4

A4

??

A4

Squaring the Rectangle

Squaring the Rectangle

Euclid, 300BC

Tiling the rectangle with squares

Can we find a smaller number of identical squares?

Can we find a smaller number of identical squares?

How?

Idea: Be greedy

How?

We are almost done!

Now what?

From Geometry to Numbers?

Euclidean algorithm for finding the GCD Greatest Common Divisor

15 40

GCD = 5

Euclid,300BC 10

15 5

10

??

The flow of ideas

2000BC Euclid, 300BC

Pythagoras 570-495 BC

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.””

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.”

How does pure

thought travel?

“ Pure thought could be just as

powerful as geomagnetism” ^S.

Communications via interaction Trade and imperialism

How does pure

thought travel?

Babylonians and Egyptians

~5000 years ago

source: wikipedia

Greek - Alexander

~2500 years ago

Roman Empire

~2000 years ago

source: wikipedia

Arabs

~1200 years ago

Muhammad ibn Mūsā al-Khwārizmī

source: wikipedia Merv, 2014

Merv, 2014

Merv, 2014

The geography and flow of ideas

number system

Roman numerals and the abacus

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

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

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?

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

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

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

by the

Babylonians

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

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

קָבָא

pebble

? ?

Source: Wikipedia

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

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?

I V

X L

C D V

M

CCCCLXXXVIIII 489 DCI 601 +

The Abacus:

It’s all About Syntax

2 1

5 1

I V

X L

C D V

M

CCCCLXXXVIIII 489 DCI 601 +

The Abacus:

It’s all About Syntax

2 1

5 1

I V

X L

C D V

M

CCCCLXXXVIIII 489 DCI 601 +

The Abacus:

It’s all About Syntax

2 1

5 1

I V

X L

C D V

M

CCCCLXXXVIIII 489 DCI 601 +

The Abacus:

It’s all About Syntax

2 1

5 1

I V

X L

C D V

M

CCCCLXXXVIIII 489 DCI 601 +

The Abacus:

It’s all About Syntax

2 1

5 1

I V

X L

C D V

M

CCCCLXXXVIIII 489 DCI 601 +

The Abacus:

It’s all About Syntax

2 1

5 1

I V

X L

C D V

M

CCCCLXXXVIIII 489 DCI 601 +

The Abacus:

It’s all About Syntax

2 1

5 1

I V

X L

C D V

M

CCCCLXXXVIIII 489 DCI 601 +

The Abacus:

It’s all About Syntax

2 1

5 1

I V

X L

C D V

M

CCCCLXXXVIIII 489 DCI 601 +

The Abacus:

It’s all About Syntax

2 1

5 1

I V

X L

C D V

M

CCCCLXXXVIIII 489 DCI 601 +

The Abacus:

It’s all About Syntax

2 1

5 1

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?

??

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?

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

positional base 10

From Physical (abacus) to Symbols

Algorizms

Algorizmi

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

Operations are done on syntax

Muhammad ibn Mūsā al-Khwārizmī

يﻱمﻡزﺯرﺭاﺍوﻭخﺥلﻝاﺍ ىﻯسﺱوﻭمﻡ نﻥبﺏ دﺩمﻡحﺡمﻡ 780-850AD A Persian mathematician, who wrote on Hindu-Arabic

numerals and was among the first to use

zero

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 ”

It is rhetorical (words) no symbols

??

computation = single digit syntax manipulation

Left to right…

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

Liber Abaci – First Chapter

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

Liber Abaci – First Chapter

Positional: order is important

Algorizmi

“brain surgery” and

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

In first grade:

We use our BRAIN for remembering

Algorizmi’s syntax

Arithmetic boxes

decimal

Algorithms – Syntax Manipulation

2 symbol adder

digit 1 digit 2

carry

sum

carry

Algorithms – Syntax Manipulation

2 symbol adder

8 6

digit 1 digit 2

carry

sum

0 carry

Algorithms – Syntax Manipulation

2 symbol adder

8 6

4

digit 1 digit 2

carry

sum

1 0 carry

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

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 = ??

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 = ??

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 = ??

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 = ??

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 = ??

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 = ??

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

Algorithm =

a procedure for syntax manipulation

Dear Algo and Fibo,

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

smaller syntax?

The Binary

When was the binary system “invented”?

(11010100111) ² (1703) ¹⁰

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

Leibniz – Binary System

Gottfried Leibniz 1646-1716

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

8

??

Binary Addition

carry

Gottfried Leibniz 1646-1716

Binary Multiplication

Addition of

shifted versions

3 6

5 5

10 20

Gottfried Leibniz 1646-1716

Leibniz:

No Need for Flash Cards!

Invented the Binary System?

Gottfried Leibniz 1646-1716

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

Leibniz – Binary System

Gottfried Leibniz 1646-1716

We learned about

gifted people….

Everyone has a gift!

"I have no special talents

I am only passionately curious"

Albert Einstein

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

Magenta Question. Extra Credit of 3 points Everyone has a gift!

MQ2

someone you know you

anyone...

•  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 Everyone has a gift!

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

1.  Diana Ardelean - The Gift of Becoming a GO Player 2.  Shalini Majumdar - The Gift of Companionship

3.  Alex Geffner-Mihlsten - The Gift of Names and Lives 4.  Sumana Mahata - The Gift of Laughter

5.  Parth Shah - My Mother has the Greatest of all Gifts!

6.  Tim Menninger – Colors

7.  Nasser Al Rayes - Boo (The Gift of Being Tall) 8.  Sophia Yuenchih Chen - The Gift of Uniqueness

1.  Christopher Haack: The gift of resilience 2.  Joon Lee: Settling is not an option

3.  Spencer Strumwasser: The gift of dyslexia 4.  Richard Zhu: The gift of memory

5.  Ashwin Hari: The gift of musical composition 6.  Jessica Nassimi: Evolution—a gift in disguise 7.  Serena Delgadillo: The gift of self-expression 8.  Megan Keehan: Gift of motherliness

9.  Zane Murphy: Grandmother and the piano

1.  May Chen - The Gift of a Bucket List 2.  Rohan Doshi - Gifts Come in Threes

3.  Nikhil Ghosh - The Gift of Not Being Number One 4.  Joey Hong – The Gift of Imperfection

5.  Suna Kim – The Gift of Playing the Piano

6.  Rachel Morton - A Gift that I Hope I Have

7.  Andrew Wang - Helping Another Realize His Gift 8.  Bianca Yang –The Gift of Art and Two Languages

Quiz time

Problem 2:

Prove that Problem 1:

Quiz #3 – 10min

is not rational.

Namely, it cannot be expressed as , with p _and q    _integers.

Solve only one of the problems

Compute the following using Euclid’s algorithm

=  ??  

Show your work