IST 4 Information and Logic

(1)

IST 4

Information and Logic

(2)

Everyone has a gift! MQ1

Due Today by 10pm

Please email PDF

lastname-firstname.pdf

to [email protected]

Have your name inside the file as well...

HW #1

Due Tuesday, 4/17

2:30pm in class

(3)

mon tue wed thr fri

2 _M₁

1

9 _M₁

16

1 2

^M²

23

2

30 _M₂

7

3

14

3 4

21

4 5

28

4

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

= office hours

^oh

= today

T oh

oh

sun

T

(4)

Quiz #1

Review

(5)

Quiz #1 – 10min

What are the 2 important properties of a language?

Question 1:

1. Verbal and Written

3. Syntax and Semantics

4. Finite Building Blocks and Separation 2. Reason about real paradigms

Reason about fictional paradigms

DNA, natural human language, mathematics, music…

(6)

sepa ration

building blocks

Separation A

Between syntax and semantics Separation B

between what is represented

and reality, feasibility, time, space, ...

l a n g u a g e s

(7)

is a space between two digits

31x60 + 31 = 1891 Base 60:

Question 2:

Write the value in decimal of the following numbers Show your work

Quiz #1 – 10min

2x60 + 19x1 = 139 1x3600 + 2x60 + 31x1 = 3751

The weights: 216,000 | 3,600 | 60 | 1

(8)

So Far…

- Lecture 1: Life – DNA sequences and evolution - Lecture 2:

The human brain – natural languages, memory, teaching - Lecture 3: Artificial languages - numbers and writing

(limited) memory and innovation process (artificial languages)

information systems

(9)

Our number sense is limited...

We are doing well because we

found a way to

(10)

language for quantities

Our number sense is limited...

(11)

Our number sense is limited...

managing time…

(12)

Our number sense is limited...

managing trade…

(13)

Our memory sense is limited...

an idea: recording our speech Writing is the recording of the

sounds of our verbal language

reb ^us

(14)

Today

- Lecture 1: Life – DNA sequences and evolution - Lecture 2:

The human brain – natural languages, memory, teaching - Lecture 3: Artificial languages - numbers and writing

- Lecture 4: the development of languages for quantities - Babylonian mathematics

- Number representation systems

(15)

Deciphering the

Babylonians

(16)

Otto Neugebauer 1899-1990

Austria (EE) –> Germany (Math/history) –> Denmark –> US (Brown U)

Mathematics –> History of Exact Sciences

His son:

Gerry Neugebauer 1932-2014

(PhD ‘60)

Millikan Professor of Physics, Emeritus

• Born in Austria

• Participated in WWI, was POW

• Germany 1933, lost his job

• Escaped Germany to Denmark in 1934

• Escaped Europe to the US in 1939, WWII

1945

(17)

Otto Neugebauer 1899-1990

Denmark –> US (PhD 1957, Brown U) -> 1961, Yale U

Asger Aaboe 1922–2007

Mathematics –> Teaching Math ->History of Exact Sciences

(18)

Otto Neugebauer 1899-1990 Asger Aaboe

1922–2007

Today, you will get to be Otto and Asger!

(19)

1 32

45 67 8 109 1112 1314

(20)

1 32

45 67 8 109

1112 1314

15 1617

1819?

20 40 50

??

9 18

2736 4554

63 1,12

1,301,21 1,481,39 1,57 2,6 1,3

2,15 2,242,33 2,422,51 3,04,30 6,07,30

72 450

81

30 270

(21)

1 32

45 67 8 109

1112 1314

15 1617

1819?

20 40 50

??

9 18

2736 4554

63 1,12

1,301,21 1,481,39 1,57 2,6 1,3

2,15 2,242,33 2,422,51 3,04,30 6,07,30

72 450

81

270

Multiplication table for 9

30

(22)

??

Multiplication

instead of division

(23)

(24)

(25)

fraction

No fractional point

(26)

fraction

7 and 11 and other numbers are missing? Why?

No fractional point

(27)

Is 24 a regular number?

A number that its prime factors are at most 5 is called a regular number

Is 900 a regular number?

Is 896 a regular number?

2x2x2x3

2x2x2x2x2x2x2x7 2x2x3x3x5x5

a must be a regular number Q: What are the possible values for a ^?

a, b are integers, must divide

(28)

language for quantities

Our number sense is limited...

time trade

geometry

(29)

The Babylonians

knew everything!

even Geometry...

3 45 9

(30)

The Babylonians

knew everything!

even Geometry...

3 45 9

Yale U. 3/2018 - YBC

(31)

The Babylonians

knew everything!

even Geometry...

3 45 9

A square hint

4

16

1

(32)

The Babylonians

knew everything!

even Geometry...

4 16 1

A square hint

A = (C/4) x (C/4) A = (C x C)/16

A = Area

C = Circumference

(33)

The Babylonians

knew everything!

even Geometry...

C = 4

CxC = 16 A=1

A= (16)/16 = 1

A square hint

A = (C/4) x (C/4) A = (C x C)/16

A = Area

C = Circumference

(34)

The Babylonians

knew everything!

even Geometry...

3 45 9

? ?

?

(35)

The Babylonians

knew everything!

even Geometry...

3 45 9

(36)

The Babylonians

knew everything!

even Geometry...

3 45 9

(37)

The Babylonians

knew everything!

even Geometry...

12 ? 2,24

12 12 again!!

(38)

Algorithms

(39)

separation

12137+35823

compute ^{using the} rules of the syntax

independent of the semantics using building blocks

building Blocks

algorithms

(40)

The language of numbers

Our first algorithm

Translation between languages

(41)

Positional number systems

10

2

60

(42)

Base-b Positional Systems

Base-10 is embedded in our language and thought

(43)

Base-b Conversion to Base-B

Translation between languages

English to Spanish

French Spanish

French to English

English

(44)

Base-b Conversion to Base-B

Successive division by B using base-10 arithmetic

Base b to base 10 Base 10 to base B

b 10 B

Sum the corresponding weights using base-10 arithmetic

Translation between languages

(45)

Successive division by B using base-10 arithmetic

Base b to base 10 Base 10 to base B

b 10 B

Sum the corresponding weights using base-10 arithmetic

Base-b Conversion to Base-B

Translation between languages

(46)

Base-b Conversion to Base-B

Translation between languages

8 16

64

128

+ + +

(47)

8 16

64 128

Base-b: Conversion from Base 10

Conversion from base-10 to binary:

4 2 1

32 32

64 16 8 4 2 1

216 108

Idea: discover the blue blocks!

Odd number – the right most block is blue Divide by 2 – to expose the next block...

If odd – subtract 1

Our first algorithm - syntax manipulation

Even number – the right most block is yellow

(48)

8 16

64 128

Base-b: Conversion from Base 10

Conversion from base-10 to binary:

4 2 1

32 32

16 8 4 2 1 32

16 8 4 2 1

64 16

8 4 2 1

4 2 1

2 1

1

216 108 54 27 13 6 3 1

The PPT COMPUTER

(49)

8 16

64 128

Base-b: Conversion from Base 10

Conversion from base-10 to binary:

4 2 1

32 32

16 8 4 2 1 32

16 8 4 2 1

64 16

8 4 2 1

4 2 1

2 1

1

216 108 54 27 13 6 3 1

The PPT COMPUTER

(50)

8 16

64 128

Base-b: Conversion from Base 10

Conversion from base-10 to binary:

4 2 1

32 32

16 8 4 2 1 32

16 8 4 2 1

64 16

8 4 2 1

4 2 1

2 1

1

216 108 54 27 13 6 3 1

A faster PPT COMPUTER

(51)

The language of numbers

weighted and positional

(52)

Babylonians and Egyptians

~5000 years ago

weighted

positional

(53)

The Egyptians Preferred 10

2x100 + 7x10 + 6x1 = 276

How will you represent a trillion

1,000,000,000,000?

separation

However, not a finite number of building blocks!

(54)

Number Systems

weighted positional system weighted system

4x60 + 36x1 = 276

276

2x100 + 7x10 + 6x1 = 276

2x100 + 1x50 + 2x10 + 1x5 + 1x1= 276

CCLXXVI

(55)

0

bounded syntax

What does a positional

number system have that is

unique?

(56)

Number Systems

weighted positional system weighted system

4x60 + 36x1 = 276

276

2x100 + 7x10 + 6x1 = 276

2x100 + 1x50 + 2x10 + 1x5 + 1x1= 276

CCLXXVI

finite alphabet unbounded alphabet

(57)

Can we represent a number in a

positional system without a 0?

Answer: Yes

How will you represent 10 without a 0?

ide A - represent 10 with a new digit: A

No 0

??

How will you represent 100 without a 0?

Assume base 10

100 = 9A

100 = 90 + 10

(58)

base 10

No 0

base 10 no-0 same weights

Different digits

(59)

base 10

base 10 no-0

Q: How many different quantities can

be represented by at most two digits? base 10 no-0 ^{base 10} 100

110 0-99

1-AA

(60)

Base-10 No-0 Positional System

Conversion from decimal to decimal no-zero?

Start from the right, translate X0 to (X-1)A

Repeat until there are no zeros

(61)

Conversion from base-10 to base-10 no zero

• Start from the right, translate X0 to (X-1)A

• Repeat until there are no zeros

You will explore this

number system in HW#2

…and now in quiz #2

(62)

Quiz #2

(63)

IST 4 Information and Logic

IST 4

Information and Logic

Everyone has a gift! MQ1

Due Today by 10pm

Please email PDF

lastname-firstname.pdf

to [email protected]

Have your name inside the file as well...

HW #1

Due Tuesday, 4/17

2:30pm in class

mon tue wed thr fri

1

1 2

2

3

3 4

4 5

5

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

Mx= MQx out Mx= MQx due

= office hours

= today

sun

Quiz #1

Review

Quiz #1 – 10min

What are the 2 important properties of a language?

Question 1:

1. Verbal and Written

3. Syntax and Semantics

4. Finite Building Blocks and Separation 2. Reason about real paradigms

Reason about fictional paradigms

DNA, natural human language, mathematics, music…

sepa ration

building blocks

Separation A

Between syntax and semantics Separation B

between what is represented

and reality, feasibility, time, space, ...

l a n g u a g e s

is a space between two digits

31x60 + 31 = 1891 Base 60:

Question 2:

Write the value in decimal of the following numbers Show your work

Quiz #1 – 10min

2x60 + 19x1 = 139 1x3600 + 2x60 + 31x1 = 3751

The weights: 216,000 | 3,600 | 60 | 1

So Far…

- Lecture 1: Life – DNA sequences and evolution - Lecture 2:

The human brain – natural languages, memory, teaching - Lecture 3: Artificial languages - numbers and writing

(limited) memory and innovation process (artificial languages)

information systems

Our number sense is limited...

We are doing well because we

found a way to

language for quantities

Our number sense is limited...

Our number sense is limited...

managing time…

Our number sense is limited...

managing trade…

Our memory sense is limited...

an idea: recording our speech Writing is the recording of the

sounds of our verbal language

reb us

Today

- Lecture 1: Life – DNA sequences and evolution - Lecture 2:

The human brain – natural languages, memory, teaching - Lecture 3: Artificial languages - numbers and writing

- Lecture 4: the development of languages for quantities - Babylonian mathematics

- Number representation systems

Deciphering the

Babylonians

Mathematics –> History of Exact Sciences

• Born in Austria

• Participated in WWI, was POW

• Germany 1933, lost his job

• Escaped Germany to Denmark in 1934

• Escaped Europe to the US in 1939, WWII

reb ^us

a must be a regular number Q: What are the possible values for a ^?