COS 111 Review Session 1

Friday, March 4, 2005

Outline

• All About Numbers• Boolean/Logic Circuits• Assignment 4• Questions

Can you say five ?

Say five

Dutch – vijf

German – fünf

French – cinq

Spanish – cinco

Hindi – paanch

Slang -- Lincoln

Math -- 5

Say five

Dutch – vijf

German – fünf

French – cinq

Spanish – cinco

Hindi – paanch

Slang – Lincoln

Math -- 5

Spoken form

Say five

Dutch – vijf

German – fünf

French – cinq

Spanish – cinco

Hindi – paanch

Slang – Lincoln

Math -- 5

Visual form

When is five not five

When using different langauges GM called one of their small cars "Nova". They didn't

sell too many in Spain where 'NoVa' means “doesn’t go”

Math has many sub-dialects – binish, tertiarist, octalish, hexadecimalish, AnyNish (I am making the names up

but that’s not the point :))

How much is 10 ?

• You need to know what language it is being spoken in– V in roman numerals refers to decimal 5 but refers to

decimal 31 in hexatridecimalish

How do we translate from one dialect to another ? We need to understand the structure of math-dialects

Closer look at Roman Numerals

• Pick a few agreed upon quantities – I, V, X, L, C, D, M

• Express all other numbers as sums and differences of above – 7 is VII, 19 is XIX, 10000 is MMMMMMMMMM

• Not very convenient as numbers become large• Structure also cumbersome – 41 is XLI or IXL

Penta System

• Instead of sums and differences, can we use multiplication to provide structure to number ?

• MMMMMMMMMM can be X-M• But a odd collection I, V, X, L, C, D, M wont do• Pick 5 symbols – 0, 1, 2, 3, 4. Why 5 ?

– Its arbitrary.

– It doesn’t matter what the base is as long as its fixed

Lets count…

0

1

2

3

4

What now ?

We need to combine our symbols to come up write bigger numbers

Lets count…

0

1

2

3

4

What now ?

We have made one pass over all symbols. So lets note down that fact. One pass and no more.

Lets count…

0

1

2

3

4

10 – lets call this a fif

We now use position of a symbol in a number to hold its value.

Lets count…

0

1

2

3

4

10

10 – fif

11 – fif one

12 – fif two

13 – fif three

14 – fif four

20 -- twofif

Lets count…

0

1

2

3

4

10

10

11

12

13

14

20

20

21

22

23

24

30

30

31

32

33

34

40

40 – fourfif

41 – fourfif one

42 – fourfif two

43 – fourfif three

44 – fourfif four

100 – fiffif

We now use position of a symbol in a number to hold its value

Penta System

• A number ABCDE is hence– A fif-fif-fif-fif +

– B fif-fif-fif +

– C fif-fif +

– D fif +

– E

– A*fif^4 + B*fif^3 + C*fif^2 + D*fif + E

b System

• A number Xk-1….X0 in base b is

– Sum of Xi-1*b^i for i from 0 to k-1

• All rules of multiplication, addition, subtraction are similar to what we normally do in base 10 numbers

Lets do some practice

• Conversion from one base to another• Subtraction, addition, multiplication in any base• Suggest numbers and operations and we work it

out together.

Before we move to next topic…

• Old number systems joke – – Why is Christmas like Halloween ?

– Because 31 oct = 25 dec

Outline

• All About Numbers• Boolean/Logic Circuits• Assignment 4• Questions

Boolean Algebra

• Shorthand for writing and thinking about logic circuits

• Notation – ' is a NOT

– . is an AND

– + is an OR

– 1 represents TRUE

– 0 represents FALSE

Some simple rules

• (A ') ' = A• (A ' + A) = 1• A + 0 = A• A + 1 = 1• (A '.A) = 0• A.0 = 0• A.1 = A• A + A = A• A.A = A

Distributive Laws

• E +(E1.E2...En) = (E+E1).(E+E2)...(E+En)• E.(E1+E2+...En) = (E.E1) + (E.E2)... + (E.En)

DeMorgan’s Laws

• (E1 + E2 + ... + En)' = E1'.E2'....En'• (E1.E2...En)' = E1' + E2' + ... + En'

Lets try some examples

• x'.y + x.y + x• x.y.z + x'.y.z + x'.y'.z + x'.y'.z + x.y'.z' + x.y'.z• x'.y + x'.y' + x.y' + x.y

Outline

• All About Numbers• Boolean/Logic Circuits• Assignment 4• Questions