COS 111 Review Session 1

1
COS 111 Review Session 1

• Friday, March 4, 2005

2
Outline

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

3
Can you say five ?
4
Say five

• Dutch vijf
• German fünf
• French cinq
• Spanish cinco
• Hindi paanch
• Slang -- Lincoln
• Math -- 5

5
Say five

• Dutch vijf
• German fünf
• French cinq
• Spanish cinco
• Hindi paanch
• Slang Lincoln
• Math -- 5

Spoken form
6
Say five

• Dutch vijf
• German fünf
• French cinq
• Spanish cinco
• Hindi paanch
• Slang Lincoln
• Math -- 5

Visual form
7
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
  doesnt go
• Math has many sub-dialects binish, tertiarist,
  octalish, hexadecimalish, AnyNish (I am making
  the names up but thats not the point ))

8
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

9
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

10
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 doesnt matter what the base is as long as its
  fixed

11
Lets count

• 0
• 1
• 2
• 3
• 4
• What now ?
• We need to combine our symbols to come up write
  bigger numbers

12
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.

13
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.

14
Lets count

• 0
• 1
• 2
• 3
• 4
• 10

10 fif 11 fif one 12 fif two 13 fif
three 14 fif four 20 -- twofif
15
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
16
Penta System

• A number ABCDE is hence
• A fif-fif-fif-fif
• B fif-fif-fif
• C fif-fif
• D fif
• E
• Afif4 Bfif3 Cfif2 Dfif E

17
b System

• A number Xk-1.X0 in base b is
• Sum of Xi-1bi for i from 0 to k-1
• All rules of multiplication, addition,
  subtraction are similar to what we normally do in
  base 10 numbers

18
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.

19
Before we move to next topic

• Old number systems joke
• Why is Christmas like Halloween ?
• Because 31 oct 25 dec

20
Outline

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

21
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

22
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

23
Distributive Laws

• E (E1.E2...En) (EE1).(EE2)...(EEn)
• E.(E1E2...En) (E.E1) (E.E2)... (E.En)

24
DeMorgans Laws

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

25
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

26
Outline

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