Chapter 0 1

Numbers & Logic

Bits & Pieces

Chapter 0 2

Base 10 example

• Decimal Number 9 7 0 1

• Place 4 3 2 1

• Place - 1 3 2 1 0

• 10(place - 1) 103 102 101 100

• =============================== • = 9*1000 + 7*100 + 0*10 + 1*1

• = 9701

Chapter 0 3

Numeric Values– The numeric value of a set of digits is

determined as:• The sum of the products of each digit and its

corresponding place value,

• where the place value is the numeric-base raised to the place - 1.

Chapter 0 4

Base 2 example

• Binary Number 0 1 0 1

• 2(place - 1) 23 22 21 20

• ===============================

• = 0*8 + 1*4 + 0*2 + 1*1

• = 5

Chapter 0 5

A general exampleBase n

• Binary Number 0 1 0 1

• n(place - 1) n3 n2 n1 n0

• ===============================

• 0*(n * n* n) + 1*(n*n) + 0* n + 1*1

Chapter 0 6

Commonly Used Systems

• Binary Base 2

• Octal Base 8

• Decimal Base 10

• Hexadecimal Base 16

Chapter 0 7

Legal Digits

• What are the legal digits?

• Start at zero and stop at the base - 1

• Binary 0, 1

• Octal 0, 1, 2, 3, 4, 5, 6, 7

• Decimal 0, 1, 2, 3, 4, 5, 6, 7, 8, 9

• Hex 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F

Chapter 0 8

What is the decimal value of?

• 10101 base 2

• 10101 base 8

• 10101 base 10

• 10101 base 16

Chapter 0 9

• 0000 00 00 0

• 0001 01 01 1

• 0010 02 02 2

• 0011 03 03 3

• 0100 04 04 4

• 0101 05 05 5

• 0110 06 06 6

• 0111 07 07 7

• 1000 10 08 8

• 1001 11 09 9

• 1010 12 10 A

• 1011 13 11 B

• 1100 14 12 C

• 1101 15 13 D

• 1110 16 14 E

• 1111 17 15 F

• Counting in Binary, Octal, Decimal and Hexadecimal

• A single Hex digit can be used to represent the value of four binary digits

Chapter 0 10

Hex = Binary Shorthand

• Hexadecimals are often used as a shorthand for large binary values.

• This shorthand is useful for specifying memory locations, e.g.

• Decimal - 16,274,482

• Binary - 111110000101010000110010

• Hex - F85432

Chapter 0 11

Binary to Hex

• Each Hexadecimal digit represents four binary digits

• 1111 1000 0101 0100 0011 0010

• F 8 5 4 3 2

Chapter 0 12

Binary to Octal

• Each Octal digit represents three binary digits

• 111 110 000 101 010 000 110 010

• 7 6 0 5 2 0 6 2

Chapter 0 13

ASCII & EBCDIC

• American Standard Code for Information Interchange– ASCII @ Wikipedia

• Binary Coded Decimals– Binary Coded Decimals

• Extended Binary Coded Decimals– EBCDIC @ Wikipedia

• Unicode– Unicode @ Wikipeda

Chapter 0 14

Boolean Logic

• A two valued logic often used in computers and information systems.

• The only legal values in Boolean Logic are– TRUE– FALSE

Chapter 0 15

Logical Values

• Logical values can only be True or False

• Similar to numeric values, logical values can be combined into logical expressions using logical operators.

• The logical operators are: not and or < <= = >= >

Chapter 0 16

Logical Expressions

• a logical-expression is any expression that evaluates to False or True

• False

• True• not logical-expression

• logical-expression and logical-expression

• logical-expression or logical-expression

Chapter 0 17

Logical Expressionsare not unlike

Numerical Expressions• A numerical-expression is any expression that

evaluates to a legal numerical value.

• Examples of numerical expressions:– 3– -4– 3 + 8 / 2– (3 + 8) / 2

Chapter 0 18

Numerical Operators

• Unary operators have only one argument– the positive and negative signs are the unary

numerical operators.– + -

• Binary operators require two arguments– addition, subtraction, multiplication, division,

and exponentiation are the binary operators– + - * / ^

Chapter 0 19

You’ve probably already used logical expressions

• The relational operators > >= = < <= evaluate to logical results.

• Example– the expression 3 + 5 <= 8 - 4 evaluates to a

value of False, so it is a logical expression.– Note that here we have combined numerical

expressions with relational operators to form a logical expression.

Chapter 0 20

Logical Operators

• Unary operators have only one argument– not is the only unary logical operator.– not

• Binary operators require two arguments– conjunction and disjunction are the binary

operators – and or

Chapter 0 21

Truth TablesA NOT A

F T

T F

A B A AND B

F F F

F T F

T F F

T T T

A B A OR B

F F F

F T T

T F T

T T T

Chapter 0 22

Operator Precedence

• Higher precedence evaluate first,• Equal precedence evaluate left to right• Parenthesis can be used to modify the order of precedence, expressions inside parenthesis are evaluated first.

Chapter 0 23

Operator Precedence

- (unary)

* / div mod numerical operators

+ -

< = >= > <= relational operators

not

and logical operators

or

Chapter 0 24

Evaluation of Logical Expressions

• A = True

• B = False

• Given the above evaluate the following:

• A or B => True

• A and B => False

• 3 > 7 or A => TRUE

• (3 < 7) and not A => False

Chapter 0 25

Complex Logical Expression

• A = True B = False C = True D =False

• A or not B and not (3 + 7 <= 10 / 2) or C and D

Chapter 0 26

Complex Logical Expression

• A = True B = False C = True D =False

• A or not B and not (3 + 7 <= 10 / 2) or C and D

• T or not F and not (3 + 7 <= 10 / 2) or T and F

Chapter 0 27

Complex Logical Expression

• A = True B = False C = True D =False

• A or not B and not (3 + 7 <= 10 / 2) or C and D

• T or not F and not (3 + 7 <= 10 / 2) or T and F

• T or not F and not ( 10 <= 5 ) or T and F

• T or not F and not ( F ) or T and F

Chapter 0 28

Complex Logical Expression

• A = True B = False C = True D =False

• A or not B and not (3 + 7 <= 10 / 2) or C and D

• T or not F and not ( F ) or T and F

• T or T and T or T and F

Chapter 0 29

Complex Logical Expression

• A = True B = False C = True D =False

• A or not B and not (3 + 7 <= 10 / 2) or C and D

• T or T and T or T and F

• T or T or F

Chapter 0 30

Complex Logical Expression

• A = True B = False C = True D =False

• A or not B and not (3 + 7 <= 10 / 2) or C and D

• T or T or F

• T or F

• T

Chapter 0 31

Normal Forms

• Conjunctive Normal Form– A and B and C and D and E– any false value makes the expression false

• Disjunctive Normal Form– A or B or C or D or E– any true value makes the expression true

Chapter 0 32

Computer Time

• millisecond 10-3 = 1/1,000

• microsecond 10-6 = 1/1,000,000

• nanosecond 10-9 = 1/1,000,000,000

• picosecond 10-12 = 1/1,000,000,000,000

• femtosecond 10-15 = 1/1,000,000,000,000,000

• Conversion of Time Units

Chapter 0 33

Computer Units

• Thousand 103 = 1,000• Kilobyte 210 = 1,024• Million 106 = 1,000,000• Megabyte 220 = 1,048,576• Billion 109 = 1,000,000,000• Gigabyte 230 = 1,073,741,824• Trillion 1012 = 1,000,000,000,000• Terabyte 240 = 1,099,511,627,776

Chapter 0 34

Bigger Units• Trillion 1012 = 1,000,000,000,000• Terabyte 240 = 1,099,511,627,776• Quadrillion 1015 = 1,000,000,000,000,000• Petabyte 250 = 1,125,899,906,842,624• Quintillion 1018 = 1,000,000,000,000,000,000• Exabyte 260 = 1,152,921,504,606,846,976• Sextillion 1021 = 1,000,000,000,000,000,000,000• Zettabyte 270 = 1,180,591,620,717,411,303,424• Septillion 1024 = 1,000,000,000,000,000,000,000,000• Yottabyte 280 = 1,208,925,819,614,629,174,706,176• Byte Converter – File Size Calculator• Million, Billion, Trillion