Fundamentals of Programming

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1 Fundamentals of Programming Lecture 2 Number Systems & Arithmetic Lecturer : Ebrahim Jahandar Some Parts borrowed from slides by IETC1011-Yourk University

2 Common Number Systems System Base Symbols Used by humans? Used in computers? Decimal 10 0, 1, 9 Yes No Binary 2 0, 1 No Yes Octal 8 0, 1, 7 No No Hexa-decimal 16 0, 1, 9, A, B, F Yes No 2

3 Quantities/Counting (1 of 3) Decimal Binary Octal Hexa-decimal

4 Quantities/Counting (2 of 3) Decimal Binary Octal Hexa-decimal A B C D E F 4

5 Quantities/Counting (3 of 3) Decimal Binary Octal Hexa-decimal

6 The possibilities: Conversion Among Bases Decimal Octal Binary Hexadecimal pp

7 Quick Example = = 31 8 = Base 7

8 Decimal to Decimal (just for fun) Decimal Octal Binary Hexadecimal Next slide 8

9 Weight => 5 x 10 0 = 5 2 x 10 1 = 20 1 x 10 2 = Base 9

10 Binary to Decimal Decimal Octal Binary Hexadecimal 10

11 Technique Binary to Decimal Multiply each bit by 2 n, where n is the weight of the bit The weight is the position of the bit, starting from 0 on the right Add the results 11

12 Bit 0 Example => 1 x 2 0 = 1 1 x 2 1 = 2 0 x 2 2 = 0 1 x 2 3 = 8 0 x 2 4 = 0 1 x 2 5 =

13 Hexadecimal to Decimal Decimal Octal Binary Hexadecimal 13

14 Technique Hexadecimal to Decimal Multiply each bit by 16 n, where n is the weight of the bit The weight is the position of the bit, starting from 0 on the right Add the results 14

15 Example ABC 16 => C x 16 0 = 12 x 1 = 12 B x 16 1 = 11 x 16 = 176 A x 16 2 = 10 x 256 =

16 Decimal to Binary Decimal Octal Binary Hexadecimal 16

17 Technique Decimal to Binary Divide by two, keep track of the remainder First remainder is bit 0 (LSB, least-significant bit) Second remainder is bit 1 Etc. 17

18 Example =? =

19 Hexadecimal to Binary Decimal Octal Binary Hexadecimal 19

20 Technique Hexadecimal to Binary Convert each hexadecimal digit to a 4-bit equivalent binary representation 20

21 10AF 16 =? 2 Example 1 0 A F AF 16 =

22 Decimal to Octal Decimal Octal Binary Hexadecimal 22

23 Technique Divide by 8 Keep track of the remainder Decimal to Octal 23

24 =? 8 Example =

25 Decimal to Hexadecimal Decimal Octal Binary Hexadecimal 25

26 Technique Decimal to Hexadecimal Divide by 16 Keep track of the remainder 26

27 =? 16 Example = D = 4D

28 Binary to Octal Decimal Octal Binary Hexadecimal 28

29 Technique Binary to Octal Group bits in threes, starting on right Convert to octal digits 29

30 =? 8 Example =

31 Binary to Hexadecimal Decimal Octal Binary Hexadecimal 31

32 Technique Binary to Hexadecimal Group bits in fours, starting on right Convert to hexadecimal digits 32

33 =? 16 Example B B = 2BB 16 33

34 Exercise Convert... Decimal Binary Octal Hexa-decimal AF 34

35 Exercise Convert Decimal Binary Octal Hexa-decimal C AF 35

36 Common Powers (1 of 2) Base 10 Power Preface Symbol pico p 10-9 nano n 10-6 micro 10-3 milli m 10 3 kilo k 10 6 mega M 10 9 giga G tera T Value

37 Common Powers (2 of 2) Base 2 Power Preface Symbol 2 10 kilo k 2 20 mega M 2 30 Giga G Value What is the value of k, M, and G? In computing, particularly w.r.t. memory, the base-2 interpretation generally applies 37

38 Review multiplying powers For common bases, add powers a b a c = a b+c = 2 16 = 65,536 or = = 64k 38

39 Two 1-bit values Binary Addition (1 of 2) A B A + B two 39

40 Two n-bit values Binary Addition (2 of 2) Add individual bits Propagate carries E.g.,

41 Decimal (just for fun) Multiplication (1 of 3) 35 x

42 Multiplication (2 of 3) Binary, two 1-bit values A B A B

43 Multiplication (3 of 3) Binary, two n-bit values As with decimal values E.g., 1110 x

44 Signed Integer Representation Negative numbers must be encode in binary number systems Well-known methods Sign-magnitude One s Complement Two s Complement Which one is better? Calculation speed Complexity of the computation circuit 44

45 Sign-magnitude One sign bit + magnitude bits Positive: s = 0 Negative: s= 1 Range = {(-127) 10.. (+127) 10 } Two ways to represent zero: (+0) ( 0) Examples: (+ 43) 10 = (- 43) 10 = s Sign Magnitude 1 0 How many positive and negative integers can be represented using N bits? Positive: 2 N-1-1 Negative: 2 N

46 Two s Complement Negative numbers: 1. Invert all the bits through the number ~(0) = 1 ~(1) = 0 2. Add one Example: +1 = =? ~( ) Only one zero ( ) Range = { } 46

47 A proposition is a statement that can be either true or false The sky is blue I is a English major x == y Not propositions: Are you Bob? x = 7 Boolean propositions 47

48 We use Boolean variables to refer to propositions Usually are lower case letters starting with p (i.e. p, q, r, s, etc.) A Boolean variable can have one of two values true (T) or false (F) A proposition can be Boolean variables A single variable: p An operation of multiple variables: p (q r) 48

49 Introduction to Logical Operators About a dozen logical operators Similar to algebraic operators + * - / In the following examples, p = Today is Friday q = Today is my birthday 49

50 Logical operators: Not A not operation switches (negates) the truth value Symbol: or ~ In C++ and Java, the operand is! p = Today is not Friday p T F p F T 50

51 Logical operators: And An and operation is true if both operands are true Symbol: It s like the A in And In C++ and Java, the operand is && p q = Today is Friday and today is my birthday p q p q T T T T F F F T F F F F 51

52 Logical operators: Or An or operation is true if either operands are true Symbol: In C++ and Java, the operand is p q = Today is Friday or today is my birthday (or possibly both) p q p q T T T T F T F T T F F F 52

53 Logical operators: Exclusive Or An exclusive or operation is true if one of the operands are true, but false if both are true Symbol: Often called XOR p q (p q) (p q) In Java, the operand is ^ (but not in C++) p q = Today is Friday or today is my birthday, but not both p q p q T T F T F T F T T F F F 53

54 ASCII Codes American Standard Code for Information Interchange First decision: 7 bits for representation Final decision: 8 bits for representation 256 characters ASCII ( P ) = (50) 16 ASCII ( = ) = (3D) 16 row number column number Hexadecimal NUL DLE P ` p 1 SOH DC1! 1 A Q a q 2 STX DC2 " 2 B R b r 3 ETX DC3 # 3 C S c s 4 EOT DC4 $ 4 D T d t 5 ENQ NAK % 5 E U e u 6 ACK SYN & 6 F V f v 7 BEL ETB ' 7 G W g w 8 BS CAN ) 8 H X h x 9 TAB EM ( 9 I Y i y A LF SUB * : J Z j z B VT ESC + ; K ] k } C FF FS, > L \ l D CR GS - = M [ m { E SO RS. < N ^ n ~ F SI US /? O _ o 54

55 Summary Numeral Systems Decimal, Binary, Octal, Hexadecimal Computer Data Storage Units Bit, Byte, Kilo byte, Giga byte, Numeral Systems Conversion Convert between different bases Calculations in Number Systems Addition and multiplication Boolean Algebra Boolean Operations Signed Integer Representation Sing-magnitude: one sign bit + magnitude bits Two s complement : (-N) = ~(N) + 1 ASCII Codes 55

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