1. Introduction to Computer Systems and Software Lecture 2 of 2 Simon Coupland simonc@dmu.ac.uk

2. Representation of Data Within the Computer Contents:  Decimal and Binary Integer Numbers  Binary Addition  Signed Binary Numbers  Overflow  Hexadecimal Numbers  Number Conversion  Real Numbers  Character Encoding

3. Introduction A few terms:  A bit – a single Binary digIT, 0 or 1  A byte – eight bits  A word – one or more bytes  Integer – whole number  Real number – a number with decimal points  Binary – Base 2 numbers  Octal – Base 8 numbers  Decimal – Base 10 numbers (everyday numbers)  Hexadecimal – Base 16 numbers

4. Decimal and Binary Numbers We all use decimal numbers Base 10 numbers Example: 124 Digit76543210 Digit Value10 7 10 6 10 5 10 4 10 3 10 2 10 1 10 0 Digit Value10M1M100k10k1k100101 Example00000124

5. Decimal and Binary Numbers Computers use binary numbers Base 2 numbers: Example: 124 Bit76543210 Bit Value2727 2626 2525 2424 23232 2121 2020 1286432168421 Example01111100

6. Decimal and Binary Numbers More binary numbers: 00010011 = 16 + 2 + 1 = 19 01011101 = 64 + 16 + 8 + 4 + 1 = 93 11111111 = 255 00000000 = 0

7. Binary Addition When adding binary numbers we use binary logic Binary Addition Truth Table 1: ABA + BCarry 0000 0110 1010 1101

8. Binary Addition Binary Addition Truth Table 2: ABCarry (in)A + BCarry (out) 00000 00110 01010 01101 10010 10101 11001 11111

9. Binary Addition Binary Addition Example: 00010001 + 00011101 =

10. Binary Addition Binary Addition Example: 00010001 + 00011101 = 0 1

11. Binary Addition Binary Addition Example: 00010001 + 00011101 = 10

12. Binary Addition Binary Addition Example: 00010001 + 00011101 = 110

13. Binary Addition Binary Addition Example: 00010001 + 00011101 = 1110

14. Binary Addition Binary Addition Example: 00010001 + 00011101 = 01110 1

15. Binary Addition Binary Addition Example: 00010001 + 00011101 = 101110

16. Binary Addition Binary Addition Example: 00010001 + 00011101 = 0101110

17. Binary Addition Binary Addition Example: 00010001 + 00011101 = 00101110

18. Binary Addition Binary Addition Q1: 00101101 + 00011001 =

19. Binary Addition Binary Addition Q1: 00101101 + 00011001 = 01000110

20. Binary Addition Binary Addition Q2: 00001110 + 00001111 =

21. Binary Addition Binary Addition Q2: 00001110 + 00001111 = 00011101

22. Binary Addition Binary Addition Q3: 01011001 + 00111011 =

23. Binary Addition Binary Addition Q3: 01011001 + 00111011 = 10010011

24. Negative Binary Numbers Sign-true Magnitude Left most bit holds sign Example: -10 Bit76543210 Bit Valuesign2626 2525 2424 23232 2121 2020 Bit Value+/-6432168421 Example10001010

25. Negative Binary Numbers Ones complement All 1’s and 0’s are switched When negative, result = -255 + value Example: -10 Bit76543210 Bit Value2727 2626 2525 2424 23232 2121 2020 1286432168421 Example11110101

26. Negative Binary Numbers Twos complement Left most bit holds sign When negative, result = -128 + result Example: -10 Bit76543210 Bit ValueSign2626 2525 2424 23232 2121 2020 Bit Value+/-1286432168421 Example11110110

27. Negative Binary Numbers Conversion to Twos complement  Ones complement the byte/word  Add 1 Example: 00001010+10 11110101 Ones complement + 00000001 = 11110110

28. Why Use Twos Complement? Because addition rules still work: 00010110 22 + 10001000+ -120 = 10011110= -98

29. Overflow Overflow is when the number of bits is too small to store the result of an arithmetic operation Example (twos complement) : 01011001 89 + 01111011+ 123 = 11010011= -45

30. Overflow Overflow can be easily detected for signed binary numbers Errors can then be corrected Adding two positive numbers should give a positive result Adding two negative numbers should give a negative result Adding a positive and negative number together can never result in overflow. Why?

31. Hexadecimal Numbers Writing code with long binary numbers would be cumbersome and error prone A hexadecimal digit can take 16 values (0-F) One hex digit can represent a four bit word Examples: DecimalHexadecimalBinary 440100 10A1010 744A01001010

32. Number Conversion Binary to Hexadecimal:  From the least significant (rightmost) bit split the binary number into groups of four bits.  Each 4 bits has a hexadecimal equivalent  Example: 0001001110011110 1 3 9 E

33. Number Conversion Hexadecimal to Binary:  Convert each hexadecimal digit into its 4 bit binary equivalent  Join the all 4 bit words together  Example: A42F 1010 0100 0010 1111

34. Number Conversion Question Hexadecimal to Binary:  Convert D36B into binary

35. Number Conversion Question Hexadecimal to Binary:  Convert D36B into binary D 3 6 B 1101 0011 0110 1011

36. Number Conversion Question Binary to Hexadecimal:  Convert 1001101001001010 into hexadecimal

37. Number Conversion Question Binary to Hexadecimal:  Convert 1001101001001010 into hexadecimal 1001 1010 0100 1010 9 A 8 A

38. Number Conversion Decimal to Binary  Use the following algorithm, begin with LSB int dec_value = some_number; int next_bit; while(dec_value > 0) { next_bit = dec_value % 2; dec_value = dec_value / 2; }

39. Number Conversion Decimal to Binary Convert 42 to binary: dec_value/2%2Result 422100 101 50010 5211010 21001010 101101010

40. Number Conversion Question Convert 39 to binary:

41. Number Conversion Question Convert 39 to binary: dec_value/2%2Result 391911 9111 941111 4200111 21000111 101100111

42. Number Conversion Binary to Decimal  Use the following algorithm, begin with MSB int dec_value = 0; int bit_value; int bit_index = MSB_index; while(bit_index >= 0) { bit_value = word[bit_index]; dec_value = dec_value * 2 + bit_value; bit_index--; }

43. Number Conversion Binary to Decimal  Convert 011010 to decimal: bit_indexbit_valuedec_value 500 411 313 206 1113 0026

44. Number Conversion Binary to Decimal  Question, convert 101110 to decimal:

45. Number Conversion Binary to Decimal  Question, convert 101110 to decimal: bit_indexbit_valuedec_value 511 402 315 2111 1123 0046

46. Number Conversion Decimal to Hexadecimal  Use the following algorithm, begin with LSB int dec_value = some_number; char hex_digit; while(dec_value > 0) { hex_digit = to_hex(dec_value % 16); dec_value = dec_value / 16; }

47. Number Conversion Decimal to Hexadecimal  Convert 1863 to hexadecimal dec_valuedec_value / 16dec_value % 16hex_digit 186311677 744 7077

48. Number Conversion Decimal to Hexadecimal  Question, convert 1437 to hexadecimal

49. Number Conversion Decimal to Hexadecimal  Question, convert 1437 to hexadecimal dec_valuedec_value / 16dec_value % 16hex_digit 14378913D 89599 5055

50. Number Conversion Hexadecimal to Decimal  Use the following algorithm, begin with MSB int dec_value = 0; char digit_value; while(!word.off) { digit_value = to_dec(hex_digit); dec_value = dec_value * 16 + digit_value; }

51. Number Conversion Hexadecimal to Decimal  Convert F2AC to decimal Indexhex_valuedec_valueresult 3F15 222242 1A103882 0C1262124

52. Number Conversion Hexadecimal to Decimal  Convert E59A to decimal

53. Number Conversion Hexadecimal to Decimal  Convert E59A to decimal Indexhex_valuedec_valueresult 3E14 255229 1993673 0A1058778

54. Real Numbers A number n is a real number if n % 1 != 0 Take a lot of processing power/time Often math co-processors are used to perform arithmetic on these numbers

55. Fixed Point Real Numbers Similar to integer representation 8 bit example: Bit87654321 value-/+23232 2121 2020 2 -1 2 -2 2 -3 value-/+84210.50.250.125

56. Fixed Point Real Numbers Maximum number is small Accuracy is limited Not used because of these reasons

57. Floating Point Real Numbers Floating point numbers are held in two parts  Mantissa  Exponent In  0.3876  10 4 The mantissa is 0.3876 The exponent is 4

58. Floating Point Real Numbers Example (not IEEE float): 00000000000000110010100111110011 mantissaexponent Mantissa = 809 Exponent = -13 809  10 -13 0.0000000000809

59. Character Encoding ASCII EBCDIC UNICODE ISO-8859-1

60. Character Encoding ASCII  American Standard Code for Information Interchange  Characters are represented by integers  Table in your notes A = 65 B = 66 a = 96 z = 123 @ = 64

61. Character Encoding ASCII  Special functions characters are also encoded  Examples: Carriage return = 13 Escape = 27 Space = 32

62. Recap Binary numbers can be signed of unsigned Twos complement gives simple addition Overflow - when the result of a arithmetic operation is too large to be represented Hexadecimal numbers – base 16 Real numbers use floating point Characters are represented as intergers