Introduction to Computer Systems and Software

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

5. Decimal and Binary Numbers • Computers use binary numbers • Base 2 numbers: • Example: 124

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:

8. Binary Addition • Binary Addition Truth Table 2:

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

25. Negative Binary Numbers • Ones complement • All 1’s and 0’s are switched • When negative, result = -255 + value • Example: -10

26. Negative Binary Numbers • Twos complement • Left most bit holds sign • When negative, result = -128 + result • Example: -10

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:

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 0001 0011 1001 1110 1 3 9 E 139E

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 A 4 2 F 1010 0100 0010 1111 1010010000101111

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 9A8A

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:

40. Number Conversion • Question • Convert 39 to binary:

41. Number Conversion • Question • Convert 39 to binary:

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:

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

45. Number Conversion • Binary to Decimal • Question, convert 101110 to decimal:

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

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

49. Number Conversion • Decimal to Hexadecimal • Question, convert 1437 to hexadecimal

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; }