Chapter1: Number Systems

1. Chapter1: Number Systems Prepared by: Department of Preparatory year Edited by: Lec. GHADER R. KURDi

2. Objectives • Understand the concept of number systems. • Describe the decimal, binary, hexadecimal and octal system. • Convert a number in binary, octal or hexadecimal to a number in the decimal system. • Convert a number in the decimal system to a number in binary, octal and hexadecimal. • Convert a number in binary to octal, hexadecimal and vice versa.

3. Introduction • Data is a combination of Numbers, Characters and special characters. • The data or Information should be in the form machine readable and understandable. • Data has to be represented in the form of electronic pulses. • The data has to be converted into electronic pulses and each pulse should be identified with a code.

4. Introduction • Everycharacter, special character and keystrokes have numerical equivalent (ASCII code). • Thus using this equivalent, the data can be interchanged into numeric format. • For this numeric conversions we use number systems. • Each number system has radix or Base number , Which indicates the number of digit in that number system.

5. The decimal system • Base (Radix): 10 • Decimal digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 • Examples:

6. The binary system • Base (Radix): 2 • Decimal digits: 0 (absence of pulses) , 1 (presence of pulse) • Examples:

7. The octal system • Base (Radix): 8 • Decimal digits: 0, 1, 2, 3, 4, 5, 6, 7 • Examples:

8. The hexadecimal system • Base (Radix): 16 • Hexadecimal digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A-10, B-11, C-12, D-13, E-14, F-15 • Examples:

9. Quantities/Counting (1 of 3)

10. Quantities/Counting (2 of 3)

11. Quantities/Counting (3 of 3) Etc.

12. Exercise – true or false...

13. Exercise – true or false...

14. Summary of the Number Systems

15. Conversion Among Bases • The possibilities: Decimal Octal Binary Hexadecimal

16. Quick Example 2510 = 110012 = 318 = 1916 Base (radix)

17. Decimal to Decimal (just for fun) Decimal Octal Binary Hexadecimal Next slide…

18. Decimal to Decimal Technique: • Multiply each bit by 10n, where n is the “weight” of the bit • The weight is the position of the bit, starting from 0 on the right Position 6th 5th 4th 3rd 2nd 1st Weight 105 104 103 102 101 100 • Add the results

19. Weight 5 x 100 = 5 + 12510 => 2 x 101 =20 + 1 x 102 = 100 125 Base

20. Binary to Decimal Decimal Octal Binary Hexadecimal

21. Binary to Decimal Technique: • Multiply each bit by 2n, where n is the “weight” of the bit • The weight is the position of the bit, starting from 0 on the right Position 6th 5th 4th 3rd 2nd 1st Weight 25 24 23 22 2120 • Add the results

22. Example Bit “0” 1010112 => 1 x 20 = 1 + 1 x 21 = 2 + 0 x 22 = 0 + 1 x 23 = 8 + 0 x 24 = 0 + 1 x 25 = 32 4310

23. Octal to Decimal Decimal Octal Binary Hexadecimal

24. Octal to Decimal Technique • Multiply each bit by 8n, where n is the “weight” of the bit • The weight is the position of the bit, starting from 0 on the right Position 6th 5th 4th 3rd 2nd 1st Weight 85 84 83 82 81 80 • Add the results

25. Example 7248=> 4 x 80 = 4 x 1 = 4 + 2 x 81 = 2 x 8= 16 + 7 x 82 = 7 x 64 = 448 46810

26. Hexadecimal to Decimal Decimal Octal Binary Hexadecimal

27. Hexadecimal to Decimal Technique • Multiply each bit by 16n, where n is the “weight” of the bit • The weight is the position of the bit, starting from 0 on the right Position 6th 5th 4th 3rd 2nd 1st Weight 165 164 163 162 161 160 • Add the results

28. Example ABC16 => C x 160 = 12 x 1 = 12 + B x 161 = 11 x 16 = 176 + A x 162 = 10 x 256 = 2560 274810

29. Decimal to Binary Decimal Octal Binary Hexadecimal

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

31. 2 125 62 1 2 31 0 2 15 1 2 3 1 2 7 1 2 0 1 2 1 1 Example 12510 = ?2 12510 = 11111012

32. Decimal to Octal Decimal Octal Binary Hexadecimal

33. Decimal to Octal Technique • Divide by 8 • Keep track of the remainder

34. 8 19 2 8 2 3 8 0 2 Example 123410 = ?8 8 1234 154 2 123410 = 23228

35. Decimal to Hexadecimal Decimal Octal Binary Hexadecimal

36. Decimal to Hexadecimal Technique • Divide by 16 • Keep track of the remainder

37. 16 1234 77 2 16 4 13 = D 16 0 4 Example 123410 = ?16 123410 = 4D216

38. Octal to Binary Decimal Octal Binary Hexadecimal

39. Octal to Binary Technique • Convert each octal digit to a 3-bit equivalent binary representation Note: The octal system has a concept of representing three binary digits as one octal numbers, thereby reducing the number of digits of binary number still maintaining the concept of binary system.

40. 7 0 5 111 000 101 Example 7058 = ?2 7058 = 1110001012

41. Hexadecimal to Binary Decimal Octal Binary Hexadecimal

42. Hexadecimal to Binary Technique • Convert each hexadecimal digit to a 4-bit equivalent binary representation Note: the hexadecimal system has a concept of representing four binary digits as one hexadecimal number.

43. 1 0 A F 0001 0000 1010 1111 Example 10AF16 = ?2 10AF16 = 00010000101011112

44. Binary to Octal Decimal Octal Binary Hexadecimal

45. Binary to Octal Technique • Group the digits of the given number into 3's starting from the right • if the number of bits is not evenly divisible by 3 then add 0's at the most significant end • Write down one octal digit for each group

46. 1 011 010 111 1 3 2 7 Example 10110101112 = ?8 10110101112 = 13278

47. Binary to Hexadecimal Decimal Octal Binary Hexadecimal

48. Binary to Hexadecimal Technique • Group bits in fours, starting on right • Convert to hexadecimal digits

49. Example 10101110112 = ?16 • 10 1011 1011 • B B 10101110112 = 2BB16

50. Octal to Hexadecimal Decimal Octal Binary Hexadecimal