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
Every character, 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)
Decimal
Binary
Octal
Hexa- decimal 1 2 10 3 11 4
100 5
101 6
110 7
111

10. Quantities/Counting (2 of 3)
Decimal
Binary
Octal
Hexa- decimal 8
1000 10 9
1001 11
1010 12 A
1011 13 B
1100 14 C
1101 15 D
1110 16 E
1111 17 F

11. Quantities/Counting (3 of 3)
Decimal
Binary
Octal
Hexa- decimal 16
10000 20 10 17
10001 21 11 18
10010 22 12 19
10011 23 13
10100 24 14
10101 25 15
10110 26
10111 27
Etc.

12. Exercise – true or false...
Number
Decimal
Binary
Octal
Hexa- decimal
100001 33
1AF
11021
67S

13. Exercise – true or false...
Number
Decimal
Binary
Octal
Hexa- decimal
100001 √ 33 ×
1AF
11021
67S

14. Summary of the Number Systems
Base
Symbols
Used by humans?
Used in computers?
Decimal 10
0, 1, … 9
Yes No
Binary 2
0, 1
Octal 8
0, 1, … 7
Hexa- decimal 16
0, 1, … 9,
A, B, … F

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

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

17. Decimal to Decimal (just for fun)
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 th th th rd nd st
Weight
Add the results

19. Weight
12510 =>
5 x 100 = 5 +
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 th th th rd nd st
Weight
Add the results

22. Example
Bit “0”
=> 1 x 20 = x 21 = x 22 = x 23 = x 24 = 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 th th th rd nd st
Weight
Add the results

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

26. Hexadecimal to 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 th th th rd nd st
Weight
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. Example
12510 = ?2
12510 =

32. Decimal to Octal
Decimal
Octal
Binary
Hexadecimal

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

34. Example
= ?8 8
19 2 8
2 3 8
0 2
= 23228

35. Decimal to Hexadecimal
Octal
Binary
Hexadecimal

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

37. Example
= ?16
77 2 16
= D
0 4
= 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. Example
7058 = ?2
7058 =

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. Example
10AF16 = ?2
A F
10AF16 =

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. Example
= ?8
= 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
= ?16
B B
= 2BB16

50. Octal to Hexadecimal
Decimal
Octal
Binary
Hexadecimal

51. Octal to Hexadecimal Technique Octal Binary Hex
Use binary as an intermediary
Octal
Binary
Hex

52. Example
10768 = ?16 E
10768 = 23E16

53. Hexadecimal to Octal
Decimal
Octal
Binary
Hexadecimal

54. Hexadecimal to Octal Technique Hex Binary Octal
Use binary as an intermediary
Hex
Binary
Octal

55. Example
1F0C16 = ?8
1 F C
1F0C16 =

56. Exercise – Convert ... Decimal Binary Octal Hexa- decimal 33 1110101
703
1AF
Don’t use a calculator!

57. Binary fractions
The decimal value is calculated in the same way as for non-fractional numbers, the exponents are now negative

58. Example
Bit “0”
=> 1 x 22 = x 21 = x 20 = x 2-1 = x 2-2 = x 2-3 =
5.7510

59. Summary of Conversion Methods
Conversion Among Bases
Conversion from the binary system to octal, hexadecimal and vice versa.
Conversion from the decimal system to binary, octal or hexadecimal
Conversion from binary, octal or hexadecimal to the decimal system

60. Chapter2: Arithmetic operations in Binary System

61. Introduction
The basic arithmetic operations which can be performed rising binary number system are:
Addition
Subtraction
Multiplication
Division

62. Binary Addition A B
A + B 1 10
“two”

63. Binary Addition Two n-bit values Add individual bits Propagate carries
E.g., 1 1 1 1 1 1

64. Binary Subtraction A B
A - B 1
With one barrow

65. Binary Subtraction Two n-bit values Subtract individual bits
If you must subtract a one from a zero, you need to “borrow” from the left
E.g.,

66. Binary Multiplication
A  B 1

67. Multiplication Binary, two n-bit values As with decimal values E.g.,x
11 x x

68. Binary Division A B
A / B
Not acceptable 1

69. Division Binary, two n-bit values As with decimal values E.g., 00
000

70. Exercises –
1000 * 1001
/ 1000