1. NUMBER SYSTEM
Prepared by:
Engr Zakria

2. Introduction
A number system defines how a number can be represented using distinct symbols. A number can be represented differently in different systems. For example, the two numbers (2A)16 and (52)8 both refer to the same quantity, (42)10, but their representations are different.

3. The decimal system (base 10)
The word decimal is derived from the Latin root decem (ten). In this system the base b = 10 and we use ten symbols
S = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}
The symbols in this system are often referred to as decimal digits or just digits.

4. The binary system (base 2)
The word binary is derived from the Latin root bini (or two by two). In this system the base b = 2 and we use only two symbols,
S = {0, 1}
The symbols in this system are often referred to as binary digits or bits (binary digit).

5. The hexadecimal system (base 16)
The word hexadecimal is derived from the Greek root hex (six) and the Latin root decem (ten). In this system the base b = 16 and we use sixteen symbols to represent a number. The set of symbols is
S = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F}
Note that the symbols A, B, C, D, E, F are equivalent to 10, 11, 12, 13, 14, and 15 respectively. The symbols in this system are often referred to as hexadecimal digits.

6. The octal system (base 8)
The word octal is derived from the Latin root octo (eight). In this system the base b = 8 and we use eight symbols to represent a number. The set of symbols is
S = {0, 1, 2, 3, 4, 5, 6, 7}

7. COMMON NUMBER SYSTEM System 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

8. Decimal
Binary
Octal
Hexa- decimal 1 2 10 3 11 4
100 5
101 6
110 7
111

9. 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

10. 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.

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

12. Quick Example
2510 = = 318 = 1916
Base

13. Decimal to Decimal (just for fun)
Octal
Binary
Hexadecimal

14. Weight
12510 => 5 x 100 = x 101 = x 102 =
Base

15. Binary to Decimal
Decimal
Octal
Binary
Hexadecimal

16. 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
Add the results

17. Example
Bit “0”
=> 1 x 20 = x 21 = x 22 = x 23 = x 24 = x 25 = 32
4310

18. Octal to Decimal
Decimal
Octal
Binary
Hexadecimal

19. 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
Add the results

20. Example
7248 => 4 x 80 = x 81 = x 82 =

21. Hexadecimal to Decimal
Octal
Binary
Hexadecimal

22. 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
Add the results

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

24. Decimal to Binary
Decimal
Octal
Binary
Hexadecimal

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

26. Example
12510 = ?2
12510 =

27. Octal to Binary
Decimal
Octal
Binary
Hexadecimal

28. Octal to Binary Technique
Convert each octal digit to a 3-bit equivalent binary representation

29. Example
7058 = ?2
7058 =

30. Hexadecimal to Binary
Decimal
Octal
Binary
Hexadecimal

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

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

33. Decimal to Octal
Decimal
Octal
Binary
Hexadecimal

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

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

36. Decimal to Hexadecimal
Octal
Binary
Hexadecimal

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

38. Example
= ?16
77 2 16
= D
0 4
= 4D216

39. Binary to Octal
Decimal
Octal
Binary
Hexadecimal

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

41. Example
= ?8
= 13278

42. Binary to Hexadecimal
Decimal
Octal
Binary
Hexadecimal

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

44. Example
= ?16
B B
= 2BB16

45. Octal to Hexadecimal
Decimal
Octal
Binary
Hexadecimal

46. Octal to Hexadecimal
Technique
Use binary as an intermediary

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

48. Hexadecimal to Octal
Decimal
Octal
Binary
Hexadecimal

49. Hexadecimal to Octal
Technique
Use binary as an intermediary

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

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

52. Exercise – Convert … Decimal Binary Octal Hexa- decimal 33 100001 41
Answer
Decimal
Binary
Octal
Hexa- decimal 33
100001 41 21
117
165 75
451
703
1C3
431
657
1AF

53. Common Powers (1 of 2) Base 10 Power Preface Symbol pico p nano n
10-12
pico p
10-9
nano n
10-6
micro 
10-3
milli m
103
kilo k
106
mega M
109
giga G
1012
tera T
Value
.001
1000

54. Common Powers (2 of 2) Base 2 What is the value of “k”, “M”, and “G”?
Preface
Symbol
210
kilo k
220
mega M
230
Giga G
Value
1024
What is the value of “k”, “M”, and “G”?
In computing, particularly w.r.t. memory, the base-2 interpretation generally applies

55. Review – multiplying powers
For common bases, add powers
ab  ac = ab+c
26  210 = 216 = 65,536
or…
26  210 = 64  210 = 64k

56. Binary Addition (1 of 2)
Two 1-bit values A B
A + B 1 10
“two”

57. Binary Addition (2 of 2) Two n-bit values Add individual bits
Propagate carries
E.g., 1 1

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

59. Multiplication (2 of 3)
Binary, two 1-bit values A B
A  B 1

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

61. Fractions Decimal to decimal (just for fun)
3.14 => 4 x 10-2 = x 10-1 = x 100 =

62. Fractions Binary to decimal
=> 1 x 2-4 = x 2-3 = x 2-2 = x 2-1 = x 20 = x 21 =

63. Fractions Decimal to binary 3.14579 11.001001...
x x x x x x
etc.

64. Exercise – Convert ... Decimal Binary Octal Hexa- decimal 29.8
3.07
C.82
Don’t use a calculator!
Skip answer
Answer

65. Exercise – Convert … Decimal Binary Octal Hexa- decimal 29.8
Answer
Decimal
Binary
Octal
Hexa- decimal
29.8 …
35.63…
1D.CC…
5.8125
5.64
5.D
3.07
3.1C
14.404
C.82