1. Lecture 2 Number Systems
ITEC 1000 “Introduction to Information Technology”
Lecture 2
Number Systems
Hexadecimal
Decimal
Octal
Binary
Prof. Peter Khaiter

2. Lecture Template: Types of number systems Number bases
Range of possible numbers
Conversion between number bases
Common powers
Arithmetic in different number bases
Shifting a number

3. Types of Number Systems
Additive: Numbers have intrinsic value: e.g.: Roman Numerals: LVIII = = 58
Positional: Value depends on position: e.g.: Decimal system: 55 = 5 x x 1
Additive Number Systems are not used much any more:
Awkward to use.
Prone to errors.

4. Definitions
The Base of a number system – how many different digits (incl. zero) are used in the system.
Base 2: 0, 1
Base 5: 0, 1, 2, 3, 4
Base 8: 0, 1, 2, 3, 4, 5, 6, 7
Base 10: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9
Base 16: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F

5. Definitions
Bit – a cell holding a single binary number (0 or 1)
Byte = 8 bits (can hold 28 = 256 different patterns/values)
Word – a fixed-sized group of bits that the computer handles together. Typical word sizes: 4, 8, 16, 32, 64, 128 bits
1K = 1024 bytes

6. Magnetic Core Memory

7. Common Number Systems 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. Positional decimal system
The number 125 means:
1 group of (100 = 102)
2 groups of (10 = 101)
5 groups of (1 = 100)

9. Place values (1 of 2)
In our usual positional number system, the meaning of a digit depends on where it is located in the number
Example:
3 groups of 1000
7 groups of 100
3 groups of 10
2 groups of 1

10. Place values (2 of 2) Weight
12510 => 5 x 100 = x 101 = x 102 =
Base

11. Representing in bases: 10, 2, 8, 16
86510 = 8 x x x 100 =
10112 = 1 x x x x 20 = = 1110
258 = 2 x x 80 = = 2110
A716 = 10 x x 160 = = 16710
Note: The subscript naming the base is itself given in base ten (10), by convention.
Base

12. Counting in bases (1 of 3) Decimal Binary Octal Hexa- decimal 1 2 10 31 2 10 3 11 4
100 5
101 6
110 7
111

13. Counting in bases (2 of 3) Decimal Binary Octal Hexa- decimal 8 10009
1001 11
1010 12 A
1011 13 B
1100 14 C
1101 15 D
1110 16 E
1111 17 F

14. Counting in bases (3 of 3) Decimal Binary Octal Hexa- decimal 16 1000020 10 17
10001 21 11 18
10010 22 12 19
10011 23 13
10100 24 14
10101 25 15
10110 26
10111 27

15. Estimating magnitude: Binary
= 21410
> ( additional bits to the right)
Place 27 26 25 24 23 22 21 20
Value
128 64 32 16 8 4 2 1
Evaluate
1 x 128
1 x 64
0 x 32
1 x16
0 x 8
1 x 4
1 x 2
0 x 1
Sum for Base 10

16. Range of possible numbers
R = BK where
R = range
B = base
K = number of digits
Example #1: Base 10, 2 digits
R = 102 = 100 different numbers (0…99)
Example #2: Base 2, 16 digits
R = 216 = 65,536 or 64K
16-bit PC can store 65,536 different number values

17. Decimal Range for Bit Widths
Bits
Digits
Range 1 0+
2 (0 and 1) 4 1+
16 (0 to 15) 8 2+
256 10 3
1,024 (1K) 16 4+
65,536 (64K) 20 6
1,048,576 (1M) 32 9+
4,294,967,296 (4G) 64
19+
Approx. 1.6 x 1019
128
38+
Approx. 2.6 x 1038
See also Ch. 3, p. 74

18. Conversion Among Bases
Hexadecimal
Decimal
Octal
Binary

19. Binary to Decimal (1 of 3)
Hexadecimal
Decimal
Octal
Binary

20. Binary to Decimal (2 of 3) 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

21. Binary to Decimal (3 of 3) Bit “0”
=> 1 x 20 = x 21 = x 22 = x 23 = x 24 = x 25 = 32
4310
Bit “0”

22. Octal to Decimal (1 of 3)
Decimal
Octal
Binary
Hexadecimal

23. Octal to Decimal (2 of 3) Technique Note: 80 = 1, 81 = 8,
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 together
Note: 80 = 1, 81 = 8,
82 = = 512, Etc.

24. Octal to Decimal (3 of 3)
7248 => 4 x 80 = 4 x 1 = x 81 = 2 x 8 = x 82 = 7 x 64 =

25. Hexadecimal to Decimal (1 of 3)
Octal
Binary
Hexadecimal

26. Hexadecimal to Decimal (2 of 3)
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
Note: 160 = 1, 161 = 16,
162 = = 4096, Etc.

27. Hexadecimal to Decimal (3 of 3)
ABC16 => C x 160 = 12 x 1 = B x 161 = 11 x 16 = A x 162 = 10 x 256 = 2560
274810

28. Decimal to Binary (1 of 3)
Decimal
Octal
Binary
Hexadecimal

29. Decimal to Binary (2 of 3) Technique
Divide by two, keep track of the remainder
First remainder is bit 0 (LSB, least-significant bit)
Second remainder is bit 1
Etc.

30. Decimal to Binary (3 of 3) 12510 = ?2 2 125 62 1 2 31 0 2 15 1 2 7 1
12510 =

31. Octal to Binary (1 of 3)
Decimal
Octal
Binary
Hexadecimal

32. Octal to Binary (2 of 3) Technique
Convert each octal digit to a 3-bit equivalent binary representation
See Table, Slides 12-14

33. Octal to Binary (3 of 3)
7058 = ?2
7058 =

34. Hexadecimal to Binary (1 of 3)
Octal
Binary
Hexadecimal

35. Hexadecimal to Binary (2 of 3)
Technique
Convert each hexadecimal digit to a 4-bit equivalent binary representation
See Table, Slides 12-14

36. Hexadecimal to Binary (3 of 3)
10AF16 = ?2
A F
10AF16 =

37. Decimal to Octal (1 of 3)
Decimal
Octal
Binary
Hexadecimal

38. Decimal to Octal (2 of 3) Technique Divide by 8
Keep track of the remainder

39. Decimal to Octal (3 of 3)
= ?8 8
19 2 8
2 3 8
0 2
= 23228

40. Decimal to Hexadecimal (1 of 3)
Octal
Binary
Hexadecimal

41. Decimal to Hexadecimal (2 of 3)
Technique
Divide by 16
Keep track of the remainder

42. Decimal to Hexadecimal (3 of 3)
= ?16
77 2 16
= D
0 4
= 4D216

43. Binary to Octal (1 of 3)
Decimal
Octal
Binary
Hexadecimal

44. Binary to Octal (2 of 3) Technique
Group bits in threes, starting on right
Convert to octal digits
See Table, Slides 12-14

45. Binary to Octal (3 of 3)
= ?8
= 13278

46. Binary to Hexadecimal (1 of 3)
Octal
Binary
Hexadecimal

47. Binary to Hexadecimal (2 of 3)
Technique
Group bits in fours, starting on right
Convert to hexadecimal digits
See Table, Slides 12-14

48. Binary to Hexadecimal (3 of 3)
= ?16
2 B B
= 2BB16

49. Octal to Hexadecimal (1 of 3)
Binary
Hexadecimal

50. Octal to Hexadecimal (2 of 3)
Technique
Use binary as an intermediary
See Table, Slides 12-14

51. Octal to Hexadecimal (3 of 3)
10768 = ?16 E
10768 = 23E16

52. Hexadecimal to Octal (1 of 3)
Binary
Hexadecimal

53. Hexadecimal to Octal (2 of 3)
Technique
Use binary as an intermediary
See Table, Slides 12-14

54. Hexadecimal to Octal (3 of 3)
1F0C16 = ?8
1 F C
1F0C16 =

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

56. Exercise – Convert (answers)
Decimal
Binary
Octal
Hexa- decimal 33
100001 41 21
117
165 75
451
703
1C3
431
657
1AF

57. 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

58. 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

59. Example
1. Double click on My Computer 2. Right click on C: 3. Click on Properties
/ 230 =

60. For common bases, add powers
Multiplying powers
For common bases, add powers
ab  ac = ab+c
26  210 = 216 = 65,536
or…
26  210 = 64  210 = 64k

61. Table of powers Power Base 8 7 6 5 4 3 2 1 256 128 64 32 16 32,768
256
128 64 32 16
32,768
4,096
512
65,536

62. Number point or radix point
Fractions
Number point or radix point
Decimal point in base 10
Binary point in base 2
No exact relationship between fractional numbers in different number bases
Exact conversion may be impossible

63. Decimal fractions Move the number point one place to the right
Effect: multiplies the number by the base number
Example:
Move the number point one place to the left
Effect: divides the number by the base number
Example:

64. Fractions: Base 10 .258910 10-1 10-2 10-3 10-4 Place 10-1 10-2 10-3
Place
10-1
10-2
10-3
10-4
Value
1/10
1/100
1/1000
1/10000
Evaluate
2 x 1/10
5 x 1/100
8 x 1/1000
9 x1/10000
Sum .2
.05
.008
.0009

65. Fractions: Base 2 .1010112 = 0.67187510 2-1 2-2 2-3 2-4 2-5 2-6 Place =
Place
2-1
2-2
2-3
2-4
2-5
2-6
Value
1/2
1/4
1/8
1/16
1/32
1/64
Evaluate
1 x 1/2
0 x 1/4
1x 1/8
0 x 1/16
1 x 1/32
1 x 1/64
Sum .5
0.125

66. Fractions: Base 10 and Base 2
No general relationship between fractions of types 1/10k and 1/2k
Therefore a number representable in base 10 may not be representable in base 2
But: the converse is true: all fractions of the form 1/2k can be represented in base 10
Fractional conversions from one base to another are stopped
If there is a rational solution, or
When the desired accuracy is attained

67. Fractions: From Base B To Base 10
Determine the appropriate weight for each fractional digit as a negative power of the base
Multiply each digit by its weight
Add the values
Example:
.A116 = 10x x16-2 = 10x x =

68. Fractions: From Base 10 To Base B
Multiply the fraction by the base value B (i.e., 2, 8 or 16)
Record the values that move to the left of the radix point and drop them
Repeat the process until
the value being multiplied is zero, or
the desired number of digits of accuracy is attained

69. Fractions: From Base 10 To Base B
= ? x

70. Fractions: From Base 2 To Base 8, 16
Group digits from left to right in groups of 3 (base 8) or 4 (base 16)
Supplement the right-most group with 0’s, if necessary
Convert each group to the desired base.
See Table, Slides 12-14

71. Fractions: From Base 8, 16 To Base 2
Convert each octal (base 8) or hexadecimal (base 16) digit to its 3-bit or 4-bit representation
See Table, Slides 12-14

72. Fractions: between Base 8 and Base 16
Use binary conversion as an intermediary
Example:
0.C816 = ?8
C816 = 0.628

73. Mixed number conversion
Convert whole part and fraction part separately
See Table, Slides 12-14

74. Arithmetic operations (1 of 14)
Binary Addition
Two 1-bit values A B
A + B 1 10
0 and carry 1 to the next more significant bit, i.e. “two”

75. Arithmetic operations (2 of 14)
Two n-bit values
Add individual bits (see Table)
Propagate carries 1 1
Note: superscripts are carried amounts.

76. Addition (different bases) (3 of 14)
Problem
Largest Single Digit
Decimal 6 +3 9
Octal +1 7
Hexadecimal +9 F
Binary 1 +0

77. Addition (different bases) (4 of 14)
Problem
Carry
Answer
Decimal 6 +4
Carry the 10 10
Octal +2
Carry the 8
Hexadecimal +A
Carry the 16
Binary 1 +1
Carry the 2

78. Addition Table: Base 10 (5 of 14)
= 910 + 1 2 3 4 5 6 7 8 9 10 11 12
8 etc 13

79. Addition Table: Base 8 (6 of 14)
= 118 + 1 2 3 4 5 6 7 10 11 12 13 14 15 16

80. Arithmetic operations (7 of 14)
Binary Subtraction
Two 1-bit values A B
A - B 1
and borrow 1 from the next more significant bit

81. Arithmetic operations (8 of 14)
Two n-bit values
Subtract individual bits (see Table)
Keep track of the borrowings
– = 10

82. Arithmetic operations (9 of 14)
Multiplication
Decimal (just for fun)
35 x

83. Arithmetic operations (10 of 14)
Binary multiplication, two 1-bit values A B
A  B 1

84. Arithmetic operations (11 of 14)
Binary multiplication, two n-bit values
As with decimal values x

85. Multiplication Table: Base 10 (12 of 14)
310 x 610 = 1810 x 1 2 3 4 5 6 7 8 9 10 12 14 16 18 15 21 24 27 20 28 32 36 25 30 35 40 45 42 48 54 49 56 63
etc.

86. Multiplication Table: Base 8 (13 of 14)
38 x 68 = 228 x 1 2 3 4 5 6 7 10 12 14 16 11 17 22 25 20 24 30 34 31 36 43 44 52 61

87. Arithmetic operations (14 of 14)
Binary division, two n-bit values
As with decimal values
100001/11 = )
quotient
1011 11
dividend
00100
divisor 11 11 11

88. Shifting a number
Shifting a decimal number to the left by one position is equivalent to multiplying by 10
Shifting a binary number to the left by one position is equivalent to multiplying by 2
General rule: shifting a number in any base left one digit multiplies its value by the base; shifting one digit right divides its value by the base

89. Thank you!
Reading: Lecture slides and notes, Chapter 3