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 A number system defines how a number can
be represented using distinct symbols. A
number can be represented differently in
different system.
 Several number systems have been used in
the past and can be categorized into two
groups: positional and non-positional
systems.
Roman numerals are a good example of a
non-positional number system. This number
system has a set of symbols
S = {I, V, X, L, C, D, M}. The values of each
symbol are shown in Table.
 To find the value of a number, we need to
add the value of symbols subject to specific
rules
 In positional number system, there are only
few symbols called digits, and these symbols
represent different values depending on the
position they occupy in the number.
Commonly used number systems are:
 Decimal
 Binary
 Octal
 Hexadecimal
Why do we need more number systems?
• Humans understand decimal
• Digital electronics (computers) understand binary
• Since computers have 32, 64, and even 128 bit
busses, displaying numbers in binary is cumbersome
• Data on a 32 bit data bus would look like the
following:
0110 1001 0111 0001 0011 0100 1100 1010
• Hexadecimal (base 16) and octal (base 8) number
systems are used to represent binary data in a more
compact form.
 The word decimal is a derivative of decem,
which is the Latin word for ten.
 The number system that we use day-to-day life
is called the Decimal number system.
 The most popular & commonly used number
system is the Decimal number system as it
supports the entire mathematical & accounting
concept in the world.
 The base is equal to ten because there are
altogether ten digits (1,2, 3, 4, 5, 6, 7, 8, 9)10
 The binary number system uses two digits to
represent numbers, the values are 0 & 1. This
numbering system is sometime called the Base 2
numbering system (0,1).
 Binary digit is often referred to by the common
abbreviation BIT. Thus, a “bit” in a computer
terminology means either a 0 or a 1.
 This number system is natural to an electronic
machines or devices as their mechanism based on the
OFF or ON switching of the circuits.
 Therefore, 0 represent the OFF & 1 represent ON
state of the circuit.
 The octal number system uses eight values to
represent numbers. The values are (0, 1, 2,
3, 4, 5, 6, 7)8 and the base of this system is
eight.
 The hexadecimal number system has 16-
digits or symbols (hexa means six & decimal
means 10 so sum is sixteen) are (0, 1, 2, 3, 4,
5,6, 7, 8, 9, A, B, C, D, E, F)16, so it has the
base 16.
 This system uses numerical values from 0 to
9 & alphabets from A to F.
 Alphabets A to F represent decimal numbers
from 10 to 15.
System Base Symbols
Used by
humans?
Used in
computers?
Decimal 10 0, 1, … 9 Yes No
Binary 2 0, 1 No Yes
Octal 8 0, 1, … 7 No No
Hexa-
decimal
16 0, 1, … 9,
A, B, … F
No No
Decimal Binary Octal
Hexa-
decimal
0 0 0 0
1 1 1 1
2 10 2 2
3 11 3 3
4 100 4 4
5 101 5 5
6 110 6 6
7 111 7 7
Decimal Binary Octal
Hexa-
decimal
8 1000 10 8
9 1001 11 9
10 1010 12 A
11 1011 13 B
12 1100 14 C
13 1101 15 D
14 1110 16 E
15 1111 17 F
Decimal Binary Octal
Hexa-
decimal
16 10000 20 10
17 10001 21 11
18 10010 22 12
19 10011 23 13
20 10100 24 14
21 10101 25 15
22 10110 26 16
23 10111 27 17
Hexadecimal
Decimal Octal
Binary
16
Successive
Division
Weighted
Multiplication
Octal8
0 1 2 3 4 5 6 7
Successive
Division
Hexadecimal16
0 1 2 3 4 5 6 7 8 9 A B C D E F
Weighted
Multiplication
Successive
Division
Weighted
Multiplication
Binary2
0 1
Decimal10
0 1 2 3 4 5 6 7 8 9
2510 = 110012 = 318 =
1916
Base
Hexadecimal
Decimal Octal
Binary
12510 => 5 x 100= 5
2 x 101= 20
1 x 102= 100
125
Base
Weight
Hexadecimal
Decimal Octal
Binary
 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
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
Bit “0”
Hexadecimal
Decimal Octal
Binary
 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
7248 => 4 x 80 = 4
2 x 81 = 16
7 x 82 = 448
46810
Hexadecimal
Decimal Octal
Binary
 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
ABC16 => C x 160 = 12 x 1 = 12
B x 161 = 11 x 16 = 176
A x 162 = 10 x 256 = 2560
274810
Hexadecimal
Decimal Octal
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.
12510 = ?2
2 125
62 12
31 02
15 1
2
7 1
2
3 12
1 12
0 1
12510 = 11111012
Hexadecimal
Decimal Octal
Binary
 Technique
 Convert each octal digit to a 3-bit equivalent
binary representation
7058 = ?2
7 0 5
111 000 101
7058 = 1110001012
Hexadecimal
Decimal Octal
Binary
 Technique
 Convert each hexadecimal digit to a 4-bit
equivalent binary representation
10AF16 = ?2
1 0 A F
0001 0000 1010 1111
10AF16 = 00010000101011112
Hexadecimal
Decimal Octal
Binary
 Technique
 Divide by 8
 Keep track of the remainder
123410 = ?8
8 1234
154 2
8
19 2
8
2 3
8
0 2
123410 = 23228
Hexadecimal
Decimal Octal
Binary
 Technique
 Divide by 16
 Keep track of the remainder
123410 = ?16
123410 = 4D216
16 1234
77 216
4 13 = D16
0 4
Hexadecimal
Decimal Octal
Binary
 Technique
 Group bits in threes, starting on right
 Convert to octal digits
10110101112 = ?8
1 011 010 111
1 3 2 7
10110101112 = 13278
Hexadecimal
Decimal Octal
Binary
 Technique
 Group bits in fours, starting on right
 Convert to hexadecimal digits
10101110112 = ?16
10 1011 1011
2 B B
10101110112 = 2BB16
Hexadecimal
Decimal Octal
Binary
 Technique
 Use binary as an intermediary
10768 = ?16
1 0 7 6
001 000 111 110
2 3 E
10768 = 23E16
Hexadecimal
Decimal Octal
Binary
 Technique
 Use binary as an intermediary
1F0C16 = ?8
1 F 0 C
0001 1111 0000 1100
1 7 4 1 4
1F0C16 = 174148
Decimal Binary Octal
Hexa-
decimal
33
1110101
703
1AF
Decimal Binary Octal
Hexa-
decimal
33 100001 41 21
117 1110101 165 75
451 111000011 703 1C3
431 110101111 657 1AF
 Base 10
Power Preface Symbol
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
.000000000001
.000000001
.000001
.001
1000
1000000
1000000000
1000000000000
 Base 2
Power Preface Symbol
210 kilo k
220 mega M
230 Giga G
Value
1024
1048576
1073741824
• What is the value of “k”, “M”, and “G”?
• In computing, particularly w.r.t. memory,
the base-2 interpretation generally applies
 For common bases, add powers
26  210 = 216 = 65,536
or…
26  210 = 64  210 = 64k
ab  ac = ab+c
 Two 1-bit values
A B A + B
0 0 0
0 1 1
1 0 1
1 1 10
“two”
 Two n-bit values
 Add individual bits
 Propagate carries
 E.g.,
10101 21
+ 11001 + 25
101110 46
11
 Binary, two 1-bit values
A B A  B
0 0 0
0 1 0
1 0 0
1 1 1
 Binary, two n-bit values
 As with decimal values
 E.g.,
1110
x 1011
1110
1110
0000
1110
10011010
 Decimal to decimal
3.14 => 4 x 10-2 = 0.04
1 x 10-1 = 0.1
3 x 100 = 3
3.14
 Binary to decimal
10.1011 => 1 x 2-4 = 0.0625
1 x 2-3 = 0.125
0 x 2-2 = 0.0
1 x 2-1 = 0.5
0 x 20 = 0.0
1 x 21 = 2.0
2.6875
 Decimal to binary
3.14579
.14579
x 2
0.29158
x 2
0.58316
x 2
1.16632
x 2
0.33264
x 2
0.66528
x 2
1.33056
etc.
11.001001...
Decimal Binary Octal
Hexa-
decimal
29.8
101.1101
3.07
C.82
Decimal Binary Octal
Hexa-
decimal
29.8 11101.110011… 35.63
…
1D.CC…
5.8125 101.1101 5.64 5.D
3.109375 11.000111 3.07 3.1C
12.5078125 1100.10000010 14.404 C.82
 Study Project
 Software Based (Verilog)
 Hardware Based (using digital ICs)
 Software & Hardware Based (using FPGA)
 Due on 4th October 2013
 One Pager Only
 Group of two students maximum
 Title
 Group members
 Objective/Scope
 Hardware/Software Requirement
 Timeline

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