
Method of representing or encoding numbers

Two main notation types
 Sign-value
 Roman numerals
 Positional (place-value)
 Modern decimal notation

Several number systems are commonly used in
computing
Decimal
 Binary
 Hexadecimal

 “Shorthand” for binary
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
We count in “decimal” or base 10 by powers of 10:






100 = 1
101 = 10
102 = 10*10 = 100
103 = 10*10*10 = 1,000
106 = 1,000,000
Computers count in “binary” or base 2:





20 = 1
21 = 2
22 = 2*2 = 4
23 = 2*2*2 = 8
210 = 1,024
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
Bit (b): binary digit -- 0 or 1 value



0 or 1 correspond to the electrical values of off or on, respectively
All computer data is represented in bits
Byte (B): grouping of 8 bits


Considered the basic unit of digital information
8 bits = 1 byte is the de facto standard (also called an octet)
 File sizes and network data are measured in bytes
 You can’t have a 5-bit file
 Physical network media transmit bits, but network protocols are based
on bytes

Unit multiples




Kilobyte: 210 = 1,024 bytes (≈ 1,000)
Megabyte: 220 = 1,048,576 bytes (≈ 1,000,000)
Gigabyte: 230 = 1,073,741,824 bytes (≈ 1,000,000,000)
Terabyte: 240 = 1,099,511,627,776 bytes (≈ 1,000,000,000,000)
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
Base (also called radix)

Number around which system is
organized
 Number of symbols used to
Number System
Decimal
Binary
Hexadecimal
Base
10
2
16
Digits
0,1,…, 9
0, 1
0, 1,…, 9, A – F
represent each digit of a number

Digits

Set of symbols used in forming
numbers
 Additional symbols may be defined
for bases > 10
 Ex: hexadecimal (see chart at right)

Positional Notation

Digit Position
Position of a digit conveys its
significance
 Rightmost digit represents the zero-
power of the base value
 Moving left, each digit represents an
increasing power of the base
number
Digital Data
Number
System
Decimal
Binary
Hexadecimal
4
3
2
1
0
104
10,000
24
16
164
65,536
103
1,000
23
8
163
4,096
102
100
22
4
162
256
101
10
21
2
161
16
100
1
20
1
160
1
5

Decimal
Digit Position (k)
Weight (bk)
Decimal Weight
Digit (ak)
Decimal Value

2
102
100
0
0
1
101
10
0
0
0
100
1
2
2
= 2,002
Binary
Digit Position (k)
Weight (bk)
Decimal Weight
Digit (ak)
Decimal Value

3
103
1,000
2
2,000
10
210
1,024
1
1,024
9
29
512
1
512
8
28
256
1
256
7
27
128
1
128
6
26
64
1
64
5
25
32
0
0
4
24
16
1
16
3
23
8
0
0
2
22
4
0
0
1
21
2
1
2
0
20
1
0
0
= 2,002
Hexadecimal
Digit Position (k)
Weight (bk)
Decimal Weight
Digit (ak)
Decimal Value
2
162
256
7
1,792
1
161
16
D
208
Digital Data
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160
1
2
2
= 2,002
6

Hexadecimal used as shorthand notation for
binary


One hex digit represents four binary digits
Hexadecimal numbers are usually prefixed
with “0x”


Tells someone reading the number it is a hex value
7D216 → 0x7D2
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Decimal
Binary
Hex
Decimal
Binary
Hex
0
0000
0
8
1000
8
1
0001
1
9
1001
9
2
0010
2
10
1010
A
3
0011
3
11
1011
B
4
0100
4
12
1100
C
5
0101
5
13
1101
D
6
0110
6
14
1110
E
7
0111
7
15
1111
F
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
Convert the decimal value 150 to binary

What decimal value does the bit pattern


110100100
Convert the binary pattern 1 0 0 1 0 1 1 1 0 0 0 1

To hex
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

Convert 2,016 to
hexadecimal (base 16)
remainder 1
7 / 16 = 0
3/2= 1
remainder 1
126 / 16 = 7
7/2= 3
remainder 1
2,016 / 16 = 126
15 / 2 = 7
remainder 1
31/ 2 = 15
remainder 1
63 / 2 = 31
remainder 1
126 / 2 = 63
remainder 0
252 / 2 = 126
remainder 0
504 / 2 = 252
remainder 0
1,008 / 2 = 504
remainder 0
2,016 / 2 = 1,008
remainder 0
Binary Digits Left to Right
1/2= 0
201610 = 111111000002
remainder 7
remainder E (14)
Remainder 0
200210 = 7E016
11
Digital Data
Hex Digits Left to Right
Convert 2,016 to
binary (base 2)

American Standard Code for Information
Interchange (ASCII)

Character-encoding scheme based on the ordering of the
English alphabet
 Maps digital bit pattern and character symbol

Includes

Printable characters
 Alphabet, numbers, symbols

Graphics characters
 ■┌┐╔╗

Control characters
 Backspace, tab, line feed, carriage return
 Many of these characters are non-printable
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Digital Data
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
Strings are a sequence of bytes


Bytes represent ASCII characters
Characters can be represented as
Character value: ‘m’
 Decimal value: 109
 Hexadecimal value: 0x6d


Try entering these URLs in a browser
http://%67%6f%6f%67%6c%65%2e%63%6f%6d
 http://%66%61%63%65%62%6f%6f%6b%2e%63%6f%6d
 ‘%’ tells the program what follows is a hex value

Digital Data
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

A file is a collection of binary data
Format determines the interpretation of bit
patterns

Specifies the type of data a file contains
 Text, image, sound, Word, Excel, etc.

Example

01010101 01010011 01000001
 USA (ASCII text file)

(24-bit bitmap)
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
Executable programs are just files
Sequence of bytes
 Computers interpret the byte sequence as instructions


Executable file formats

Windows – Portable Executable (PE) format
 .exe, .dll (libraries), .sys (device drivers)

UNIX – Executable and Linkable Format (ELF)
 Also used by UNIX variants
 Linux, BSD, Solaris

Mac OS X – Mach object file (Mach-O)
 Earlier Mac OSs use Preferred Executable Format (PEF)
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
Convention for naming files

<filename>.<extension>
 Extension is usually three letters
 .txt (text), .jpg (JPEG image), .mp3 (MP3 audio), .doc (Word), .xls (Excel)
 Can be more or less
 .c (C source code), .gz (GNU zip), .docx (Word), .xlsx (Excel)

Extension DOES NOT determine bit interpretation


Operating systems and applications make decisions based
on file extension


Determined by file format
.doc *ought* to be a Word document…I should use Microsoft
Word to open it
Extensions can be changed without changing the file format

Example: change .zip to .txt to email a ZIP file
 Can still open the “.txt” file with a program that recognizes the ZIP file
format
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

What string is represented by the ASCII values
: 85 83 78 65
What exactly is displayed when we open the
file yay.txt in notepad which consists of the
following bytes:

1010101
1010011
1001101
1000011
What exactly would be displayed in Frhed when we
open yay.txt?
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
Number Systems Decoded
http://www.globalknowledge.ca/articles/whitepaper
detail.asp?wpid=315

RFC 20: ASCII format for Network Interchange
http://tools.ietf.org/html/rfc20

File Signatures Table
http://www.garykessler.net/library/file_sigs.html

Comparison of Executable File Formats
http://en.wikipedia.org/wiki/Comparison_of_execut
able_file_formats
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