 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 Digital Data2

 We count in “decimal” or base 10 by powers of 10:  10 0 = 1  10 1 = 10  10 2 = 10*10 = 100  10 3 = 10*10*10 = 1,000  10 6 = 1,000,000  Computers count in “binary” or base 2:  2 0 = 1  2 1 = 2  2 2 = 2*2 = 4  2 3 = 2*2*2 = 8  2 10 = 1,024 Digital Data3

 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: 2 10 = 1,024 bytes (≈ 1,000)  Megabyte: 2 20 = 1,048,576 bytes (≈ 1,000,000)  Gigabyte: 2 30 = 1,073,741,824 bytes (≈ 1,000,000,000)  Terabyte: 2 40 = 1,099,511,627,776 bytes (≈ 1,000,000,000,000) Digital Data4

 Base (also called radix )  Number around which system is organized  Number of symbols used to 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  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 Number SystemBaseDigits Decimal 100,1,…, 9 Binary 20, 1 Hexadecimal 160, 1,…, 9, A – F Digit Position Number System Decimal , , Binary Hexadecimal , , Digital Data5

 Decimal  Binary  Hexadecimal Digit Position ( k ) 3210 Weight (b k ) Decimal Weight1, Digit (a k ) 2002 Decimal Value 2, = 2,002 Digit Position ( k ) Weight (b k ) Decimal Weight1, Digit (a k ) Decimal Value 1, = 2,002 Digit Position ( k ) 210 Weight (b k ) Decimal Weight Digit (a k ) 7D2 Decimal Value 1, = 2,002 Digital Data6

 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  7D2 16 → 0x7D2 Digital Data7

8 DecimalBinaryHex DecimalBinaryHex A B C D E F

1. Divide the value by the given base x  The remainder will be the least significant (right- most) digit of the base x number 2. Divide the result by the base  The remainder becomes the most significant (left- most) digit of the base x number 3. Repeat step 2 until the result is zero  The remainder is the most significant digit of the base x number Digital Data9

 Convert the decimal value 150 to binary  What decimal value does the bit pattern   Convert the binary pattern  To hex Digital Data10

 Convert 2,016 to binary (base 2)  Convert 2,016 to hexadecimal (base 16) 7 / 16 =0remainder / 16 =7remainder E (14) 2,016 / 16 =126Remainder 0 Digital Data 11 1 / 2 =0remainder 1 3 / 2 =1remainder 1 7 / 2 =3remainder 1 15 / 2 =7remainder 1 31/ 2 =15remainder 1 63 / 2 =31remainder / 2 =63remainder / 2 =126remainder / 2 =252remainder 0 1,008 / 2 =504remainder 0 2,016 / 2 =1,008remainder 0 Binary Digits Left to Right = = 7E0 16 Hex Digits Left to Right

 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 Digital Data12

Digital Data13

 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    ‘%’ tells the program what follows is a hex value Digital Data14

 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   USA (ASCII text file)  (24-bit bitmap) Digital Data15

 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) Digital Data16

 Convention for naming files .  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  Determined by file format  Operating systems and applications make decisions based on file extension .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 a ZIP file  Can still open the “.txt” file with a program that recognizes the ZIP file format Digital Data17

 What string is represented by the ASCII values :  What exactly is displayed when we open the file yay.txt in notepad which consists of the following bytes:  What exactly would be displayed in Frhed when we open yay.txt? Digital Data18

 Number Systems Decoded detail.asp?wpid=315  RFC 20: ASCII format for Network Interchange  File Signatures Table  Comparison of Executable File Formats able_file_formats Digital Data19