Information Coding. Content

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1 Information Coding Lionel Morel Computer Science and Information Technology - INSA Lyon Automne 2016 AC - Coding Automne / 35 Introduction Content 1 Introduction 2 Natural Numbers in base 2 3 Conversion 4 5 Notations and Other Interpretations AC - Coding Automne / 35

2 Introduction Where are we? AC - Coding Automne / 35 Introduction Where are we? Programmers usually manipulate numbers (Relative Numbers, Reals, etc) Machines only understand binary We need to make the link between the two. AC - Coding Automne / 35

3 Introduction Bit vectors Basic information: the bit {0, 1} (for binary digit) Example of word: We work with finite words! l is the number of bits: in practice l = 8 n n is the number of bytes (octets in french) When programming (on a 32 or 64 bits machine): n = 1 is called a byte n = 2 is called a short n = 4 is called an int (or float if it represents a pseudo-real number) n = 8 is called long long (or double for a pseudo-real) Beware, these are naming conventions only!!! So, always make sure you know what you are talking about! AC - Coding Automne / 35 Introduction (Parenthesis: good language design) 1 Consider the names chosen for the numeric types in C: char (the 8-bit integer) is an abbreviated noun (character) from typography unsigned char??? AB you can add two char int is an abbreviated noun (integer) from mathematics although = short and long are adjectives float is a verb, at least it is a computer term double means double what? long double is not even syntactically correct in english After so much nonsense, if you re lost, it is not your fault 1 courtesy of F. de Dinechin AC - Coding Automne / 35

4 Natural Numbers in base 2 Content 1 Introduction 2 Natural Numbers in base 2 3 Conversion 4 5 Notations and Other Interpretations AC - Coding Automne / 35 Natural Numbers in base 2 Natural numbers in base 2 (Unsigned integers) Let x be a vector of n bits: x n 1, x n 2,, x 1, x 0, with x i 0, 1 (in base 2). We interpret the value of x as a natural number: x = n 1 i=0 x i 2 i 2 n different values can be represented: : 0 x 2 n 1 see the poly for a generalization to base β > 1 AC - Coding Automne / 35

5 Natural Numbers in base 2 Blackboard examples AC - Coding Automne / 35 Natural Numbers in base 2 LSB: Least Significant Bit (Byte) LSBit is the bit position in a binary integer giving the units value, that is, determining whether the number is even or odd 2. LSByte is the byte in that position of a multi-byte number which has the least potential value. 2 AC - Coding Automne / 35

6 Natural Numbers in base 2 MSB: Most Significant Bit (Byte) MSBit is the bit position in a binary number having the greatest value 3. MSByte is the byte (or octet) in that position of a multi-byte number which has the greatest potential value. 3 AC - Coding Automne / 35 Natural Numbers in base 2 Some notations and convenient values We write: (x n 1, x n 2,..., x 1, x 0 ) β when writing x in base β. eg: (101) 2 = (5) 10 (1010) 2 = (10) 10 for n = 8: max: 2 n 1 = different values for n = 16: max: 2 n 1 = different values AC - Coding Automne / 35

7 Conversion Content 1 Introduction 2 Natural Numbers in base 2 3 Conversion 4 5 Notations and Other Interpretations AC - Coding Automne / 35 Conversion Binary Decimal Conversion Any number p N can be represented in a unique positional form in base 2, using n bits: (x n 1 x n 2 x 1 x 0 ) β := n 1 i=0 x i 2 i. NB: bits are numbered from 0 to n 1 AC - Coding Automne / 35

8 Conversion Blackboard examples AC - Coding Automne / 35 Conversion Binary Decimal Conversion The remainder of the euclidean division of x by 2 gives the right-most digit of its representation in base 2: x = x n 1 2 n 1 + x n 2 2 n x x x 0 = ( x n 1 2 n 2 + x p 2 2 n x ) + x x }{{}}{{} 0 quotient remainder so, x 0 = n mod 2. We get all the digits of the binary representation of a natural number n by applying euclidean divisions to the successive quotients until we reach 0 as a quotient. NB: This gives least-significant bits first! Again, see poly for a generalization to any base β > 1 AC - Coding Automne / 35

9 Conversion Binary Decimal Conversion - example To Convert n = (423) 10 to binary, we 423 = = = = = = = = = From this we deduce that: (423) 10 = ( ) 2 AC - Coding Automne / 35 Conversion Blackboard examples AC - Coding Automne / 35

10 Conversion Intermission AC - Coding Automne / 35 Content 1 Introduction 2 Natural Numbers in base 2 3 Conversion 4 5 Notations and Other Interpretations AC - Coding Automne / 35

11 2 ways of representing x : 1 bit for the sign, the rest for x x n 1 = { 0 if x 0, 1 if x < 0. and (x n 2, x n 3,..., x 1, x 0 ) = x Simple to understand 2 writings for 0 hardware implementation unsigned integers Two s complement Less easy to understand numbering and hardware implementation is unchanged Used in 99.9% of digital circuits AC - Coding Automne / 35 Two s complement - intuition AC - Coding Automne / 35

12 Two s Complement - intuition credit: Benoît Lopez AC - Coding Automne / 35 2 s Complement Let x be a vector of n bits: x n 1, x n 2,, x 1, x 0, with x i 0, 1 The value of x interpreted as a signed integer is: x = x n 1 2 n 1 + n 2 i=0 x i 2 i. AC - Coding Automne / 35

13 Blackboard examples AC - Coding Automne / 35 2 s Complement The complement of a bit a {0, 1} is: a = { 1, when a = 0 0, when a = 1 if x = x n 1, x n 2,, x 1, x 0 then x = (x n 1, x n 2,, x 1, x 0 ) + 1 AC - Coding Automne / 35

14 Blackboard examples AC - Coding Automne / 35 Sign Extension How do we assign a n-bits vector to an m-bits vector? n > m, truncate lose most significant bits n < m, requires a sign extension Consider (x n 1, x n 2,..., x 1, x 0 ). Let s write it as (y m 1, y m 2,..., yn, y n 1, y n 2,..., y 1, y 0 ). The value of x is preserved if we take: y m 1 = x n 1, y m 2 = x n 1,..., y n = x n 1, y n 1 = x n 1, y n 2 = x n 2,..., y 1 = x 1, y 0 = x 0 AC - Coding Automne / 35

15 Notations and Other Interpretations Content 1 Introduction 2 Natural Numbers in base 2 3 Conversion 4 5 Notations and Other Interpretations AC - Coding Automne / 35 Notations and Other Interpretations Useful notations Reading binary is a misery! In practice, we often use hexadecimal: binary hexa decimal binary hexa decimal A B C D E F 15 Max values: on 1 byte: FF on 2 bytes: FFFF on 3 bytes: FFFFFF on 4 bytes: FFFFFFFF Useful Notation: 0x... AC - Coding Automne / 35

16 Notations and Other Interpretations Other interpretation of bit vectors Bit fields, indicators, booleans,... Characters/symbols: encodings such as ASCII or ISO or UTF-8. Useful for writing human-readable messages Encoded/compressed data (bzip,... ), music (mp3, ogg,...), video (mpeg2, h264,...) Encrypted data Instructions = programs in their final form (that which is interpreted by HW). All this necessitates interpreting these bit vectors! AC - Coding Automne / 35 Notations and Other Interpretations Bit field AC - Coding Automne / 35

17 Notations and Other Interpretations Character encoding UTF-8 AC - Coding Automne / 35 Notations and Other Interpretations Program Instructions AC - Coding Automne / 35

18 Notations and Other Interpretations La prochaine fois (je ne vous le chanterai pas...) Combinatorial Logic and Circuits!! ie, how can we treat the information that we encode as bits. AC - Coding Automne / 35