Content. Information Coding. Where are we? Where are we? Bit vectors (aka Words )

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1 Information Coding Lecturer: Guillaume Beslon Original Author: Lionel Morel Computer Science and Information Technology - INSA Lyon Fall / 36 2 / 36 Where are we? Where are we? Programmers usually manipulate numbers (Relative Numbers, Reals, etc) Machines only understand binary ) We need to make the link between the two. 3 / 36 4 / 36 Bit vectors (aka Words ) Basic information: the bit 2 {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): I n = 1 is called a byte or a char I n = 2 is called a short I n = 4 is usually called an int (or float if it represents a pseudo-real number) I 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! 5 / 36 (Parenthesis: good language design) 1 Consider the names chosen for the numeric types in C: I char (the 8-bit integer) is an abbreviated noun (character) from typography I unsigned char??? I you can add two char AB I int is an abbreviated noun (integer) from mathematics I although = I short and long are adjectives I float is a verb, at least it is a computer term I double means double what? I long double is not even syntactically correct in english (and what about long long!) After so much nonsense, if you re lost, it is not your fault 1 courtesy of F. de Dinechin 6 / 36

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 2 0, 1 (in base 2). Using positional notation we can interpret the value of x as a natural number: Xn 1 x = x i 2 i i=0 2 n different values can be represented: 0 apple x apple 2 n 1 7 / 36 see the poly for a generalization to base > 1 8 / 36 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 / / 36 MSB: Most Significant Bit (Byte) Warning: MSB/LSB not to be confused with little endian and big endian 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. I MSB/LSB are to be understood relatively to the Binary code. I Little-Endian and Big-Endian are ways to organise the bytes into the physical memory of a computer...! Comics vs. Mangas I Big-Endian : The most significant byte is stored first in the lowest memory address (or big in first). I Little-Endian : Stores the least significant bytes in the lowest memory address.! When you directly look at the memory content, be careful to the endianness / / 36

3 Notations and important values 4 We write: (x n 1, x n 2,...,x 1, x 0 ) when writing x in base. eg: I (101) 2 =(5) 10 I (1010) 2 =(10) 10 for n = 8: I max: 2 n 1 = 255 I 256 different values for n = 32: I 2 n 1 = I 4 billions different values for n = 16: I 2 n 1 = I different values for n = 64: I 2 n 1 = a lot! I Still not a infinite number of values... 4 Probably one of the rare things to be known by heart 13 / / 36 Binary! Decimal Binary Decimal Any number p 2 N can be represented in a unique positional form in base 2, using n bits (with n = blog 2 (p)c + 1): Xn 1 (x n 1 x n 2 x 1 x 0 ) 2 := x i 2 i. NB: bits are numbered from 0 to n 1 i=0 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 {z } x {z} 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! 15 / 36 Again, see poly for a generalization to any base > 1 16 / 36 Binary Decimal - example To convert n =(423) 10 to binary, we 423 = = = = = = = = = From this we deduce that: (423) 10 =( ) 2 17 / / 36

4 Fast equivalence: the trick! / / 36 Two ways of representing x : I 1 bit for the sign, the rest for x 0 if x 0, x n 1 = 1 if x < 0. and (x n 2, x n 3,...,x 1, x 0 )= x I Pros: Simple to understand I Cons: 2 writings for 0 I Cons: hardware implementation 6= unsigned integers I Two s complement I Cons: Less easy to understand I Pros: numbering and hardware implementation is unchanged! Used in 99.99% of digital circuits 5 5 But other solutions are also used, e.g. for floating values / / 36 Two s Complement - equation Two s Complement - intuition 6 Let x be a vector of n bits: x n 1, x n 2,, x 1, x 0, with x i 2 0, 1 The value of x interpreted as a signed integer is: Xn 2 x = x n 1 2 n 1 + x i 2 i. 2 n different values can be represented: i= n 1 apple x < 2 n credit: Benoît Lopez 23 / / 36

5 Two s Complement - compute the opposite of x I The complement of a bit a 2 {0, 1} is: 1, when a = 0 a = 0, when a = 1 I 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 25 / / 36 Number Extension How do we assign a n-bits vector to an m-bits vector? I n > m, truncate! lose most significant bits I 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,...,y n, 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 27 / / 36 Useful notations Reading binary is a pain in the ass! In practice, everybody uses hexadecimal: Max values: binary hexa decimal binary hexa decimal x x x x x xA x xB x xC x xD x xE x xF 15 Useful Notation: 0x... Other interpretation of bit vectors I (Pseudo-)real numbers I Bit fields, indicators, booleans,... I Characters/symbols: encodings such as ASCII or ISO or UTF-8. Useful for writing human-readable messages I Encoded/compressed data (bzip,... ), music (mp3, ogg,...), video (mpeg2, h264,...) I Encrypted data I Instructions = programs in their final form (that which is interpreted by HW). I... I on 1 byte: 0xFF = I on 2 bytes: 0xFFFF = All this necessitates interpreting these bit vectors! I on 4 bytes: 0xFFFF.FFFF = / / 36

Floating points Floating points Integers have a constant precision but a very low dynamics... I Difficult to represent very large values (e.g. mass of the universe in kilograms) I Impossible to represent very small values (e.

6 Floating points Floating points Integers have a constant precision but a very low dynamics... I Difficult to represent very large values (e.g. mass of the universe in kilograms) I Impossible to represent very small values (e.g. mass of an electron in kilograms) I Encoding of useless (and sometimes misleading) LSB Solution: scientific notation ( and 9, ) Floating points are the binary analogs to scientific notation. Pseudo-real numbers are represented by a mantissa (or fraction) and an exponent coded by (still finite) bit vectors. Floating points are normalised by the IEEE 754 format. Mainly two formats are used: I Single precision (32 bits): sign (1 bit), exponent (8 bits, including exponent sign), fraction (23 bits)! float I Double precision (64 bits): sign (1 bit), exponent (11 bits, including exponent sign), fraction (52 bits)! double Floating points are NOT reals. The maximum number of values one can encode with a float is still 2 32 = !! Floating points have precision issues that are (1) too difficult to be listed here and (2) extremely dangerous! (e.g., in general a + b 6= b + a and b + a b a 6= 0) 31 / / 36 Bit field Character encoding UTF-8 33 / / 36 Program Instructions Next lecture... Combinatorial Logic and Circuits!! ie, how to process information encoded in bit vectors? 35 / / 36

Information Coding. Content

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