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ICC Module Computation – Information Representation Information, Computation, and Communication Representation of Information 1

ICC Module Computation – Information Representation Computation works with information Scientific Google computation/simulation datacenter --> numbers Information management --> text, photos, movies… Control process -->signals (measurements, control...) 2

ICC Module Computation – Information Representation Objectives § Which ways can we use to represent numbers and symbols? § Is it possible to build an exact representation of the real world? 3

ICC Module Computation – Information Representation Agenda § Representation of the information § Natural Numbers (e.g., 2 4 5 6): operations/domain § Integers (e.g., -1 -5 4 45698) - Videos § Reals (e.g., 3.4 4.756): fix and floating point - Videos § From the alphabet to the ideograms (next week) 4

ICC Module Computation – Information Representation A Convention A representation is a convention 1 billion of people understand it… It is not difficult to assign meaning to a symbol but to agree on a common way to interpret it. 5

ICC Module Computation – Information Representation Towards the elementary unit of information 214 graphical motifs , called keys, were used to build ~100.000 Chinese ideograms The 26 letters of the Latin alphabet have been A B C … used to build/create ~1.000.000 words of western languages The 10 Arabic digits allow us to create an infinity of 0 1 2 3… numbers (and even encrypt all the words!) Question: which one is the simplest system of signs allowing to keep/save/preserve the same expression wealth/richness like the 10 digits? Answer: a system of 2 symbols 6

ICC Module Computation – Information Representation A Bit All information can be represented with the help of a set of binary elements In agreement, a binary element is worth 0 or 1 . In agreement we use the English expression “ binary digit ” or bit in short (abbreviated) Shortcut for bit: b or bit 7

ICC Module Computation – Information Representation 0 2 1 distinct pieces 1 bit of information 1 How to represent more information? 0 0 2 2 0 1 0 distinct pieces 2 bits of information 1 0 1 1 1 8

ICC Module Computation – Information Representation n bits allow us to represent 2 n distinct pieces of information 2 n distinct pieces of information can be represented by 2 1 3 5 log 2 ( 2 n ) = n log 2 ( 2 ) = n bits 4 The uppercase letters of the latin alphabet The digits from 0 to 9 Exercise: The days of the week How many bits do we need to represent 2 4 • the days of the week: • the digits 0 to 9 : 8 • the letters of the alphabet: • upper case 10 • upper + lower case 7 16 • upper + lower case + signs ... General rule: For K distinct information , the number of bits n needed to represent this information 26 is the integer higher or equal to log 2 K 9 32

ICC Module Computation – Information Representation n bits allow us to represent 2 n distinct pieces of information n 2 n 1 2 Good practice for fast estimation: 2 4 3 8 2 10 = Kb (Ki) ≈ 10 3 = kilo (k) 4 16 2 20 = Mb (Mi) ≈ 10 6 = mega (M) 5 32 6 64 2 30 = Gb (Gi) ≈ 10 9 = giga (G) 7 128 8 256 10 1'024 2 32 = 2 30+2 = 2 30 . 2 2 ≈ 4 G 20 1'048'576 30 1'073'741'824 32 4'294'967'296 10

ICC Module Computation – Information Representation Organization of the information Convention: a byte (octet in French) is a group of 8 bits. Shortcut for byte: B Recall shortcut for bit: b (or simply bit) The most common information representations use a byte or a sequence of bytes. byte 11

ICC Module Computation – Information Representation Quiz § What is the simplest system to represent information? § How many bits to you need to store all the days of a year? § How many pieces of information can you store with 2B (bytes)? 12

ICC Module Computation – Information Representation Agenda § Representation of the information § Natural Numbers (e.g., 2 4 5 6): operations/domain § Integers (e.g., -1 -5 4 45698) - Videos § Reals (e.g., 3.4 4.756): fix and floating point - Videos § From the alphabet to the ideograms (next week) 13

ICC Module Computation – Information Representation Representing Natural Numbers (Entier naturel) 14

ICC Module Computation – Information Representation How to represent a natural number? Reminder: all numbers can be represented with a set of binary elements. Definition: a sequence of 0’s and 1’s is called a binary pattern A binary pattern on its own is not enough to understand what is encoded. We need an interpretation method of the binary pattern given as data One solution: the positional notation of numbers 15

ICC Module Computation – Information Representation Positional notation of numbers Example of an integer in base 10: 703 The number 703 is the abbreviated notation of the expression: 7 . 10 2 + 0 . 10 1 + 3 . 10 0 the digit on the right is always multiplied to the base (10) § raised to the power 0 the power of the base increases by one from digit to digit, § going from right to left this convention of positional notation can be used with § any base 16

ICC Module Computation – Information Representation Positional representation in base 2 § Depends on the same conventions as in base 10 (decimal) MSB LSB 2 7 2 6 2 5 2 4 2 3 2 2 2 1 2 0 § Most significant (bit) on the left (MSB) § Least significant (bit) on the right (LSB) 17

ICC Module Computation – Information Representation Practice: Conversions § From binary to decimal: 1 16 8 4 2 128 64 32 2 7 2 6 2 5 2 4 2 3 2 2 2 1 2 0 1 0 0 1 0 1 0 0 § add the powers of two present in the binary pattern + + + + + + + 0 1 = 11 0 0 0 0 8 2 18

ICC Module Computation – Information Representation Practice: Conversions From decimal to binary : decompose an integer X to a sum of powers of two: § 11 dec = 2 3 + 2 1 + 2 0 = 1011 bin Algorithm: take the integer division of the number by 2 as long as the result it larger than or equal to 2 11 = 2 . 5 + 1 11 div 2 = 5 + 1 rest = 2 . (2 . 2 + 1) + 1 5 div 2 = 2 + 1 rest = 2 . (2 . (2 . 1 + 0) +1) + 1 2 div 2 = 1 + 0 rest = 1 . 2 3 + 0 . 2 2 + 1 . 2 1 + 1 . 2 0 = 1011 1011 19

ICC Module Computation – Information Representation Natural numbers: covered domain (1) A representation for a computer is associated to a fix capacity expressed in number of bits (or of bytes). Example: 32-bit computer. This device has instructions to implement the base operations (addition, multiplication, etc.) very rapidly for numbers represented with 4 bytes (32 bits). So the limit to the number of different numbers that can be represented is 2 32 20

ICC Module Computation – Information Representation Natural Numbers: covered domain (2) If the representation is of natural numbers (= non-negative integers), its covered domain for 32 bits is: Min = binary pattern with 0 everywhere = zero 2 32 2 31 2 0 2 30 2 1 Max = binary pattern with 1 32 bits everywhere = 2 32 --1 21

ICC Module Computation – Information Representation Quiz § If we represent natural numbers with 8 bits using the positional representation in base 2, which numbers can we represent? § Which number does the binary patter 1001001 represent, if we interpret it using positional representation in base 2? § What is the representation of number 156 using 8 bits in positional representation in base 2? 22

ICC Module Computation – Information Representation Natural Numbers: covered domain (3) Computations on natural numbers are correct if the desired result is a natural number and belongs to the covered domain The chosen representation must take into account all possible outcomes Reasons for capacity overflow: integer division: Loss of fractional part • multiplication, addition, subtraction: propagation of the • carry beyond 2 31 23

ICC Module Computation – Information Representation Addition in Decimal vs Binary § Addition tables with numbers 0-9 and 0-1 a b a+b a b a+b 0 0 0 0 0 0 0 1 1 0 1 1 0 2 2 1 0 1 .. .. .. 1 1 1 0 0 9 9 2 1 2 0 1 0 1 1 1 2 .. .. .. 2 0 2 2 1 3 .. .. .. 9 9 1 8 24

ICC Module Computation – Information Representation Examples of capacity overflow (1) Example 1: addition of 2 integer of capacity of one bit 0 0 1 1 +0 +1 +0 +1 -- -- -- -- 0 1 1 10 Only one bit for position 2 0 , therefore position 2 1 is lost. the carry is lost 25

ICC Module Computation – Information Representation Examples of capacity overflow (2) Example 2: addition with 32 bits (2 32 – 1 ) + 1 = ? 0 2 31 2 32 2 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 26

ICC Module Computation – Information Representation Unsigned integers: covered domain and overflow Example: 000 on 3 bits 001 111 0 1 7 010 110 2 6 3 5 4 101 011 100 27