1 Computing System 2. 2 Data Representation Number Systems 22
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1 Chapter 4: Computing System & Data Representation Christian Jacob 1 Computing System Abacus 3 2 Data Representation 19 3 Number Systems Important Number Systems for Computers Decimal Defined Binary Defined Octal Defined Hexadecimal Defined Comparison of Number Systems 32 TOC 1 Back
2 1 Computing System Input Processing Output ALU Arithmetic Logical Unit Control Unit Memory Computation process: input processing of information / data output Aspects of computing systems: Number System: Computing Technology: Program / Algorithm: representation of data (numbers, strings, ) technical implementation, computing device rules for applying the computing device to perform calculations First Back TOC 2 Prev Next Last
3 1.1 Abacus ALU: Abacus Input/Output: Control Unit: Memory: Human Human Human, paper First Back TOC 3 Prev Next Last
4 Number System: Computing Technology: Program / Algorithm: digital system; each digit is described by a pair of symbols (upper deck, lower deck) frame with vertical rods which represent the symbol pairs by the positions of the beads rules for moving the beads in order to perform calculations = 24 + First Back TOC 4 Prev Next Last
5 Number System: Computing Technology: Program / Algorithm: digital system; each digit is described by a pair of symbols (upper deck, lower deck) frame with vertical rods which represent the symbol pairs by the positions of the beads rules for moving the beads in order to perform calculations = 24 + First Back TOC Prev Next Last
6 Number System: Computing Technology: Program / Algorithm: digital system; each digit is described by a pair of symbols (upper deck, lower deck) frame with vertical rods which represent the symbol pairs by the positions of the beads rules for moving the beads in order to perform calculations = First Back TOC 6 Prev Next Last
7 Number System: Computing Technology: Program / Algorithm: digital system; each digit is described by a pair of symbols (upper deck, lower deck) frame with vertical rods which represent the symbol pairs by the positions of the beads rules for moving the beads in order to perform calculations = First Back TOC 7 Prev Next Last
8 Number System: Computing Technology: Program / Algorithm: digital system; each digit is described by a pair of symbols (upper deck, lower deck) frame with vertical rods which represent the symbol pairs by the positions of the beads rules for moving the beads in order to perform calculations = 2 + First Back TOC 8 Prev Next Last
9 Number System: Computing Technology: Program / Algorithm: digital system; each digit is described by a pair of symbols (upper deck, lower deck) frame with vertical rods which represent the symbol pairs by the positions of the beads rules for moving the beads in order to perform calculations = 2 + First Back TOC 9 Prev Next Last
10 Number System: Computing Technology: Program / Algorithm: digital system; each digit is described by a pair of symbols (upper deck, lower deck) frame with vertical rods which represent the symbol pairs by the positions of the beads rules for moving the beads in order to perform calculations = 30 + First Back TOC 10 Prev Next Last
11 Number System: Computing Technology: Program / Algorithm: digital system; each digit is described by a pair of symbols (upper deck, lower deck) frame with vertical rods which represent the symbol pairs by the positions of the beads rules for moving the beads in order to perform calculations = 30 + First Back TOC 11 Prev Next Last
12 Number System: Computing Technology: Program / Algorithm: digital system; each digit is described by a pair of symbols (upper deck, lower deck) frame with vertical rods which represent the symbol pairs by the positions of the beads rules for moving the beads in order to perform calculations = 30 + First Back TOC 12 Prev Next Last
13 Number System: Computing Technology: Program / Algorithm: digital system; each digit is described by a pair of symbols (upper deck, lower deck) frame with vertical rods which represent the symbol pairs by the positions of the beads rules for moving the beads in order to perform calculations = 30 + First Back TOC 13 Prev Next Last
14 Number System: Computing Technology: Program / Algorithm: digital system; each digit is described by a pair of symbols (upper deck, lower deck) frame with vertical rods which represent the symbol pairs by the positions of the beads rules for moving the beads in order to perform calculations = 0 + First Back TOC 14 Prev Next Last
15 Number System: Computing Technology: Program / Algorithm: digital system; each digit is described by a pair of symbols (upper deck, lower deck) frame with vertical rods which represent the symbol pairs by the positions of the beads rules for moving the beads in order to perform calculations = 0 + First Back TOC 1 Prev Next Last
16 Number System: Computing Technology: Program / Algorithm: digital system; each digit is described by a pair of symbols (upper deck, lower deck) frame with vertical rods which represent the symbol pairs by the positions of the beads rules for moving the beads in order to perform calculations = 0 + First Back TOC 16 Prev Next Last
17 Number System: Computing Technology: Program / Algorithm: digital system; each digit is described by a pair of symbols (upper deck, lower deck) frame with vertical rods which represent the symbol pairs by the positions of the beads rules for moving the beads in order to perform calculations = 0 + First Back TOC 17 Prev Next Last
18 Number System: Computing Technology: Program / Algorithm: digital system; each digit is described by a pair of symbols (upper deck, lower deck) frame with vertical rods which represent the symbol pairs by the positions of the beads rules for moving the beads in order to perform calculations = 60 First Back TOC 18 Prev Next Last
19 Data Representation Chapter 4: Computing System & Data Representation Christian Jacob 2 Data Representation Bit: smallest unit of information yes / no, on / off, L / 0, 1 / 0, V / 0V Byte: group of 8 bits --> 2 8 = 26 different states Word: the number of bits (word length) which can be processed by a computer in a single step (e.g., 32 or 64) --> machine dependent Representation: N N Word size in any given computer is fixed 16-bit word --> every word (memory location) can hold a 16-bit pattern, with each bit either 0 or 1 First Back TOC 19 Prev Next Last
20 Data Representation Chapter 4: Computing System & Data Representation Christian Jacob How many distinct patterns are there in a 16-bit word? Each bit has 2 possible values: 0 or 1 --> 1 bit has 2 distinct patterns With 2 bits, each one has 2 possibilities: 00, 01, 10, 11 --> 2 2 = 4 distinct bit patterns With 3 bits, again each one has 2 possibilities: > 2 3 = 8 distinct bit patterns In general, for N bits (a word of length N) we have 2 N distinct bit patterns. NOTE: What these bit patterns mean depends entirely on the context in which the patterns are used. First Back TOC 20 Prev Next Last
21 Data Representation Chapter 4: Computing System & Data Representation Christian Jacob Powers of 2: N 2 N N 2 N , , , ,048, ,097, ,194, ,388, ,777, ,4, ,108,864, ,217, , ,43, , ,870, , ,073,741, , ,147,483, , ,294,967, , ,89,934,92 First Back TOC 21 Prev Next Last
22 Number Systems Chapter 4: Computing System & Data Representation Christian Jacob 3 Number Systems Given: basic set Z of digits (or letters); basis B = Z B = 2, 8, 10, 16 Number = linear sequence of digits The value of a digit at a specific position depends on its value and on its position. The value of a number is the sum of these values. Examples: Z = { 0, 1, 2, 3, 4,, 6, 7, 8, 9} Z = {,,,,,,,,, } --> B = 10 --> B = = 9 * + = * + 4 * * + 1 * * + * * + 2 * 10 * First Back TOC 22 Prev Next Last
23 Number Systems Chapter 4: Computing System & Data Representation Christian Jacob Rational Numbers Number --> value R B = 0.z 1 z 2 z m 1 z m m R B = z i B i = z 1 B 1 + z 2 B z m 1 B m z m B m i = 1 4 R 10 = = z i 10 i = i = 0 NOTE: Today we use a positional system for number representation. The Roman number system, however, works almost totally different: MDCLXIV = M + D + C + L + X + V - I = = 1664 First Back TOC 23 Prev Next Last
24 Number Systems Chapter 4: Computing System & Data Representation Christian Jacob 3.1 Important Number Systems for Computers Name Base Digits dual, binary 2 0, 1 octal 8 0, 1, 2, 3, 4,, 6, 7 decimal 10 0, 1, 2, 3, 4,, 6, 7, 8, 9 sedecimal / hexadecimal 16 0, 1, 2, 3, 4,, 6, 7, 8, 9, A, B, C, D, E, F First Back TOC 24 Prev Next Last
25 Number Systems Chapter 4: Computing System & Data Representation Christian Jacob 3.2 Decimal Defined Z = { ,,,,,,,,, }; B = 10 Each place to the left of a digit in a string increases by a power of 10. Each place to the right of a digit in a string decreases by a power of = = First Back TOC 2 Prev Next Last
26 Number Systems Chapter 4: Computing System & Data Representation Christian Jacob 3.3 Binary Defined Z = { 01, }; B = 2 Each place to the left of a digit in a string increases by a power of 2. Each place to the right of a digit in a string decreases by a power of = = = First Back TOC 26 Prev Next Last
27 Number Systems Chapter 4: Computing System & Data Representation Christian Jacob Counting in Binary Decimal Dual Decimal Dual First Back TOC 27 Prev Next Last
28 Number Systems Chapter 4: Computing System & Data Representation Christian Jacob 3.4 Octal Defined Z = { ,,,,,,, }; B = 8 Each place to the left of a digit in a string increases by a power of 8. Each place to the right of a digit in a string decreases by a power of = = = First Back TOC 28 Prev Next Last
29 Number Systems Chapter 4: Computing System & Data Representation Christian Jacob Counting in Octal Decimal Octal Decimal Octal First Back TOC 29 Prev Next Last
30 Number Systems Chapter 4: Computing System & Data Representation Christian Jacob 3. Hexadecimal Defined Z = { A,,,,,,,,,,, B, C, D, E, F} ; B = 16 Each place to the left of a digit in a string increases by a power of 16. Each place to the right of a digit in a string decreases by a power of 16. FB40A 16 = = = 1,029, First Back TOC 30 Prev Next Last
31 Number Systems Chapter 4: Computing System & Data Representation Christian Jacob Counting in Hexadecimal (Sedecimal) Decimal Hexadecimal Decimal Hexadecimal A 26 1A 11 0B 27 1B 12 0C 28 1C 13 0D 29 1D 14 0E 30 1E 1 0F 31 1F First Back TOC 31 Prev Next Last
32 Number Systems Chapter 4: Computing System & Data Representation Christian Jacob 3.6 Comparison of Number Systems Decimal Dual Octal Hexadecimal Decimal Dual Octal Hexadecima l A A B B C C D D E E F F First Back TOC 32 Prev Next Last
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