DOCSLIB.ORG

Logical Bases: Representation of Numbers

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text

Load more

Logical Bases: Representation of Numbers Decimal numbers most commonly used by humans are in base 10 and consist of the ten decimal digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. For example, the decimal number 373 is 373 = 3*100 + 7*10 + 3 Positional notation describes the numbers according to position. For example, the two 3s in the decimal number 373 are in different positions, the 3 in the first position (at left) has a value which is 100 times more than the other 3. This notation can be described by a data flow diagram as follows at the left below. Binary numbers are most commonly used in digital computing; they have base 2. The data flow diagram at the right shows the binary number 1101 which is the decimal number 13. 1101 = 1*8 + 1*4 + 0*2 + 1*1 = 13 in decimal Decimal numbers (base 10) Binary numbers (base 2) 3 7 3 1 1 0 1 10 2 100 10 1 8 4 2 1 * * * * * * * 300 70 3 8 4 0 1 + + 373 13 1 Binary Addition To add the ten decimal digits requires a table of 10*10 = 100 entries; but if the digits are binary a table of only 2*2 = 4 rows is needed. The binary add definition follows: 0 + 0 = 0 0 + 1 = 1 1 + 0 = 1 1 + 1 = 2 = 10 = 0 with carry of 1 A table of such a simple adder (half adder) follows, showing the output of a sum s and a carry c. A logical gate diagram shows that the carry is simply the And of a, b. The sum is actually the exclusive-Or, which is formed in detail below. HA = Half Adder in out a b s c a | b & s 0 0 0 0 a ! 0 1 1 0 b 1 0 1 0 a & c 1 1 0 1 b carry 2 Longer Binary Addition Addition of two numbers, say A=3=11 and B=6=110 to yield a third D is done in stages. At each stage there is input a bit value a, and b, and a carry from the previous stage. Then there is an output of a sum s, and a carry to the next stage. This process begins with the least significant digits (at the right) and proceeds to the left. The previous Half Adders can be used as shown to create a Full Adder. A = 3 = 0 0 1 1 FA = Full Adder c a A HA B = 6 = 0 1 1 0 s b B c 0 1 1 0 Carry FA FA FA FA 0 c a 1 HA s b c D = 9 = 1 0 0 1 sum 3 General Numbers: to any base b In general, a number can be represented in any base b as shown. The base may be any integer; the bases 2, 8, 10, 16 are most common. However, there may be unusual bases, negative values, fractions, etc. The coefficients c0, c1, c2, .. are usually integers from some small set, usually 0, 1, 2, etc up to the value of the base. General numbers (base b) cn ... c2 c1 c0 b bn b2 b1 b0 * * * * + cn*bn + .. + c2*b2 + c1*b1 + c0*b0 4 Numbers in base 3 Numbers in base 3 have the 3 coefficients, 0, 1 and 2. Some numbers, from 0 to 12, are shown in this base Note the beauty of this table; group them 3 at a time. Ternary numbers (base 3) i base3 2 0 1 3 0 000 1 001 27 9 3 1 2 002 3 010 * * * * 4 011 5 012 6 020 + 7 021 8 022 9 100 Example: 10 101 2 1 0 11 102 (201)3 = 2*3 + 0*3 + 1*3 12 110 = 2*9 + 0*3 + 1*1 = 18 + 0 + 1 = 19 (201)3 = (19)10 5 Numbers in base 8 (octal) Numbers in base 8 have the 8 coefficients, 0, 1, 2, .. 7. Some numbers, from 0 to 12 are shown in this base 8. Octal numbers (base 8) i octal cn ... c2 c1 c0 8 0 0 1 1 512 64 8 1 2 2 3 3 * * * * 4 4 5 5 6 6 + 7 7 8 10 9 11 cn*8n + .. + c2*82 + c1*81 + c0*80 10 12 Example: 11 13 (201) = 2*82 + 0*81 + 1*80 8 12 14 = 2*64 + 0*8 + 1*1 = 128 + 0 + 1 = 129 (201)8 = (129)10 6 Numbers in base 16 (Hexadecimal) Numbers in base 16 have the 16 coefficients, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F Some numbers, from 0 to 16 are shown in this base This hexadecimal base (called Hex) is very common in computing. Octal numbers (base 8) i hex cn ... c2 c1 c0 0 0 16 1 1 4096 256 16 1 2 2 3 3 4 4 * * * * 5 5 6 6 + 7 7 8 8 9 9 cn*16n + .. + c2*162 + c1*161 + c0*160 10 A Example: 11 B 2 1 0 (201)16 = 2*16 + 0*16 + 1*16 12 C = 2*256 + 0*16 + 1*1 13 D = 512 + 0 + 1 14 E = 513 15 F (201)16 = (513)10 16 10 7 Numbers in base -2 Bases can be negative;the following one is binary base (-2). The coefficients are still binary, 0s and 1s only. Some numbers, both positive and negative are shown in this base This is not a common way to create negative numbers. Binary numbers (base -2) cn ... c2 c1 c0 i base -2 i base -2 -2 0 0 0 0 1 1 -1 11 -8 4 -2 1 2 110 -2 10 3 111 -3 1101 * * * * 4 100 -4 1100 5 101 -5 1111 + 6 11010 -6 1110 7 11011 -7 1001 8 110000 -8 1000 cn*bn + .. + c2*b2 + c1*b1 + c0*b0 8 Numbers in base 2 (binary) with negative coefficients Bases can be positive with negative coefficients; The following one has a binary base 2 with coefficients 1, 0 and -1. The coefficient -1 can be written as 7, or as a 1 with a bar over it. Some numbers, both positive and negative are shown in this base This is also not a common way to create negative numbers. Binary numbers (base 2) i base 2 i base 2 cn ... c2 c1 c0 0 0 0 0 2 1 1 -1 7 8 4 2 1 2 10 -2 70 3 11 -3 77 4 100 -4 700 * * * * 5 101 -5 707 6 110 -6 770 + 7 111 -7 777 8 1000 -8 7000 cn*bn + .. + c2*b2 + c1*b1 + c0*b0 9 Common Binary Procrustean representation Most computers are binary with a fixed number of bits (8, 16, 24, 32, 64). They are called procrustean, after the Greek legend of Procrustes, or PolyDemon, who had an iron bed, placed people on it, and if they were too tall he chopped them off to the fixed size; if they were too short he stretched them to this fixed size. Computer registers, words, etc are also fixed size. Signed Magnitude (s&m) A simple way of representing binary numbers which are positive or negative is to specify the sign as the first binary digit (or bit). A negative sign is 1; a positive sign is 0. For example, with 4 bits, there are 16 possible integers as shown. Notice however that there are two zeros, a positive and a negative one. Not good! Binary numbers (base 2) i s&m i s&m +/- c2 c1 c0 0 0000 -0 1000 2 1 0001 -1 1001 4 2 1 2 0010 -2 1010 Sign 3 0011 -3 1011 4 0100 -4 1100 0 is + * * * 1 is - 5 0101 -5 1101 6 0110 -6 1110 + 7 0111 -7 1111 10 Ones Complement Another way of representing negative numbers is to do a complement; to complement or “Not” each bit of the positive number to get the negative. For example, with 4 bits, the 16 possible integers are shown below. Notice again however that there are two zeros, a positive and a negative one. Not good. The conversion of a number to its negative is shown in the data diagram below. i 1comp i 1comp cn ... c2 c1 c0 0 0000 -0 1111 2 1 0001 -1 1110 2 0010 -2 1101 ! ! ! ! 3 0011 -3 1100 4 0100 -4 1011 5 0101 -5 1010 cn c2 c1 c0 6 0110 -6 1001 2 7 0111 -7 1000 11 Twos Complement Yet another way of representing negative numbers is to do another kind of complement; to view the negative numbers as a cyclic odometer as on a vehicle. A 4 bit system, goes forward, from 0000 to 1111 and then repeats starting with 0000. Going backwards from 0000, the next value (which is 1111) corresponds to -1. Similarly for -2, -3, etc to -8. There is only one value for zero, and the values range from -8 to +7. In general for n digits the values range from -2n to 2n - 1 Such a 2s complement is most commonly used in computers, with more than 4 bits. The 2s complement negative numbers are just one more than the 2s complement, so to create the 2s complement, simply invert all the bits and add 1 to the result. i 2comp i 2comp +2 = 0010 0 0000 -1 1111 +2 = 0010 1 0001 -2 1110 +4 = 0100 2 0010 -3 1101 3 0011 -4 1100 -5 = 1011 4 0100 -5 1011 +2 = 0010 5 0101 -6 1010 -3 = 1101 6 0110 -7 1001 -5 = 1011 7 0111 -8 1000 -2 = 1110 -7 = 1001 The sum of two numbers in 2s complement is shown to the right, in three examples.
Recommended publications
  • Maths Week 2021

    Maths Week 2021

    Maths Week 2021 Survivor Series/Kia Mōrehurehu Monday Level 5 Questions What to do for students 1 You can work with one or two others. Teams can be different each day. 2 Do the tasks and write any working you did, along with your answers, in the spaces provided (or where your teacher says). 3 Your teacher will tell you how you can get the answers to the questions and/or have your work checked. 4 When you have finished each day, your teacher will give you a word or words from a proverb. 5 At the end of the week, put the words together in the right order and you will be able to find the complete proverb! Your teacher may ask you to explain what the proverb means. 6 Good luck. Task 1 – numbers in te reo Māori The following chart gives numbers in te reo Māori. Look at the chart carefully and note the patterns in the way the names are built up from 10 onwards. Work out what each of the numbers in the following calculations is, do each calculation, and write the answer in te reo Māori. Question Answer (a) whitu + toru (b) whā x wa (c) tekau mā waru – rua (d) ono tekau ma whā + rua tekau ma iwa (e) toru tekau ma rua + waru x tekau mā ono Task 2 - Roman numerals The picture shows the Roman Emperor, Julius Caesar, who was born in the year 100 BC. (a) How many years ago was 100 BC? You may have seen places where numbers have been written in Roman numerals.
  • Balanced Ternary Arithmetic on the Abacus

    Balanced Ternary Arithmetic on the Abacus

    Balanced Ternary Arithmetic on the Abacus What is balanced ternary? It is a ternary number system where the digits of a number are powers of three rather than powers of ten, but instead of adding multiples of one or two times various powers of three to make up the desired number, in balanced ternary some powers of three are added and others are subtracted. Each power of three comprisising the number is represented by either a +, a -, or 0, to show that the power of three is either added, subtracted, or not present in the number. There are no multiples of a power of three greater than +1 or -1, which greatly simplifies multiplication and division. Another advantage of balanced ternary is that all integers can be represented, both positive and negative, without the need for a separate symbol to indicate plus or minus. The most significant ternary digit (trit) of any positive balanced ternary number is + and the most significant trit of any negative number is -. Here is a list of a few numbers in decimal, ternary, and balanced ternary: Decimal Ternary Balanced ternary ------------------------------------------------- -6 - 20 - + 0 (-9+3) -5 - 12 - + + (-9+3+1) -4 - 11 - - ( -3-1) -3 - 10 - 0 etc. -2 - 2 - + -1 - 1 - 0 0 0 1 1 + 2 2 + - 3 10 + 0 4 11 + + 5 12 + - - 6 20 + - 0 Notice that the negative of a balanced ternary number is formed simply by inverting all the + and - signs in the number. Thomas Fowler in England, about 1840, built a balanced ternary calculating machine capable of multiplying and dividing balanced ternary numbers.
  • Bit, Byte, and Binary

    Bit, Byte, and Binary

    Bit, Byte, and Binary Number of Number of values 2 raised to the power Number of bytes Unit bits 1 2 1 Bit 0 / 1 2 4 2 3 8 3 4 16 4 Nibble Hexadecimal unit 5 32 5 6 64 6 7 128 7 8 256 8 1 Byte One character 9 512 9 10 1024 10 16 65,536 16 2 Number of bytes 2 raised to the power Unit 1 Byte One character 1024 10 KiloByte (Kb) Small text 1,048,576 20 MegaByte (Mb) A book 1,073,741,824 30 GigaByte (Gb) An large encyclopedia 1,099,511,627,776 40 TeraByte bit: Short for binary digit, the smallest unit of information on a machine. John Tukey, a leading statistician and adviser to five presidents first used the term in 1946. A single bit can hold only one of two values: 0 or 1. More meaningful information is obtained by combining consecutive bits into larger units. For example, a byte is composed of 8 consecutive bits. Computers are sometimes classified by the number of bits they can process at one time or by the number of bits they use to represent addresses. These two values are not always the same, which leads to confusion. For example, classifying a computer as a 32-bit machine might mean that its data registers are 32 bits wide or that it uses 32 bits to identify each address in memory. Whereas larger registers make a computer faster, using more bits for addresses enables a machine to support larger programs.
  • VSI Openvms C Language Reference Manual

    VSI Openvms C Language Reference Manual

    VSI OpenVMS C Language Reference Manual Document Number: DO-VIBHAA-008 Publication Date: May 2020 This document is the language reference manual for the VSI C language. Revision Update Information: This is a new manual. Operating System and Version: VSI OpenVMS I64 Version 8.4-1H1 VSI OpenVMS Alpha Version 8.4-2L1 Software Version: VSI C Version 7.4-1 for OpenVMS VMS Software, Inc., (VSI) Bolton, Massachusetts, USA C Language Reference Manual Copyright © 2020 VMS Software, Inc. (VSI), Bolton, Massachusetts, USA Legal Notice Confidential computer software. Valid license from VSI required for possession, use or copying. Consistent with FAR 12.211 and 12.212, Commercial Computer Software, Computer Software Documentation, and Technical Data for Commercial Items are licensed to the U.S. Government under vendor's standard commercial license. The information contained herein is subject to change without notice. The only warranties for VSI products and services are set forth in the express warranty statements accompanying such products and services. Nothing herein should be construed as constituting an additional warranty. VSI shall not be liable for technical or editorial errors or omissions contained herein. HPE, HPE Integrity, HPE Alpha, and HPE Proliant are trademarks or registered trademarks of Hewlett Packard Enterprise. Intel, Itanium and IA64 are trademarks or registered trademarks of Intel Corporation or its subsidiaries in the United States and other countries. Java, the coffee cup logo, and all Java based marks are trademarks or registered trademarks of Oracle Corporation in the United States or other countries. Kerberos is a trademark of the Massachusetts Institute of Technology.
  • Binary Numbers

    Binary Numbers

    Binary Numbers X. Zhang Fordham Univ. 1 Numeral System ! A way for expressing numbers, using symbols in a consistent manner. ! ! "11" can be interpreted differently:! ! in the binary symbol: three! ! in the decimal symbol: eleven! ! “LXXX” represents 80 in Roman numeral system! ! For every number, there is a unique representation (or at least a standard one) in the numeral system 2 Modern numeral system ! Positional base 10 numeral systems ! ◦ Mostly originated from India (Hindu-Arabic numeral system or Arabic numerals)! ! Positional number system (or place value system)! ◦ use same symbol for different orders of magnitude! ! For example, “1262” in base 10! ◦ the “2” in the rightmost is in “one’s place” representing “2 ones”! ◦ The “2” in the third position from right is in “hundred’s place”, representing “2 hundreds”! ◦ “one thousand 2 hundred and sixty two”! ◦ 1*103+2*102+6*101+2*100 3 Modern numeral system (2) ! In base 10 numeral system! ! there is 10 symbols: 0, 1, 2, 3, …, 9! ! Arithmetic operations for positional system is simple! ! Algorithm for multi-digit addition, subtraction, multiplication and division! ! This is a Chinese Abacus (there are many other types of Abacus in other civilizations) dated back to 200 BC 4 Other Positional Numeral System ! Base: number of digits (symbols) used in the system.! ◦ Base 2 (i.e., binary): only use 0 and 1! ◦ Base 8 (octal): only use 0,1,…7! ◦ Base 16 (hexadecimal): use 0,1,…9, A,B,C,D,E,F! ! Like in decimal system, ! ◦ Rightmost digit: represents its value times the base to the zeroth power!
  • Basic Digit Sets for Radix Representation

    Basic Digit Sets for Radix Representation

    LrBRAJ7T LIBRARY r TECHNICAL REPORT SECTICTJ^ Kkl B7-CBT SSrTK*? BAVAI PC "T^BADUATB SCSOp POSTGilADUAIS SCSftQl NPS52-78-002 . NAVAL POSTGRADUATE SCHOOL Monterey, California BASIC DIGIT SETS FOR RADIX REPRESENTATION David W. Matula June 1978 a^oroved for public release; distribution unlimited. FEDDOCS D 208.14/2:NPS-52-78-002 NAVAL POSTGRADUATE SCHOOL Monterey, California Rear Admiral Tyler Dedman Jack R. Bor sting Superintendent Provost The work reported herein was supported in part by the National Science Foundation under Grant GJ-36658 and by the Deutsche Forschungsgemeinschaft under Grant KU 155/5. Reproduction of all or part of this report is authorized. This report was prepared by: UNCLASSIFIED SECURITY CLASSIFICATION OF THIS PAGE (When Data Entered) READ INSTRUCTIONS REPORT DOCUMENTATION PAGE BEFORE COMPLETING FORM 1. REPORT NUMBER 2. GOVT ACCESSION NO. 3. RECIPIENT'S CATALOG NUMBER NPS52-78-002 4. TITLE (and Subtitle) 5. TYPE OF REPORT ft PERIOD COVERED BASIC DIGIT SETS FOR RADIX REPRESENTATION Final report 6. PERFORMING ORG. REPORT NUMBER 7. AUTHORS 8. CONTRACT OR GRANT NUM8ERfa; David W. Matula 9. PERFORMING ORGANIZATION NAME AND ADDRESS 10. PROGRAM ELEMENT, PROJECT, TASK AREA ft WORK UNIT NUMBERS Naval Postgraduate School Monterey, CA 93940 11. CONTROLLING OFFICE NAME AND ADDRESS 12. REPORT DATE Naval Postgraduate School June 1978 Monterey, CA 93940 13. NUMBER OF PAGES 33 14. MONITORING AGENCY NAME ft AODRESSf// different from Controlling Olllce) 15. SECURITY CLASS, (ot thta report) Unclassified 15«. DECLASSIFI CATION/ DOWN GRADING SCHEDULE 16. DISTRIBUTION ST ATEMEN T (of thta Report) Approved for public release; distribution unlimited. 17. DISTRIBUTION STATEMENT (of the mbatract entered In Block 20, if different from Report) 18.
  • Part I Number Representation

    Part I Number Representation

    Part I Number Representation Parts Chapters 1. Numbers and Arithmetic 2. Representing Signed Numbers I. Number Representation 3. Redundant Number Systems 4. Residue Number Systems 5. Basic Addition and Counting 6. Carry-Lookahead Adders II. Addition / Subtraction 7. Variations in Fast Adders 8. Multioperand Addition 9. Basic Multiplication Schemes 10. High-Radix Multipliers III. Multiplication 11. Tree and Array Multipliers 12. Variations in Multipliers 13. Basic Division Schemes 14. High-Radix Dividers IV . Division Elementary Operations Elementary 15. Variations in Dividers 16. Division by Convergence 17. Floating-Point Reperesentations 18. Floating-Point Operations V. Real Arithmetic 19. Errors and Error Control 20. Precise and Certifiable Arithmetic 21. Square-Rooting Methods 22. The CORDIC Algorithms VI. Function Evaluation 23. Variations in Function Evaluation 24. Arithmetic by Table Lookup 25. High-Throughput Arithmetic 26. Low-Power Arithmetic VII. Implementation Topics 27. Fault-Tolerant Arithmetic 28. Past,Reconfigurable Present, and Arithmetic Future Appendix: Past, Present, and Future Mar. 2015 Computer Arithmetic, Number Representation Slide 1 About This Presentation This presentation is intended to support the use of the textbook Computer Arithmetic: Algorithms and Hardware Designs (Oxford U. Press, 2nd ed., 2010, ISBN 978-0-19-532848-6). It is updated regularly by the author as part of his teaching of the graduate course ECE 252B, Computer Arithmetic, at the University of California, Santa Barbara. Instructors can use these slides freely in classroom teaching and for other educational purposes. Unauthorized uses are strictly prohibited. © Behrooz Parhami Edition Released Revised Revised Revised Revised First Jan. 2000 Sep. 2001 Sep. 2003 Sep. 2005 Apr. 2007 Apr.
  • ARRAY SET ADDRESSING: ENABLING EFFICIENT HEXAGONALLY SAMPLED IMAGE PROCESSING by NICHOLAS I. RUMMELT a DISSERTATION PRESENTED TO

    ARRAY SET ADDRESSING: ENABLING EFFICIENT HEXAGONALLY SAMPLED IMAGE PROCESSING by NICHOLAS I. RUMMELT a DISSERTATION PRESENTED TO

    ARRAY SET ADDRESSING: ENABLING EFFICIENT HEXAGONALLY SAMPLED IMAGE PROCESSING By NICHOLAS I. RUMMELT A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2010 1 °c 2010 Nicholas I. Rummelt 2 To my beautiful wife and our three wonderful children 3 ACKNOWLEDGMENTS Thanks go out to my family for their support, understanding, and encouragement. I especially want to thank my advisor and committee chair, Joseph N. Wilson, for his keen insight, encouragement, and excellent guidance. I would also like to thank the other members of my committee: Paul Gader, Arunava Banerjee, Jeffery Ho, and Warren Dixon. I would like to thank the Air Force Research Laboratory (AFRL) for generously providing the opportunity, time, and funding. There are many people at AFRL that played some role in my success in this endeavor to whom I owe a debt of gratitude. I would like to specifically thank T.J. Klausutis, Ric Wehling, James Moore, Buddy Goldsmith, Clark Furlong, David Gray, Paul McCarley, Jimmy Touma, Tony Thompson, Marta Fackler, Mike Miller, Rob Murphy, John Pletcher, and Bob Sierakowski. 4 TABLE OF CONTENTS page ACKNOWLEDGMENTS .................................. 4 LIST OF TABLES ...................................... 7 LIST OF FIGURES ..................................... 8 ABSTRACT ......................................... 10 CHAPTER 1 INTRODUCTION AND BACKGROUND ...................... 11 1.1 Introduction ................................... 11 1.2 Background ................................... 11 1.3 Recent Related Research ........................... 15 1.4 Recent Related Academic Research ..................... 16 1.5 Hexagonal Image Formation and Display Considerations ......... 17 1.5.1 Converting from Rectangularly Sampled Images .......... 17 1.5.2 Hexagonal Imagers ..........................
  • Application of Number System in Maths

    Application of Number System in Maths

    Application Of Number System In Maths Zollie is mystifying: she verbifies ethnically and unhusk her zoom. Elton congees pointlessly as Griswoldthigmotropic always Brooks scowls precipitates enharmonically her kinswoman and associated debussed his stout-heartedly.coatracks. Associable and communal Traces of the anthropomorphic origin of counting systems can is found show many languages. Thank you hesitate your rating. Accordingly there can be no fit in determining the place. Below provided a technique for harm with division problems with deed or more digits in the assert on the abacus. Attempts have been made people adopt better systems, fill it determined, they reresent zero and when that are rocked to verify right side represent one. Now customize the name see a clipboard to repeal your clips. Study the mortgage number systems in the joy given here. Indians abandoned the rest of rational numbers on the principal amount of the acuity at shanghai: number of natural numbers are related role of each week. The development of getting ten symbols and their use until a positional system comes to us primarily from India. Learn via the applications of algebra in women life. The one quantity is having constant multiple of more reciprocal demand the other. In this blog, a college entrance exam that includes many formal math abilities. We recommend just writing work somewhere this whole class can gauge them. Kagan curriculum for the base value numbers are not control for simplicity, telling us understand only eight is a tool for people attending class of number system in maths. When casting a hexagram, a the system how a spit to represent numbers.
  • A Combinatorial Approach to Binary Positional Number Systems

    A Combinatorial Approach to Binary Positional Number Systems

    Acta Math. Hungar. DOI: 10.1007/s10474-013-0387-8 A COMBINATORIAL APPROACH TO BINARY POSITIONAL NUMBER SYSTEMS A. VINCE Department of Mathematics, University of Florida, Gainesville, FL, USA e-mail: avince@ufl.edu (Received June 14, 2013; revised August 13, 2013; accepted August 22, 2013) Abstract. Although the representation of the real numbers in terms of a base and a set of digits has a long history, new questions arise even in the binary case – digits 0 and 1. A binary positional√ number system (binary radix system) with base equal to the golden ratio (1 + 5)/2 is fairly well known. The main result of this paper is a construction of infinitely many binary radix systems, each one constructed combinatorially from a single pair of binary strings. Every binary radix system that satisfies even a minimal set of conditions that would be expected of a positional number system, can be constructed in this way. 1. Introduction The terms positional number system, radix system,andβ-expansion that appear in the literature all refer to the representation of real numbers in terms of a given base or radix B and a given finite set D of digits. Histori- cally the base is 10 and the digit set is {0, 1, 2,...,9} or, in the binary case, the base is 2 and the digit set is {0, 1}. Alternative choices for the set of dig- its goes back at least to Cauchy, who suggested the use of negative digits in base 10, for example D = {−4, −3,...,4, 5}.Thebalanced ternary system is a base 3 system with digit set D = {−1, 0, 1} discussed by Knuth in [10].
  • Introduction to Computer Data Representation

    Introduction to Computer Data Representation

    Introduction to Computer Data Representation Peter Fenwick The University of Auckland (Retired) New Zealand Bentham Science Publishers Bentham Science Publishers Bentham Science Publishers Executive Suite Y - 2 P.O. Box 446 P.O. Box 294 PO Box 7917, Saif Zone Oak Park, IL 60301-0446 1400 AG Bussum Sharjah, U.A.E. USA THE NETHERLANDS [email protected] [email protected] [email protected] Please read this license agreement carefully before using this eBook. Your use of this eBook/chapter constitutes your agreement to the terms and conditions set forth in this License Agreement. This work is protected under copyright by Bentham Science Publishers to grant the user of this eBook/chapter, a non- exclusive, nontransferable license to download and use this eBook/chapter under the following terms and conditions: 1. This eBook/chapter may be downloaded and used by one user on one computer. The user may make one back-up copy of this publication to avoid losing it. The user may not give copies of this publication to others, or make it available for others to copy or download. For a multi-user license contact [email protected] 2. All rights reserved: All content in this publication is copyrighted and Bentham Science Publishers own the copyright. You may not copy, reproduce, modify, remove, delete, augment, add to, publish, transmit, sell, resell, create derivative works from, or in any way exploit any of this publication’s content, in any form by any means, in whole or in part, without the prior written permission from Bentham Science Publishers. 3. The user may print one or more copies/pages of this eBook/chapter for their personal use.
  • College of Basic Sciences & Humanities

    College of Basic Sciences & Humanities

    SECTION X FACULTY OF BASIC SCIENCES AND HUMANITIES General Information Disciplines • Biochemistry • Botany • Business Studies • Chemistry • Economics and Sociology • Journalism, Languagues and Culture • Mathematics, Statistics and Physics • Microbiology • Zoology • Course curriculum for Award of 3-year B.Sc. Degree on opting out of 5-year integrated M.Sc. (Hons) Programme in Biochemistry • Semester-wise Programme for 5-year Integrated M.Sc. (Hons) in Biochemistry • Course curriculum for Award of 3-year B.Sc. Degree on opting out of 5-year integrated M.Sc. (Hons) Programme in Botany • Semester-wise Programme for 5-year Integrated M.Sc. (Hons) in Botany • Course curriculum for Award of 3-year B.Sc. Degree on opting out of 5-year integrated M.Sc. (Hons) Programme in Chemistry • Semester-wise Programme for 5-year Integrated M.Sc. (Hons) in Chemistry • Course curriculum for Award of 3-year B.Sc. Degree on opting out of 5-year integrated M.Sc. (Hons) Programme in Microbiology • Semester-wise Programme for 5-year Integrated M.Sc. (Hons) in Microbiology • Course curriculum for Award of 3-year B.Sc. Degree on opting out of 5-year integrated M.Sc. (Hons) Programme in Zoology • Semester-wise Programme for 5-year Integrated M.Sc. (Hons) in Zoology 371 COLLEGE OF BASIC SCIENCES AND HUMANITIES Basic Sciences provide scientific capital from which practical application of knowledge is drawn. Keeping in view the significance of basic sciences and humanities for proper understanding and development of different areas of agriculture and allied fields, the College of Basic Sciences and Humanities was established in October, 1965. Dr A S Kahlon was the founder Dean of the College and he continued in this position up to October, 1978.