Multiplication Tables of Various Bases by Michael Th Omas De Vlieger
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13054-Duodecimal.Pdf
Universal Multiple-Octet Coded Character Set International Organization for Standardization Organisation Internationale de Normalisation Международная организация по стандартизации Doc Type: Working Group Document Title: Proposal to encode Duodecimal Digit Forms in the UCS Author: Karl Pentzlin Status: Individual Contribution Action: For consideration by JTC1/SC2/WG2 and UTC Date: 2013-03-30 1. Introduction The duodecimal system (also called dozenal) is a positional numbering system using 12 as its base, similar to the well-known decimal (base 10) and hexadecimal (base 16) systems. Thus, it needs 12 digits, instead of ten digits like the decimal system. It is used by teachers to explain the decimal system by comparing it to an alternative, by hobbyists (see e.g. fig. 1), and by propagators who claim it being superior to the decimal system (mostly because thirds can be expressed by a finite number of digits in a "duodecimal point" presentation). • Besides mathematical and hobbyist publications, the duodecimal system has appeared as subject in the press (see e.g. [Bellos 2012] in the English newspaper "The Guardian" from 2012-12-12, where the lack of types to represent these digits correctly is explicitly stated). Such examples emphasize the need of the encoding of the digit forms proposed here. While it is common practice to represent the extra six digits needed for the hexadecimal system by the uppercase Latin capital letters A,B.C,D,E,F, there is no such established convention regarding the duodecimal system. Some proponents use the Latin letters T and E as the first letters of the English names of "ten" and "eleven" (which obviously is directly perceivable only for English speakers). -
Vigesimal Numerals on Ifẹ̀ (Togo) and Ifẹ̀ (Nigeria) Dialects of Yorùbá
Vigesimal Numerals on Ifẹ̀ (Togo) and Ifẹ̀ (Nigeria) Dialects of Yorùbá Felix Abídèmí Fábùnmi (Ilé-Ifẹ̀) Abstract This study intends to bring Ifẹ̀ (Togo) into a linguistic limelight using the numeral systems. Numerals are a very important aspect of the day to day socio-economic and linguistic life of Ifẹ̀ (Togo) people. The traditional Ifẹ̀ (Togo) number system is vigesimal. In this study, forty- two different number words are listed for Yorùbá Ifẹ̀ (Nigeria) and Yorùbá Ifẹ̀ (Togo) and compared with Standard Yorùbá. We compared the Ifẹ̀ (Togo) number words and counting patterns with that of the Standard Yorùbá and Ifẹ̀ (Nigeria) and discovered that, by the nature of the components of these numbers, majority of the basic number words are either bisyllabic or trisyllabic, each syllable having the form VCV for the cardinals, and CVCV for the ordinals. There are irregularities in tonality; there are also alternations in the sequences of the vowel (oral and nasalized) and consonant sounds. This work finds out that Ifẹ̀ (Togo) has two counting patterns. In the first pattern, it uses addition solely to derive the number words but with a counting pattern where 'ten', 'twenty' and the added number units are taken as a whole. In the second counting pattern, subtraction is used to derive number words but this is applicable only to three numbers i. e. seventeen – /mɛ́ɛtadínóɡú/, eighteen – /méèʤìdínóɡu/ and nineteen – /mɔ̀kɔ̃dínoɡ́ u/. The Ifẹ̀ (Togo) dialect of Yorùbá mostly uses additive number positions. The dialect favours additive number positions more than the subtractive and the multiplicative positions. In other words, higher numbers are frequently used as bases for addition not as bases for multiplication in Ifẹ̀ (Togo). -
The Hexadecimal Number System and Memory Addressing
C5537_App C_1107_03/16/2005 APPENDIX C The Hexadecimal Number System and Memory Addressing nderstanding the number system and the coding system that computers use to U store data and communicate with each other is fundamental to understanding how computers work. Early attempts to invent an electronic computing device met with disappointing results as long as inventors tried to use the decimal number sys- tem, with the digits 0–9. Then John Atanasoff proposed using a coding system that expressed everything in terms of different sequences of only two numerals: one repre- sented by the presence of a charge and one represented by the absence of a charge. The numbering system that can be supported by the expression of only two numerals is called base 2, or binary; it was invented by Ada Lovelace many years before, using the numerals 0 and 1. Under Atanasoff’s design, all numbers and other characters would be converted to this binary number system, and all storage, comparisons, and arithmetic would be done using it. Even today, this is one of the basic principles of computers. Every character or number entered into a computer is first converted into a series of 0s and 1s. Many coding schemes and techniques have been invented to manipulate these 0s and 1s, called bits for binary digits. The most widespread binary coding scheme for microcomputers, which is recog- nized as the microcomputer standard, is called ASCII (American Standard Code for Information Interchange). (Appendix B lists the binary code for the basic 127- character set.) In ASCII, each character is assigned an 8-bit code called a byte. -
Bits, Data Types, and Operations
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Agenda 1. Data Types 2. Unsigned & Signed Integers 3. Arithmetic Operations Chapter 2 4. Logical Operations 5. Shifting Bits, Data Types, 6. Hexadecimal & Octal Notation and Operations 7. Other Data Types COMPSCI210 S1C 2009 2 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 1. Data Types Computer is a binary digital system. How do we represent data in a computer? Digital system: Binary (base two) system: At the lowest level, a computer is an electronic machine. • finite number of symbols • has two states: 0 and 1 • works by controlling the flow of electrons Easy to recognize two conditions: 1. presence of a voltage – we’ll call this state “1” 2. absence of a voltage – we’ll call this state “0” Basic unit of information is the binary digit, or bit. Values with more than two states require multiple bits. • A collection of two bits has four possible states: Could base state on value of voltage, 00, 01, 10, 11 but control and detection circuits more complex. • A collection of three bits has eight possible states: • compare turning on a light switch to 000, 001, 010, 011, 100, 101, 110, 111 measuring or regulating voltage • A collection of n bits has 2n possible states. COMPSCI210 S1C 2009 3 COMPSCI210 S1C 2009 4 1 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. -
Abstract of Counting Systems of Papua New Guinea and Oceania
Abstract of http://www.uog.ac.pg/glec/thesis/ch1web/ABSTRACT.htm Abstract of Counting Systems of Papua New Guinea and Oceania by Glendon A. Lean In modern technological societies we take the existence of numbers and the act of counting for granted: they occur in most everyday activities. They are regarded as being sufficiently important to warrant their occupying a substantial part of the primary school curriculum. Most of us, however, would find it difficult to answer with any authority several basic questions about number and counting. For example, how and when did numbers arise in human cultures: are they relatively recent inventions or are they an ancient feature of language? Is counting an important part of all cultures or only of some? Do all cultures count in essentially the same ways? In English, for example, we use what is known as a base 10 counting system and this is true of other European languages. Indeed our view of counting and number tends to be very much a Eurocentric one and yet the large majority the languages spoken in the world - about 4500 - are not European in nature but are the languages of the indigenous peoples of the Pacific, Africa, and the Americas. If we take these into account we obtain a quite different picture of counting systems from that of the Eurocentric view. This study, which attempts to answer these questions, is the culmination of more than twenty years on the counting systems of the indigenous and largely unwritten languages of the Pacific region and it involved extensive fieldwork as well as the consultation of published and rare unpublished sources. -
2 1 2 = 30 60 and 1
Math 153 Spring 2010 R. Schultz SOLUTIONS TO EXERCISES FROM math153exercises01.pdf As usual, \Burton" refers to the Seventh Edition of the course text by Burton (the page numbers for the Sixth Edition may be off slightly). Problems from Burton, p. 28 3. The fraction 1=6 is equal to 10=60 and therefore the sexagesimal expression is 0;10. To find the expansion for 1=9 we need to solve 1=9 = x=60. By elementary algebra this means 2 9x = 60 or x = 6 3 . Thus 6 2 1 6 40 1 x = + = + 60 3 · 60 60 60 · 60 which yields the sexagsimal expression 0; 10; 40 for 1/9. Finding the expression for 1/5 just amounts to writing this as 12/60, so the form here is 0;12. 1 1 30 To find 1=24 we again write 1=24 = x=60 and solve for x to get x = 2 2 . Now 2 = 60 and therefore we can proceed as in the second example to conclude that the sexagesimal form for 1/24 is 0;2,30. 1 One proceeds similarly for 1/40, solving 1=40 = x=60 to get x = 1 2 . Much as in the preceding discussion this yields the form 0;1,30. Finally, the same method leads to the equation 5=12 = x=60, which implies that 5/12 has the sexagesimal form 0;25. 4. We shall only rewrite these in standard base 10 fractional notation. The answers are in the back of Burton. (a) The sexagesimal number 1,23,45 is equal to 1 3600 + 23 60 + 45. -
1. Understanding Decimal, Binary, Octal and Hexadecimal Numbers
Objectives: 1. Understanding decimal, binary, octal and hexadecimal numbers. 2. Counting in decimal, binary, octal and hexadecimal systems. 3. Convert a number from one number system to another system. 4. Advantage of octal and hexadecimal systems. 1. Understanding decimal, binary, octal and hexadecimal numbers Decimal number systems: Decimal numbers are made of decimal digits: (0,1,2,3,4,5,6,7,8,9 --------10-base system) The decimal system is a "positional-value system" in which the value of a digit depends on its position. Examples: 453→4 hundreds, 5 tens and 3 units. 4 is the most weight called "most significant digit" MSD. 3 carries the last weight called "least significant digit" LSD. number of items that a decimal number represent: 9261= (9× )+(2× )+(6× )+(1× ) The decimal fractions: 3267.317= (3× )+(2× )+(6× )+(7× )+ (3× ) + (6× ) + (1× ) Decimal point used to separate the integer and fractional part of the number. Formal notation→ . Decimal position values of powers of (10). Positional values "weights" 2 7 7 8 3 . 2 3 4 5 MSD LSD Binary numbers: . Base-2 system (0 or 1). We can represent any quantity that can be represented in decimal or other number systems using binary numbers. Binary number is also positional–value system (power of 2). Example: 1101.011 1 1 0 1 . 0 1 1 MSD LSD Notes: . To find the equivalent of binary numbers in decimal system , we simply take the sum of products of each digit value (0,1)and its positional value: Example: = (1× ) + (0× ) + (1× ) + (1× )+ (1× )+ (0× ) +(1× ) = 8 + 0 + 2 + 1 + + 0 + = In general, any number (decimal, binary, octal and hexadecimal) is simply the sum of products of each digit value and its positional value. -
Chapter 6 MISCELLANEOUS NUMBER BASES. the QUINARY
The Number Concept: Its Origin and Development By Levi Leonard Conant Ph. D. Chapter 6 MISCELLANEOUS NUMBER BASES. THE QUINARY SYSTEM. The origin of the quinary mode of counting has been discussed with some fulness in a preceding chapter, and upon that question but little more need be said. It is the first of the natural systems. When the savage has finished his count of the fingers of a single hand, he has reached this natural number base. At this point he ceases to use simple numbers, and begins the process of compounding. By some one of the numerous methods illustrated in earlier chapters, he passes from 5 to 10, using here the fingers of his second hand. He now has two fives; and, just as we say “twenty,” i.e. two tens, he says “two hands,” “the second hand finished,” “all the fingers,” “the fingers of both hands,” “all the fingers come to an end,” or, much more rarely, “one man.” That is, he is, in one of the many ways at his command, saying “two fives.” At 15 he has “three hands” or “one foot”; and at 20 he pauses with “four hands,” “hands and feet,” “both feet,” “all the fingers of hands and feet,” “hands and feet finished,” or, more probably, “one man.” All these modes of expression are strictly natural, and all have been found in the number scales which were, and in many cases still are, in daily use among the uncivilized races of mankind. In its structure the quinary is the simplest, the most primitive, of the natural systems. -
Number Systems and Number Representation Aarti Gupta
Number Systems and Number Representation Aarti Gupta 1 For Your Amusement Question: Why do computer programmers confuse Christmas and Halloween? Answer: Because 25 Dec = 31 Oct -- http://www.electronicsweekly.com 2 Goals of this Lecture Help you learn (or refresh your memory) about: • The binary, hexadecimal, and octal number systems • Finite representation of unsigned integers • Finite representation of signed integers • Finite representation of rational numbers (if time) Why? • A power programmer must know number systems and data representation to fully understand C’s primitive data types Primitive values and the operations on them 3 Agenda Number Systems Finite representation of unsigned integers Finite representation of signed integers Finite representation of rational numbers (if time) 4 The Decimal Number System Name • “decem” (Latin) => ten Characteristics • Ten symbols • 0 1 2 3 4 5 6 7 8 9 • Positional • 2945 ≠ 2495 • 2945 = (2*103) + (9*102) + (4*101) + (5*100) (Most) people use the decimal number system Why? 5 The Binary Number System Name • “binarius” (Latin) => two Characteristics • Two symbols • 0 1 • Positional • 1010B ≠ 1100B Most (digital) computers use the binary number system Why? Terminology • Bit: a binary digit • Byte: (typically) 8 bits 6 Decimal-Binary Equivalence Decimal Binary Decimal Binary 0 0 16 10000 1 1 17 10001 2 10 18 10010 3 11 19 10011 4 100 20 10100 5 101 21 10101 6 110 22 10110 7 111 23 10111 8 1000 24 11000 9 1001 25 11001 10 1010 26 11010 11 1011 27 11011 12 1100 28 11100 13 1101 29 11101 14 1110 30 11110 15 1111 31 11111 .. -
1 Evolution and Base Conversion
Indian Institute of Information Technology Design and Manufacturing, Kancheepuram logo.png Chennai { 600 127, India Instructor An Autonomous Institute under MHRD, Govt of India N.Sadagopan http://www.iiitdm.ac.in Computational Engineering Objectives: • To learn to work with machines (computers, ATMs, Coffee vending machines). • To understand how machines think and work. • To design instructions which machines can understand so that a computational task can be performed. • To learn a language which machines can understand so that human-machine interaction can take place. • To understand the limitations of machines: what can be computed and what can not be computed using machines. Outcome: To program a computer for a given computational task involving one or more of arithmetic, algebraic, logical, relational operations. 1 Evolution and Base Conversion Why Machines ? We shall now discuss the importance of automation; human-centric approach (manual) versus machines. Although machines are designed by humans, for many practical reasons, machines are superior to humans as far as problem solving is concerned. We shall highlight below commonly observed features that make machines powerful. • Reliable, accurate and efficient. • Can do parallel tasks if trained. • Efficient while peforming computations with large numbers. If designed correctly, then there is no scope for error. • Good at micro-level analysis with precision. • Consistent It is important to highlight that not all problems can be solved using machines (computers). For example, (i) whether a person is happy (ii) whether a person is lying (not speaking the truth) (iii) reciting the natural number set (iv) singing a song in a particular raga. Computer: A computational machine In this lecture, we shall discuss a computational machine called computer. -
About Numbers How These Basic Tools Appeared and Evolved in Diverse Cultures by Allen Klinger, Ph.D., New York Iota ’57
About Numbers How these Basic Tools Appeared and Evolved in Diverse Cultures By Allen Klinger, Ph.D., New York Iota ’57 ANY BIRDS AND Representation of quantity by the AUTHOR’S NOTE insects possess a The original version of this article principle of one-to-one correspondence 1 “number sense.” “If is on the web at http://web.cs.ucla. was undoubtedly accompanied, and per- … a bird’s nest con- edu/~klinger/number.pdf haps preceded, by creation of number- mtains four eggs, one may be safely taken; words. These can be divided into two It was written when I was a fresh- but if two are removed, the bird becomes man. The humanities course had an main categories: those that arose before aware of the fact and generally deserts.”2 assignment to write a paper on an- the concept of number unrelated to The fact that many forms of life “sense” thropology. The instructor approved concrete objects, and those that arose number or symmetry may connect to the topic “number in early man.” after it. historic evolution of quantity in differ- At a reunion in 1997, I met a An extreme instance of the devel- classmate from 1954, who remem- ent human societies. We begin with the bered my paper from the same year. opment of number-words before the distinction between cardinal (counting) As a pack rat, somehow I found the abstract concept of number is that of the numbers and ordinal ones (that show original. Tsimshian language of a tribe in British position as in 1st or 2nd). -
Curiosities Regarding the Babylonian Number System
Curiosities Regarding the Babylonian Number System Sherwin Doroudi April 12, 2007 By 2000 bce the Babylonians were already making significant progress in the areas of mathematics and astronomy and had produced many elaborate mathematical tables on clay tablets. Their sexagecimal (base-60) number system was originally inherited from the Sumerian number system dating back to 3500 bce. 1 However, what made the Babylonian number system superior to its predecessors, such as the Egyptian number system, was the fact that it was a positional system not unlike the decimal system we use today. This innovation allowed for the representation of numbers of virtually any size by reusing the same set of symbols, while also making it easier to carry out arithmetic operations on larger numbers. Its superiority was even clear to later Greek astronomers, who would use the sexagecimal number system as opposed to their own native Attic system (the direct predecessor to Roman numerals) when making calculations with large numbers. 2 Most other number systems throughout history have made use of a much smaller base, such as five (quinary systems), ten (decimal systems), or in the case of the Mayans, twenty (vigesimal systems), and such choices for these bases are all clearly related to the fingers on the hands. The choice (though strictly speaking, it's highly unlikely that any number system was directly \chosen") of sixty is not immediately apparent, and at first, it may even seem that sixty is a somewhat large and unwieldy number. Another curious fact regarding the ancient number system of the Babylonians is that early records do not show examples of a \zero" or null place holder, which is an integral part of our own positional number system.