Conversions Between Percents, Decimals, and Fractions
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Nondecimal Positional Systems
M03_LONG0847_05_AIE_C03.qxd 9/17/07 10:38 AM Page 162 Not for Sale 162 CHAPTER 3 Numeration and Computation 3.2 Nondecimal Positional Systems In the preceding section, we discussed several numeration systems, including the decimal system in common use today. As already mentioned, the decimal system is a positional system based on 10. This is probably so because we have ten fingers and historically, as now, people often counted on their fingers. In the Did You Know box at the end of this section, we will give applications of positional systems that are based on whole numbers other than 10. One very useful application of bases other than ten is as a way to deepen our understanding of number systems, arithmetic in general, and our own decimal system. Although bases other than ten were in the K–5 curriculum in the past, they are not present currently. However, nondecimal positional systems are studied in the “math methods” course that is a part of an elementary education degree and may return to elementary school in the future. We will discuss addition and subtraction in base five in the next section and multiplication in base six in the following one. Base Five Notation For our purposes, perhaps the quickest route to understanding base five notation is to reconsider the abacus of Figure 3.1. This time, however, we allow only 5 beads to be moved forward on each wire. As before, we start with the wire to the students’ right (our left) and move 1 bead forward each time as we count 1, 2, 3, and so on. -
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). -
Zero Displacement Ternary Number System: the Most Economical Way of Representing Numbers
Revista de Ciências da Computação, Volume III, Ano III, 2008, nº3 Zero Displacement Ternary Number System: the most economical way of representing numbers Fernando Guilherme Silvano Lobo Pimentel , Bank of Portugal, Email: [email protected] Abstract This paper concerns the efficiency of number systems. Following the identification of the most economical conventional integer number system, from a solid criteria, an improvement to such system’s representation economy is proposed which combines the representation efficiency of positional number systems without 0 with the possibility of representing the number 0. A modification to base 3 without 0 makes it possible to obtain a new number system which, according to the identified optimization criteria, becomes the most economic among all integer ones. Key Words: Positional Number Systems, Efficiency, Zero Resumo Este artigo aborda a questão da eficiência de sistemas de números. Partindo da identificação da mais económica base inteira de números de acordo com um critério preestabelecido, propõe-se um melhoramento à economia de representação nessa mesma base através da combinação da eficiência de representação de sistemas de números posicionais sem o zero com a possibilidade de representar o número zero. Uma modificação à base 3 sem zero permite a obtenção de um novo sistema de números que, de acordo com o critério de optimização identificado, é o sistema de representação mais económico entre os sistemas de números inteiros. Palavras-Chave: Sistemas de Números Posicionais, Eficiência, Zero 1 Introduction Counting systems are an indispensable tool in Computing Science. For reasons that are both technological and user friendliness, the performance of information processing depends heavily on the adopted numbering system. -
RATIO and PERCENT Grade Level: Fifth Grade Written By: Susan Pope, Bean Elementary, Lubbock, TX Length of Unit: Two/Three Weeks
RATIO AND PERCENT Grade Level: Fifth Grade Written by: Susan Pope, Bean Elementary, Lubbock, TX Length of Unit: Two/Three Weeks I. ABSTRACT A. This unit introduces the relationships in ratios and percentages as found in the Fifth Grade section of the Core Knowledge Sequence. This study will include the relationship between percentages to fractions and decimals. Finally, this study will include finding averages and compiling data into various graphs. II. OVERVIEW A. Concept Objectives for this unit: 1. Students will understand and apply basic and advanced properties of the concept of ratios and percents. 2. Students will understand the general nature and uses of mathematics. B. Content from the Core Knowledge Sequence: 1. Ratio and Percent a. Ratio (p. 123) • determine and express simple ratios, • use ratio to create a simple scale drawing. • Ratio and rate: solve problems on speed as a ratio, using formula S = D/T (or D = R x T). b. Percent (p. 123) • recognize the percent sign (%) and understand percent as “per hundred” • express equivalences between fractions, decimals, and percents, and know common equivalences: 1/10 = 10% ¼ = 25% ½ = 50% ¾ = 75% find the given percent of a number. C. Skill Objectives 1. Mathematics a. Compare two fractional quantities in problem-solving situations using a variety of methods, including common denominators b. Use models to relate decimals to fractions that name tenths, hundredths, and thousandths c. Use fractions to describe the results of an experiment d. Use experimental results to make predictions e. Use table of related number pairs to make predictions f. Graph a given set of data using an appropriate graphical representation such as a picture or line g. -
A Ternary Arithmetic and Logic
Proceedings of the World Congress on Engineering 2010 Vol I WCE 2010, June 30 - July 2, 2010, London, U.K. A Ternary Arithmetic and Logic Ion Profeanu which supports radical ontological dualism between the two Abstract—This paper is only a chapter, not very detailed, of eternal principles, Good and Evil, which oppose each other a larger work aimed at developing a theoretical tool to in the course of history, in an endless confrontation. A key investigate first electromagnetic fields but not only that, (an element of Manichean doctrine is the non-omnipotence of imaginative researcher might use the same tool in very unusual the power of God, denying the infinite perfection of divinity areas of research) with the stated aim of providing a new perspective in understanding older or recent research in "free that has they say a dual nature, consisting of two equal but energy". I read somewhere that devices which generate "free opposite sides (Good-Bad). I confess that this kind of energy" works by laws and principles that can not be explained dualism, which I think is harmful, made me to seek another within the framework of classical physics, and that is why they numeral system and another logic by means of which I will are kept far away from public eye. So in the absence of an praise and bring honor owed to God's name; and because adequate theory to explain these phenomena, these devices can God is One Being in three personal dimensions (the Father, not reach the design tables of some plants in order to produce them in greater number. -
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. -
4Fractions and Percentages
Fractions and Chapter 4 percentages What Australian you will learn curriculum 4A What are fractions? NUMBER AND ALGEBRA (Consolidating) 1 Real numbers 4B Equivalent fractions and Compare fractions using equivalence. Locate and represent fractions and simplifi ed fractions mixed numerals on a number line (ACMNA152) 4C Mixed numbers Solve problems involving addition and subtraction of fractions, including (Consolidating) those with unrelated denominators (ACMNA153) 4D Ordering fractions Multiply and divide fractions and decimals using ef cient written 4E Adding fractions strategies and digital technologies (ACMNA154) 4F Subtracting fractions Express one quantity as a fraction of another with and without the use of UNCORRECTEDdigital technologies (ACMNA155) 4G Multiplying fractions 4H Dividing fractions Connect fractions, decimals and percentages and carry out simple 4I Fractions and percentages conversions (ACMNA157) 4J Percentage of a number Find percentages of quantities and express one quantity as a percentage 4K Expressing aSAMPLE quantity as a of another, with and without digital PAGES technologies. (ACMNA158) proportion Recognise and solve problems involving simple ratios. (ACMNA173) 2 Money and nancial mathematics Investigate and calculate ‘best16x16 buys’, with and without digital32x32 technologies (ACMNA174) Uncorrected 3rd sample pages • Cambridge University Press © Greenwood et al., 2015 • 978-1-107-56882-2 • Ph 03 8671 1400 COP_04.indd 2 16/04/15 11:34 AM Online 173 resources • Chapter pre-test • Videos of all worked examples • Interactive widgets • Interactive walkthroughs • Downloadable HOTsheets • Access to HOTmaths Australian Curriculum courses Ancient Egyptian fractions 1 The ancient Egyptians used fractions over 4000 years thought (eyebrow closest to brain) 8 ago. The Egyptian sky god Horus was a falcon-headed 1 hearing (pointing to ear) man whose eyes were believed to have magical healing 16 powers. -
I – Mathematics in Design – SMTA1103
SCHOOL OF SCIENCE AND HUMANITIES DEPARTMENT OF MATHEMATICS UNIT – I – Mathematics in Design – SMTA1103 UNIT1- MATHEMATICS IN DESIGN Golden Ratio The ratio of two parts of a line such that the longer part divided by the smaller part is also equal to the whole length divided by the longer part. Results: i) The ratio of the circumference of a circle to its diameter is π . ii) The ratio of the width of a rectangular picture frame to its height is not necessarily equal to he golden ratio, though some artists and architects believe a frame made using the golden ratio makes the most pleasing and beautiful shape.iii) The ratio of two successive numbers of the Fibonacci sequence give an approximate value of the golden ratio,the bigger the pair of Fibonacci Numbers, the closer the approximation. Example1: The Golden Ratio = 1.61803398874989484820... = 1.618 correct to 3 decimal places. If x n are terms of the Fibonacci sequence, then what is the least value of n for which ( x n +1 / x n = 1.618 correct to 3 decimal places. Example2: A rectangle divided into a square and a smaller rectangle. If (x + y) /y = x / y ,find the value of golden ratio. Example3: The Golden Ratio = 1.61803398874989484820... = 1.6180 correct to 4 decimal places. If xn are terms of the Fibonacci sequence, then what is the least value of n for which (x n +1) / x n = 1.618 0 correct to 4 decimal places. Example4: Obtain golden ratio in a line segment. Proportion Proportion is an equation which defines that the two given ratios are equivalent to each other. -
The Counting System of the Incas in General, There Is a Lack of Material on the History of Numbers from the Pre-Columbian Americas
History of Numbers 1b. I can determine the number represented on an Inca counting board, and a quipu cord. The Counting System of the Incas In general, there is a lack of material on the history of numbers from the pre-Columbian Americas. Most of the information available concentrates on the eastern hemisphere, with Europe as the focus. Why? First, there has been a false assumption that indigenous peoples in the Americas lacked any specialized mathematics. Second, it seems at least some of their ancient mathematics has been intentionally kept secret. And Einally, when Europeans invaded and colonized the Americas, much of that evidence was most likely destroyed. In the 1960's, two researchers attempted to discover what mathematical knowledge was known by the Incas and how they used the Peruvian quipu, a counting system using cords and knots, in their mathematics. These researchers have come to certain beliefs about the quipu that are summarized below: Counting Boards It seems the Incas did not have a complicated system of computation. Whereas other peoples in the region, such as the Mayans, were doing computations related to their rituals and calendars, the Incas seem to have been more concerned with the task of record-keeping. To do this, they used what are called “quipu”(or khipu) to record quantities of items. However, they Eirst often needed to do computations whose results would be recorded on a quipu. To do these computations, they sometimes used a counting board or Yupana (from the Quechua word "yupay": count) made from a slab of wood or stone. -
Continued Fractions: Properties and Invariant Measure (Kettingbreuken: Eigenschappen En Invariante Maat)
Delft University of Technology Faculty Electrical Engineering, Mathematics and Computer Science Delft Institute of Applied Mathematics Continued fractions: Properties and invariant measure (Kettingbreuken: Eigenschappen en invariante maat) Bachelor of science thesis by Angelina C. Lemmens Delft, Nederland 14 februari 2014 Inhoudsopgave 1 Introduction 1 2 Continued fractions 2 2.1 Regular continued fractions . .2 2.1.1 Convergence of the regular continued fraction . .4 2.2 N-expansion . .9 2.3 a/b-continued fractions . 11 3 Invariant measures 12 3.1 Definition of invariant measure . 12 3.2 Invariant measure of the regular continued fraction . 13 4 The Ergodic Theorem 15 4.1 Adler's Folklore Theorem . 15 4.2 Ergodicity of the regular continued fraction . 17 5 A new continued fraction 19 5.1 Convergence of the new continued fraction . 20 5.2 Invariant measure and ergodicity of the new continued fraction . 25 5.3 Simulation of the density of the new continued fraction . 28 6 Appendix 30 1 Introduction An irrational number can be represented in many ways. For example the repre- sentation could be a rational approximation and could be more convenient to work with than the original number. Representations can be for example the decimal expansion or the binary expansion. These representations are used in everyday life. Less known is the representation by continued fractions. Continued fractions give the best approximation of irrational numbers by rational numbers. This is something that Christiaan Huygens already knew (and used to construct his planetarium). Number representations like the decimal expansion and continued fractions have relations with probability theory and dynamical systems. -
Maths Revision & Practice Booklet
Maths Revision & Practice Booklet Name: Fractions, Decimals and Percentages visit twinkl.com Maths Revision & Practice Booklet Revise Using Common Factors to Simplify Fractions Fractions that have the same value but represent this using different denominators and numerators are equivalent. We can recognise and find equivalent fractions by multiplying or dividing the numerator and denominator by the same amount. When we simplify a fraction, we use the highest common factor of the numerator and denominator to reduce the fraction to the lowest term equivalent fraction (simplest form). Factors of 21: 1 3 7 The highest 21 common factor 36 Factors of 36: 1 2 3 4 6 9 12 18 36 is 3. 21 ÷ 3 = 7 7 36 ÷ 3 = 12 12 Using Common Multiples to Express Fractions in the Same Denomination To compare or calculate with fractions, we often need to give them a common denominator. We do this by looking at the denominators and finding their lowest common multiple. 3 4 5 × 7 = 35 5 7 35 is the lowest common multiple. Remember that whatever we do 3 3 × 7 = 21 4 4 × 5 = 20 to the denominator, we have to 5 × 7 = 35 7 × 5 = 35 5 7 do to the numerator. Page 2 of 11 visit twinkl.com Maths Revision & Practice Booklet Comparing and Ordering Fractions, Including Fractions > 1 To compare and order ...look at the fractions with the same 5 2 numerators. denominators… 9 > 9 …change the 5 3 fractions into and To compare and order 9 5 equivalent fractions with different 5 9 fractions with the × × denominators… lowest common 25 27 denominator. -
Maths Module 3 Ratio, Proportion and Percent
Maths Module 3 Ratio, Proportion and Percent This module covers concepts such as: • ratio • direct and indirect proportion • rates • percent www.jcu.edu.au/students/learning-centre Module 3 Ratio, Proportion and Percent 1. Ratio 2. Direct and Indirect Proportion 3. Rate 4. Nursing Examples 5. Percent 6. Combine Concepts: A Word Problem 7. Answers 8. Helpful Websites 2 1. Ratio Understanding ratio is very closely related to fractions. With ratio comparisons are made between equal sized parts/units. Ratios use the symbol “ : “ to separate the quantities being compared. For example, 1:3 means 1 unit to 3 units, and the units are the same size. Whilst ratio can be expressed as a fraction, the ratio 1:3 is NOT the same as , as the rectangle below illustrates 1 3 The rectangle is 1 part grey to 3 parts white. Ratio is 1:3 (4 parts/units) The rectangle is a total of 4 parts, and therefore, the grey part represented symbolically is and not 1 1 4 3 AN EXAMPLE TO BEGIN: My two-stroke mower requires petrol and oil mixed to a ratio of 1:25. This means that I add one part oil to 25 parts petrol. No matter what measuring device I use, the ratio must stay the same. So if I add 200mL of oil to my tin, I add 200mL x 25 = 5000mL of petrol. Note that the total volume (oil and petrol combined) = 5200mL which can be converted to 5.2 litres. Ratio relationships are multiplicative. Mathematically: 1:25 is the same as 2:50 is the same as 100:2500 and so on.