1 Introduction 2 Place Value Representation
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Exploring Unit Fractions
Page 36 Exploring Unit Fractions Alan Christison [email protected] INTRODUCTION In their article “The Beauty of Cyclic Numbers”, Duncan Samson and Peter Breetzke (2014) explored, among other things, unit fractions and their decimal expansions6. They showed that for a unit fraction to terminate, its denominator must be expressible in the form 2푛. 5푚 where 푛 and 푚 are non-negative integers, both not simultaneously equal to zero. In this article I take the above observation as a starting point and explore it further. FURTHER OBSERVATIONS 1 For any unit fraction of the form where 푛 and 푚 are non-negative integers, both not 2푛.5푚 simultaneously equal to zero, the number of digits in the terminating decimal will be equal to the larger 1 1 of 푛 or 푚. By way of example, has 4 terminating decimals (0,0625), has 5 terminating decimals 24 55 1 (0,00032), and has 6 terminating decimals (0,000125). The reason for this can readily be explained by 26.53 1 converting the denominators into a power of 10. Consider for example which has six terminating 26.53 decimals. In order to convert the denominator into a power of 10 we need to multiply both numerator and denominator by 53. Thus: 1 1 53 53 53 125 = × = = = = 0,000125 26. 53 26. 53 53 26. 56 106 1000000 1 More generally, consider a unit fraction of the form where 푛 > 푚. Multiplying both the numerator 2푛.5푚 and denominator by 5푛−푚 gives: 1 1 5푛−푚 5푛−푚 5푛−푚 = × = = 2푛. 5푚 2푛. -
0.999… = 1 an Infinitesimal Explanation Bryan Dawson
0 1 2 0.9999999999999999 0.999… = 1 An Infinitesimal Explanation Bryan Dawson know the proofs, but I still don’t What exactly does that mean? Just as real num- believe it.” Those words were uttered bers have decimal expansions, with one digit for each to me by a very good undergraduate integer power of 10, so do hyperreal numbers. But the mathematics major regarding hyperreals contain “infinite integers,” so there are digits This fact is possibly the most-argued- representing not just (the 237th digit past “Iabout result of arithmetic, one that can evoke great the decimal point) and (the 12,598th digit), passion. But why? but also (the Yth digit past the decimal point), According to Robert Ely [2] (see also Tall and where is a negative infinite hyperreal integer. Vinner [4]), the answer for some students lies in their We have four 0s followed by a 1 in intuition about the infinitely small: While they may the fifth decimal place, and also where understand that the difference between and 1 is represents zeros, followed by a 1 in the Yth less than any positive real number, they still perceive a decimal place. (Since we’ll see later that not all infinite nonzero but infinitely small difference—an infinitesimal hyperreal integers are equal, a more precise, but also difference—between the two. And it’s not just uglier, notation would be students; most professional mathematicians have not or formally studied infinitesimals and their larger setting, the hyperreal numbers, and as a result sometimes Confused? Perhaps a little background information wonder . -
Numeration 2016
Numeration 2016 Prague, Czech Republic, May 23–27 2016 Department of Mathematics FNSPE Czech Technical University in Prague Numeration 2016 Prague, Czech Republic, May 23–27 2016 Department of Mathematics FNSPE Czech Technical University in Prague Print: Česká technika — nakladatelství ČVUT Editor: Petr Ambrož [email protected] Katedra matematiky Fakulta jaderná a fyzikálně inženýrská České vysoké učení technické v Praze Trojanova 13 120 00 Praha 2 List of Participants Rafael Alcaraz Barrera [email protected] IME, University of São Paulo Karam Aloui [email protected] Institut Elie Cartan de Nancy / Faculté des Sciences de Sfax Petr Ambrož petr.ambroz@fjfi.cvut.cz FNSPE, Czech Technical University in Prague Hamdi Ammar [email protected] Sfax University Myriam Amri [email protected] Sfax University Hamdi Aouinti [email protected] Université de Tunis Simon Baker [email protected] University of Reading Christoph Bandt [email protected] University of Greifswald Attila Bérczes [email protected] University of Debrecen Anne Bertrand-Mathis [email protected] University of Poitiers Dávid Bóka [email protected] Eötvös Loránd University Horst Brunotte [email protected] Marta Brzicová [email protected] FNSPE, Czech Technical University in Prague Péter Burcsi [email protected] Eötvös Loránd University Francesco Dolce [email protected] Université Paris-Est Art¯urasDubickas [email protected] Vilnius University Lubomira Dvořáková [email protected] FNSPE, Czech -
A Theorem on Repeating Decimals
University of Nebraska - Lincoln DigitalCommons@University of Nebraska - Lincoln Faculty Publications, Department of Mathematics Mathematics, Department of 6-1967 A THEOREM ON REPEATING DECIMALS William G. Leavitt University of Nebraska - Lincoln Follow this and additional works at: https://digitalcommons.unl.edu/mathfacpub Part of the Mathematics Commons Leavitt, William G., "A THEOREM ON REPEATING DECIMALS" (1967). Faculty Publications, Department of Mathematics. 48. https://digitalcommons.unl.edu/mathfacpub/48 This Article is brought to you for free and open access by the Mathematics, Department of at DigitalCommons@University of Nebraska - Lincoln. It has been accepted for inclusion in Faculty Publications, Department of Mathematics by an authorized administrator of DigitalCommons@University of Nebraska - Lincoln. The American Mathematical Monthly, Vol. 74, No. 6 (Jun. - Jul., 1967), pp. 669-673. Copyright 1967 Mathematical Association of America 19671 A THEOREM ON REPEATING DECIMALS A THEOREM ON REPEATING DECIMALS W. G. LEAVITT, University of Nebraska It is well known that a real number is rational if and only if its decimal ex- pansion is a repeating decimal. For example, 2/7 =.285714285714 . Many students also know that if n/m is a rational number reduced to lowest terms (that is, n and m relatively prime), then the number of repeated digits (we call this the length of period) depends only on m. Thus all fractions with denominator 7 have length of period 6. A sharp-eyed student may also notice that when the period (that is, the repeating digits) for 2/7 is split into its two half-periods 285 and 714, then the sum 285+714=999 is a string of nines. -
Repeating Decimals Warm up Problems. A. Find the Decimal Expressions for the Fractions 1 7 , 2 7 , 3 7 , 4 7 , 5 7 , 6 7 . What
Repeating Decimals Warm up problems. a. Find the decimal expressions for the fractions 1 2 3 4 5 6 ; ; ; ; ; : 7 7 7 7 7 7 What interesting things do you notice? b. Repeat the problem for the fractions 1 2 12 ; ;:::; : 13 13 13 What is interesting about these answers? Every fraction has a decimal representation. These representations either terminate (e.g. 3 8 = 0:375) or or they do not terminate but are repeating (e.g. 3 = 0:428571428571 ::: = 0:428571; ) 7 where the bar over the six block set of digits indicates that that block repeats indefinitely. 1 1. Perform the by hand, long divisions to calculate the decimal representations for 13 2 and 13 . Use these examples to help explain why these fractions have repeating decimals. 2. With these examples can you predict the \interesting things" that you observed in 1 warm up problem b.? Look at the long division for 7 . Does this example confirm your reasoning? 1 3. It turns out that 41 = 0:02439: Write out the long division that shows this and then find (without dividing) the four other fractions whose repeating part has these five digits in the same cyclic order. 4. As it turns out, if you divide 197 by 26, you get a quotient of 7 and a remainder of 15. How can you use this information to find the result when 2 · 197 = 394 is divided by 26? When 5 · 197 = 985 is divided by 26? 5. Suppose that when you divide N by D, the quotient is Q and the remainder is R. -
Residue Arithmetic in Digital Computers
r* 6 l't ¡r "RESIDUT ARITHMETIC IN DIGITAL COIIPUTERS" RAMESH CHANDRA DEBNATH/ M.SC. BEING A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY IN THE DEPARTMENT OF ELECTRICAL ENGINEERING THE UNIVERSITY OF ADELAIDE IVIARCl-i I979 SUIq¡4ARY The number systems used have great impact on the speed of a computer, At present at least 4 number systems (binary, negabinary, ternary and residue) are in use either for general purposes or for special purposes. The calîry plîopagation delay which j-s inherent to all weighted systems (binary, negabi-nary and ternary) is a limiting factor on the speed of computations. tlnlike the weighted system, the residue system is carry free which is the m¿rln attractj-ve feature of this system and therefore its potentiality in a general purpose computer, in the light of present day tech- nology, has been investigated. As the background of the investigation the binary and negabinary arithmetic have been thoroughly studied. A ne',r method for fractional negabinary division, and a correction on negabinary multiplì-cation have been proposed. It has been observed that the binary arithmetic operations are fas- ter thair thcse ir. negabinary syslem. The residue system has got. problems too. The sign detectj-on, operations j-nvolving sign detection (comparison, overflow detection, division) and the resj-due-to--decimal- con- version are difficult. Moreover, the residue systemT being an i.nteger: systetn, makes floati-ng poinc arithmetic operatlcns very difficult. The def:ails of the existing algorithms,/ methods to solve those problems have been thoroughly inves- tigated and drawbacks of those algoritlrms/methods have been di scussed. -
Irrational Numbers and Rounding Off*
OpenStax-CNX module: m31341 1 Irrational Numbers and Rounding Off* Rory Adams Free High School Science Texts Project Mark Horner Heather Williams This work is produced by OpenStax-CNX and licensed under the Creative Commons Attribution License 3.0 1 Introduction You have seen that repeating decimals may take a lot of paper and ink to write out. Not only is that impossible, but writing numbers out to many decimal places or a high accuracy is very inconvenient and rarely gives practical answers. For this reason we often estimate the number to a certain number of decimal places or to a given number of signicant gures, which is even better. 2 Irrational Numbers Irrational numbers are numbers that cannot be written as a fraction with the numerator and denominator as integers. This means that any number that is not a terminating decimal number or a repeating decimal number is irrational. Examples of irrational numbers are: p p p 2; 3; 3 4; π; p (1) 1+ 5 2 ≈ 1; 618 033 989 tip: When irrational numbers are written in decimal form, they go on forever and there is no repeated pattern of digits. If you are asked to identify whether a number is rational or irrational, rst write the number in decimal form. If the number is terminated then it is rational. If it goes on forever, then look for a repeated pattern of digits. If there is no repeated pattern, then the number is irrational. When you write irrational numbers in decimal form, you may (if you have a lot of time and paper!) continue writing them for many, many decimal places. -
**4************ from the Original Document
DoCUMENT/ RESUME ED 220 67)9 CE 033 642 TITLE Military Curriculum Materials for Vocational and (14Technical Educaticin. Fundamentals of Digital Logic, 7-1. INSTITUTION Marine Corps, Washington, D.C.; Ohio State Univ., Columbus. National Center for Research in Vocational Education, SPONS AGENCY Office of Education (DHEW), Washington, D.C4. PUB DATE (76) NOTE 146p. EDR's PRICE MF01/P606 PlusPostase. DESCRIPTORS Arithmetic; *Digital'IComputers; *Electric Circuits; *Electronic Technicians; !Equipment Maintenance; Individualized Instruction; Instructional Materials; *Mathematical Lpgic; Mathematics; Military Personnel; Military TrainVng; Postsecondary Educationp Secondary Education; *Technical Education IDENTIFIERS Binary Arithmetic; Boolean Algebra; *Computer Logic; Military Curriculum Project ABSTRACT 'Tarqeted for grades0'through adult, these military-developed curriculum mater als consist ofa student lesson book with text readings and review pierVes designed to prepare electronic.per,sonnel for further trainin in digital'techniques. Covered,in the five lessons are binary arithmeti8 (number systems, decimal systemi, the mathematical form of notation, number system conversion, matheinatical operat4ons, specially coded systems); Boolean.a,lgebra (basic 'functions, signal levels, application of L Boolean algebra, Boolean equations, drawing logic diagrams, theorems and truth tables, simplifying Boolean equations); logic gates (diode logic circuits, transistor logic ,ci.rcuits, NOT circuits, EXCLUSIVE OR circu0s, operational analysis); logic -
142857, and More Numbers Like It
142857, and more numbers like it John Kerl January 4, 2012 Abstract These are brief jottings to myself on vacation-spare-time observations about trans- posable numbers. Namely, 1/7 in base 10 is 0.142857, 2/7 is 0.285714, 3/7 is 0.428571, and so on. That’s neat — can we find more such? What happens when we use denom- inators other than 7, or bases other than 10? The results presented here are generally ancient and not essentially original in their particulars. The current exposition takes a particular narrative and data-driven approach; also, elementary group-theoretic proofs preferred when possible. As much I’d like to keep the presentation here as elementary as possible, it makes the presentation far shorter to assume some first-few-weeks-of-the-semester group theory Since my purpose here is to quickly jot down some observations, I won’t develop those concepts in this note. This saves many pages, at the cost of some accessibility, and with accompanying unevenness of tone. Contents 1 The number 142857, and some notation 3 2 Questions 4 2.1 Are certain constructions possible? . .. 4 2.2 Whatperiodscanexist? ............................. 5 2.3 Whencanfullperiodsexist?........................... 5 2.4 Howdodigitsetscorrespondtonumerators? . .... 5 2.5 What’s the relationship between add order and shift order? . ...... 5 2.6 Why are half-period shifts special? . 6 1 3 Findings 6 3.1 Relationship between expansions and integers . ... 6 3.2 Periodisindependentofnumerator . 6 3.3 Are certain constructions possible? . .. 7 3.4 Whatperiodscanexist? ............................. 7 3.5 Whencanfullperiodsexist?........................... 8 3.6 Howdodigitsetscorrespondtonumerators? . .... 8 3.7 What’s the relationship between add order and shift order? . -
Renormalization Group Calculation of Dynamic Exponent in the Models E and F with Hydrodynamic Fluctuations M
Vol. 131 (2017) ACTA PHYSICA POLONICA A No. 4 Proceedings of the 16th Czech and Slovak Conference on Magnetism, Košice, Slovakia, June 13–17, 2016 Renormalization Group Calculation of Dynamic Exponent in the Models E and F with Hydrodynamic Fluctuations M. Dančoa;b;∗, M. Hnatiča;c;d, M.V. Komarovae, T. Lučivjanskýc;d and M.Yu. Nalimove aInstitute of Experimental Physics SAS, Watsonova 47, 040 01 Košice, Slovakia bBLTP, Joint Institute for Nuclear Research, Dubna, Russia cInstitute of Physics, Faculty of Sciences, P.J. Safarik University, Park Angelinum 9, 041 54 Košice, Slovakia d‘Peoples’ Friendship University of Russia (RUDN University) 6 Miklukho-Maklaya St, Moscow, 117198, Russian Federation eDepartment of Theoretical Physics, St. Petersburg University, Ulyanovskaya 1, St. Petersburg, Petrodvorets, 198504 Russia The renormalization group method is applied in order to analyze models E and F of critical dynamics in the presence of velocity fluctuations generated by the stochastic Navier–Stokes equation. Results are given to the one-loop approximation for the anomalous dimension γλ and fixed-points’ structure. The dynamic exponent z is calculated in the turbulent regime and stability of the fixed points for the standard model E is discussed. DOI: 10.12693/APhysPolA.131.651 PACS/topics: 64.60.ae, 64.60.Ht, 67.25.dg, 47.27.Jv 1. Introduction measurable index α [4]. The index α has been also de- The liquid-vapor critical point, λ transition in three- termined in the framework of the renormalization group dimensional superfluid helium 4He belong to famous ex- (RG) approach using a resummation procedure [5] up to amples of continuous phase transitions. -
Appendix: the Cost of Computing: Big-O and Recursion
Appendix: The Cost of Computing: Big-O and Recursion Big-O otation In the previous section we have seen a chart comparing execution times for various sorting algorithms. The chart, however, did not show any absolute numbers to tell us the actual time it took, say, to sort an array of 10,000 double numbers using the insertion sort algorithm. That is because the actual time depends, among other things, on the processor of the machine the program is running on, not only on the algorithm used. For example, the worst possible sorting algorithm executing on a Pentium II 450 MHz chip may finish a lot quicker in some situations than the best possible sorting algorithm running on an old Intel 386 chip. Even when two algorithms execute on the same machine, other processes that are running on the system can considerably skew the results one way or another. Therefore, execution time is not really an adequate measurement of the "goodness" of an algorithm. We need a more mathematical way to determine how good an algorithm is, and which of two algorithms is better. In this section, we will introduce the "big-O" notation, which does provide just that: a mathematical description that allows you to rate an algorithm. The Math behind Big-O There is a little background mathematics required to work with this concept, but it is not much, and you should quickly understand the major concepts involved. The most important concept you may need to review is that of the limit of a function. Definition: Limit of a Function We say that a function f converges to a limit L as x approaches a number c if f(x) gets closer and closer to L if x gets closer and closer to c, and we use the notation lim f (x) = L x→c Similarly, a function f converges to a limit L as x approaches positive (negative) infinity if f(x) gets closer and closer to L as x is positive (negative) and gets bigger (smaller) and bigger (smaller). -
On Expansions in Non-Integer Base
SAPIENZA UNIVERSITA` DI ROMA DOTTORATO DI RICERCA IN “MODELLI E METODI MATEMATICI PER LA TECNOLOGIA E LA SOCIETA` ” XXII CICLO AND UNIVERSITE´ PARIS 7-PARIS DIDEROT DOCTORAT D’INFORMATIQUE On expansions in non-integer base Anna Chiara Lai Dipartimento di Metodi e Modelli Matematici per le Scienze applicate Via Antonio Scarpa, 16 - 00161 Roma. and Laboratoire d’Informatique Algorithmique: Fondements et Applications CNRS UMR 7089, Universit´eParis Diderot - Paris 7, Case 7014 75205 Paris Cedex 13 A. A. 2008-2009 Contents Introduction 3 Chapter 1. Background results on expansions in non-integer base 6 1. Our toolbox 6 2. Positional number systems 10 3. Positional numeration systems with positive integer base 11 4. Expansions in non-integer bases 12 Chapter 2. Expansions in complex base 17 1. Introduction 17 2. Geometrical background 18 3. Characterization of the convex hull of representable numbers 24 4. Representability in complex base 35 5. Overview of original contributions, conclusions and further developments 37 Chapter 3. Expansions in negative base 40 1. Introduction 40 2. Preliminaries on expansions in non-integer negative base 41 3. Symbolic dynamical systems and the alternate order 44 4. A characterization of sofic ( q)-shifts and ( q)-shifts of finite type 46 − − 5. Entropy of the ( q)-shift 47 − 6. The Pisot case 49 7. A conversion algorithm from positive to negative base 52 8. Overview of original contributions, conclusions and further developments 56 Chapter4. GeneralizedGoldenMeanforternaryalphabets 58 1. Introduction 58 2. Expansions in non-integer base with alphabet with deleted digits 58 3. Critical bases 60 4. Normal ternary alphabets 62 5.