6 Number Theory II: Modular Arithmetic, Cryptography, and Randomness
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The Meter Greeters
Journal of Applied Communications Volume 59 Issue 2 Article 3 The Meter Greeters C. Hamilton Kenney Follow this and additional works at: https://newprairiepress.org/jac This work is licensed under a Creative Commons Attribution-Noncommercial-Share Alike 4.0 License. Recommended Citation Kenney, C. Hamilton (1976) "The Meter Greeters," Journal of Applied Communications: Vol. 59: Iss. 2. https://doi.org/10.4148/1051-0834.1951 This Article is brought to you for free and open access by New Prairie Press. It has been accepted for inclusion in Journal of Applied Communications by an authorized administrator of New Prairie Press. For more information, please contact [email protected]. The Meter Greeters Abstract The United States and Canada became meter greeters away back in the 1800's. The U.S. Congress passed an act in 1866 legalizing the metric system for weights and measures use, and metric units were on the law books of the Dominion of Canada in 1875. This article is available in Journal of Applied Communications: https://newprairiepress.org/jac/vol59/iss2/3 Kenney: The Meter Greeters The Meter Greeters C. Hamilton Kenney The United States and Canada became meter greeters away back in the 1800's. The U.S. Congress passed an act in 1866 legalizing the metric system for weights and measures use, and metric units were on the law books of the Dominion of Canada in 1875. The U.S. A. was a signatory to the Treaty of the Meter l signed in Paris, France. in 1875, establishing the metric system as an international measurement system, but Canada did not become a signatory nation until 1907. -
Number Theory
“mcs-ftl” — 2010/9/8 — 0:40 — page 81 — #87 4 Number Theory Number theory is the study of the integers. Why anyone would want to study the integers is not immediately obvious. First of all, what’s to know? There’s 0, there’s 1, 2, 3, and so on, and, oh yeah, -1, -2, . Which one don’t you understand? Sec- ond, what practical value is there in it? The mathematician G. H. Hardy expressed pleasure in its impracticality when he wrote: [Number theorists] may be justified in rejoicing that there is one sci- ence, at any rate, and that their own, whose very remoteness from or- dinary human activities should keep it gentle and clean. Hardy was specially concerned that number theory not be used in warfare; he was a pacifist. You may applaud his sentiments, but he got it wrong: Number Theory underlies modern cryptography, which is what makes secure online communication possible. Secure communication is of course crucial in war—which may leave poor Hardy spinning in his grave. It’s also central to online commerce. Every time you buy a book from Amazon, check your grades on WebSIS, or use a PayPal account, you are relying on number theoretic algorithms. Number theory also provides an excellent environment for us to practice and apply the proof techniques that we developed in Chapters 2 and 3. Since we’ll be focusing on properties of the integers, we’ll adopt the default convention in this chapter that variables range over the set of integers, Z. 4.1 Divisibility The nature of number theory emerges as soon as we consider the divides relation a divides b iff ak b for some k: D The notation, a b, is an abbreviation for “a divides b.” If a b, then we also j j say that b is a multiple of a. -
FROM HARMONIC ANALYSIS to ARITHMETIC COMBINATORICS: a BRIEF SURVEY the Purpose of This Note Is to Showcase a Certain Line Of
FROM HARMONIC ANALYSIS TO ARITHMETIC COMBINATORICS: A BRIEF SURVEY IZABELLA ÃLABA The purpose of this note is to showcase a certain line of research that connects harmonic analysis, speci¯cally restriction theory, to other areas of mathematics such as PDE, geometric measure theory, combinatorics, and number theory. There are many excellent in-depth presentations of the vari- ous areas of research that we will discuss; see e.g., the references below. The emphasis here will be on highlighting the connections between these areas. Our starting point will be restriction theory in harmonic analysis on Eu- clidean spaces. The main theme of restriction theory, in this context, is the connection between the decay at in¯nity of the Fourier transforms of singu- lar measures and the geometric properties of their support, including (but not necessarily limited to) curvature and dimensionality. For example, the Fourier transform of a measure supported on a hypersurface in Rd need not, in general, belong to any Lp with p < 1, but there are positive results if the hypersurface in question is curved. A classic example is the restriction theory for the sphere, where a conjecture due to E. M. Stein asserts that the Fourier transform maps L1(Sd¡1) to Lq(Rd) for all q > 2d=(d¡1). This has been proved in dimension 2 (Fe®erman-Stein, 1970), but remains open oth- erwise, despite the impressive and often groundbreaking work of Bourgain, Wol®, Tao, Christ, and others. We recommend [8] for a thorough survey of restriction theory for the sphere and other curved hypersurfaces. Restriction-type estimates have been immensely useful in PDE theory; in fact, much of the interest in the subject stems from PDE applications. -
Water Heater Formulas and Terminology
More resources http://waterheatertimer.org/9-ways-to-save-with-water-heater.html http://waterheatertimer.org/Figure-Volts-Amps-Watts-for-water-heater.html http://waterheatertimer.org/pdf/Fundamentals-of-water-heating.pdf FORMULAS & FACTS BTU (British Thermal Unit) is the heat required to raise 1 pound of water 1°F 1 BTU = 252 cal = 0.252 kcal 1 cal = 4.187 Joules BTU X 1.055 = Kilo Joules BTU divided by 3,413 = Kilowatt (1 KW) FAHRENHEIT CENTIGRADE 32 0 41 5 To convert from Fahrenheit to Celsius: 60.8 16 (°F – 32) x 5/9 or .556 = °C. 120.2 49 140 60 180 82 212 100 One gallon of 120°F (49°C) water BTU output (Electric) = weighs approximately 8.25 pounds. BTU Input (Not exactly true due Pounds x .45359 = Kilogram to minimal flange heat loss.) Gallons x 3.7854 = Liters Capacity of a % of hot water = cylindrical tank (Mixed Water Temp. – Cold Water – 1⁄ 2 diameter (in inches) Temp.) divided by (Hot Water Temp. x 3.146 x length. (in inches) – Cold Water Temp.) Divide by 231 for gallons. % thermal efficiency = Doubling the diameter (GPH recovery X 8.25 X temp. rise X of a pipe will increase its flow 1.0) divided by BTU/H Input capacity (approximately) 5.3 times. BTU output (Gas) = GPH recovery x 8.25 x temp. rise x 1.0 FORMULAS & FACTS TEMP °F RISE STEEL COPPER Linear expansion of pipe 50° 0.38˝ 0.57˝ – in inches per 100 Ft. 100° .076˝ 1.14˝ 125° .092˝ 1.40˝ 150° 1.15˝ 1.75˝ Grain – 1 grain per gallon = 17.1 Parts Per million (measurement of water hardness) TC-092 FORMULAS & FACTS GPH (Gas) = One gallon of Propane gas contains (BTU/H Input X % Eff.) divided by about 91,250 BTU of heat. -
Modular Arithmetic
CS 70 Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 5 Modular Arithmetic One way to think of modular arithmetic is that it limits numbers to a predefined range f0;1;:::;N ¡ 1g, and wraps around whenever you try to leave this range — like the hand of a clock (where N = 12) or the days of the week (where N = 7). Example: Calculating the day of the week. Suppose that you have mapped the sequence of days of the week (Sunday, Monday, Tuesday, Wednesday, Thursday, Friday, Saturday) to the sequence of numbers (0;1;2;3;4;5;6) so that Sunday is 0, Monday is 1, etc. Suppose that today is Thursday (=4), and you want to calculate what day of the week will be 10 days from now. Intuitively, the answer is the remainder of 4 + 10 = 14 when divided by 7, that is, 0 —Sunday. In fact, it makes little sense to add a number like 10 in this context, you should probably find its remainder modulo 7, namely 3, and then add this to 4, to find 7, which is 0. What if we want to continue this in 10 day jumps? After 5 such jumps, we would have day 4 + 3 ¢ 5 = 19; which gives 5 modulo 7 (Friday). This example shows that in certain circumstances it makes sense to do arithmetic within the confines of a particular number (7 in this example), that is, to do arithmetic by always finding the remainder of each number modulo 7, say, and repeating this for the results, and so on. -
8.6 Modular Arithmetic
“mcs” — 2015/5/18 — 1:43 — page 263 — #271 8.6 Modular Arithmetic On the first page of his masterpiece on number theory, Disquisitiones Arithmeticae, Gauss introduced the notion of “congruence.” Now, Gauss is another guy who managed to cough up a half-decent idea every now and then, so let’s take a look at this one. Gauss said that a is congruent to b modulo n iff n .a b/. This is j written a b.mod n/: ⌘ For example: 29 15 .mod 7/ because 7 .29 15/: ⌘ j It’s not useful to allow a modulus n 1, and so we will assume from now on that moduli are greater than 1. There is a close connection between congruences and remainders: Lemma 8.6.1 (Remainder). a b.mod n/ iff rem.a; n/ rem.b; n/: ⌘ D Proof. By the Division Theorem 8.1.4, there exist unique pairs of integers q1;r1 and q2;r2 such that: a q1n r1 D C b q2n r2; D C “mcs” — 2015/5/18 — 1:43 — page 264 — #272 264 Chapter 8 Number Theory where r1;r2 Œ0::n/. Subtracting the second equation from the first gives: 2 a b .q1 q2/n .r1 r2/; D C where r1 r2 is in the interval . n; n/. Now a b.mod n/ if and only if n ⌘ divides the left side of this equation. This is true if and only if n divides the right side, which holds if and only if r1 r2 is a multiple of n. -
Pure Mathematics
Why Study Mathematics? Mathematics reveals hidden patterns that help us understand the world around us. Now much more than arithmetic and geometry, mathematics today is a diverse discipline that deals with data, measurements, and observations from science; with inference, deduction, and proof; and with mathematical models of natural phenomena, of human behavior, and social systems. The process of "doing" mathematics is far more than just calculation or deduction; it involves observation of patterns, testing of conjectures, and estimation of results. As a practical matter, mathematics is a science of pattern and order. Its domain is not molecules or cells, but numbers, chance, form, algorithms, and change. As a science of abstract objects, mathematics relies on logic rather than on observation as its standard of truth, yet employs observation, simulation, and even experimentation as means of discovering truth. The special role of mathematics in education is a consequence of its universal applicability. The results of mathematics--theorems and theories--are both significant and useful; the best results are also elegant and deep. Through its theorems, mathematics offers science both a foundation of truth and a standard of certainty. In addition to theorems and theories, mathematics offers distinctive modes of thought which are both versatile and powerful, including modeling, abstraction, optimization, logical analysis, inference from data, and use of symbols. Mathematics, as a major intellectual tradition, is a subject appreciated as much for its beauty as for its power. The enduring qualities of such abstract concepts as symmetry, proof, and change have been developed through 3,000 years of intellectual effort. Like language, religion, and music, mathematics is a universal part of human culture. -
From Arithmetic to Algebra
From arithmetic to algebra Slightly edited version of a presentation at the University of Oregon, Eugene, OR February 20, 2009 H. Wu Why can’t our students achieve introductory algebra? This presentation specifically addresses only introductory alge- bra, which refers roughly to what is called Algebra I in the usual curriculum. Its main focus is on all students’ access to the truly basic part of algebra that an average citizen needs in the high- tech age. The content of the traditional Algebra II course is on the whole more technical and is designed for future STEM students. In place of Algebra II, future non-STEM would benefit more from a mathematics-culture course devoted, for example, to an understanding of probability and data, recently solved famous problems in mathematics, and history of mathematics. At least three reasons for students’ failure: (A) Arithmetic is about computation of specific numbers. Algebra is about what is true in general for all numbers, all whole numbers, all integers, etc. Going from the specific to the general is a giant conceptual leap. Students are not prepared by our curriculum for this leap. (B) They don’t get the foundational skills needed for algebra. (C) They are taught incorrect mathematics in algebra classes. Garbage in, garbage out. These are not independent statements. They are inter-related. Consider (A) and (B): The K–3 school math curriculum is mainly exploratory, and will be ignored in this presentation for simplicity. Grades 5–7 directly prepare students for algebra. Will focus on these grades. Here, abstract mathematics appears in the form of fractions, geometry, and especially negative fractions. -
SE 015 465 AUTHOR TITLE the Content of Arithmetic Included in A
DOCUMENT' RESUME ED 070 656 24 SE 015 465 AUTHOR Harvey, John G. TITLE The Content of Arithmetic Included in a Modern Elementary Mathematics Program. INSTITUTION Wisconsin Univ., Madison. Research and Development Center for Cognitive Learning, SPONS AGENCY National Center for Educational Research and Development (DHEW/OE), Washington, D.C. BUREAU NO BR-5-0216 PUB DATE Oct 71 CONTRACT OEC-5-10-154 NOTE 45p.; Working Paper No. 79 EDRS PRICE MF-$0.65 HC-$3.29 DESCRIPTORS *Arithmetic; *Curriculum Development; *Elementary School Mathematics; Geometric Concepts; *Instruction; *Mathematics Education; Number Concepts; Program Descriptions IDENTIFIERS Number Operations ABSTRACT --- Details of arithmetic topics proposed for inclusion in a modern elementary mathematics program and a rationale for the selection of these topics are given. The sequencing of the topics is discussed. (Author/DT) sJ Working Paper No. 19 The Content of 'Arithmetic Included in a Modern Elementary Mathematics Program U.S DEPARTMENT OF HEALTH, EDUCATION & WELFARE OFFICE OF EDUCATION THIS DOCUMENT HAS SEEN REPRO Report from the Project on Individually Guided OUCED EXACTLY AS RECEIVED FROM THE PERSON OR ORGANIZATION ORIG INATING IT POINTS OF VIEW OR OPIN Elementary Mathematics, Phase 2: Analysis IONS STATED 00 NOT NECESSARILY REPRESENT OFFICIAL OFFICE Of LOU Of Mathematics Instruction CATION POSITION OR POLICY V lb. Wisconsin Research and Development CENTER FOR COGNITIVE LEARNING DIE UNIVERSITY Of WISCONSIN Madison, Wisconsin U.S. Office of Education Corder No. C03 Contract OE 5-10-154 Published by the Wisconsin Research and Development Center for Cognitive Learning, supported in part as a research and development center by funds from the United States Office of Education, Department of Health, Educa- tion, and Welfare. -
Forests Commission Victoria-Australia
VICTORIA, 1971 FORESTS COMMISSION VICTORIA-AUSTRALIA FIFTY SECOND ANNUAL REPORT FINANCIAL YEAR 1970-71 PRESENTED TO BOTH HOUSES OF PARLIAMENT PURSUANT TO ACT No. 6254, SECTION 35 . .Approximate Cosl of llrport.-Preparation, not given. Printing (250 copies), $1,725.00. No. 14-9238/71.-Price 80 cents FORESTS COMMISSION, VICTORIA TREASURY GARDENS, MELBOURNE, 3002 ANNUAL REPORT 1970-71 In compliance with the provisions of section 35 of the Forests Act 1958 (No. 6254) the Forests Commission has the honour to present to Parliament the following report of its activities and financial statements for the financial year 1970-71. F. R. MOULDS, Chainnan. C. W. ELSEY, Commissioner. A. J. THREADER, Commissioner. F. H. TREYV AUD, Secretary. CONTENTS PAGE 6 FEATURES. 8 fvlANAGEMENT- Forest Area, Surveys, fvlapping, Assessment, Recreation, fvlanagement Plans, Plantation Extension Planning, Forest Land Use Planning, Public Relations. 12 0PERATIONS- Silviculture of Native Forests, Seed Collection, Softwood Plantations, Hardwood Plantations, Total Plantings, Extension Services, Utilization, Grazing, Forest Engineering, Transport, Buildings, Reclamation and Conservation Works, Forest Prisons, Legal, Search and Rescue Operations. 24 ECONOMICS AND fvlARKETING- Features, The Timber Industry, Sawlog Production, Veneer Timber, Pulpwood, Other Forest Products, Industrial Undertakings, Other Activities. 28 PROTBCTION- Fire, Radio Communications, Biological, Fire Research. 32 EDUCATION AND RESEARCH- Education-School of Forestry, University of fvlelbourne, Overseas and Other Studies ; Research-Silviculture, Hydrology, Pathology, Entomology, Biological Survey, The Sirex Wood Wasp; Publications. 38 CONFERENCES. 39 ADMINISTRATJON- Personnel-Staff, Industrial, Number of Employees, Worker's Compensation, Staff Training ; fvlethods ; Stores ; Finance. APPENDICES- 43 I. Statement of Output of Produce. 44 II. Causes of Fires. 44 III. Summary of Fires and Areas Burned. -
Fundamentals of Math CHAPTER 1
© Jones and Bartlett Publishers, LLC. NOT FOR SALE OR DISTRIBUTION Fundamentals of Math CHAPTER 1 OBJECTIVES ■ Understand the difference between the Arabic and Roman numeral systems ■ Translate Arabic numerals to Roman numerals ■ Translate Roman numerals to Arabic numerals ■ Understand the metric system ■ Understand the apothecary system ■ Be able to convert metric to apothecary ■ Be able to convert apothecary to metric ARABIC NUMERALS The Arabic number system uses the numerals 1, 2, 3, 4, 5, 6, 7, 8, 9, and zero (0). It is also known as the decimal system. Depending on how these numbers are arranged determines the value of the number. For example, digits 4, 7, and 2 placed together (472) represent the number four hundred seventy-two. A decimal point (.) separates whole numbers, or units, from fractional num- bers, or fractional units. All numbers on the left side of the decimal point are considered whole numbers. All numbers placed on the right of the decimal point are considered fractional units, or less than one whole unit. The following num- ber line shows the relationship of Arabic numerals based on their position in a number. Ten-thousands hundreds ones tenths thousandths hundred-thousandths -----5------8------2-----4-----3---- . ----6------7------9------3------2-------------- thousands tens hundredths ten-thousandths The number 43.6 contains the numerals 4, 3, and 6. This represents forty-three units of one and six-tenths of one unit. Decimals will be covered in more detail in Chapter 2. 1 59612_CH01_FINAL.indd 1 8/20/09 7:38:45 PM © Jones and Bartlett Publishers, LLC. NOT FOR SALE OR DISTRIBUTION 2 Chapter 1 ■ Fundamentals of Math ROMAN NUMERALS The Roman numeral system does not utilize numerals. -
Outline of Mathematics
Outline of mathematics The following outline is provided as an overview of and 1.2.2 Reference databases topical guide to mathematics: • Mathematical Reviews – journal and online database Mathematics is a field of study that investigates topics published by the American Mathematical Society such as number, space, structure, and change. For more (AMS) that contains brief synopses (and occasion- on the relationship between mathematics and science, re- ally evaluations) of many articles in mathematics, fer to the article on science. statistics and theoretical computer science. • Zentralblatt MATH – service providing reviews and 1 Nature of mathematics abstracts for articles in pure and applied mathemat- ics, published by Springer Science+Business Media. • Definitions of mathematics – Mathematics has no It is a major international reviewing service which generally accepted definition. Different schools of covers the entire field of mathematics. It uses the thought, particularly in philosophy, have put forth Mathematics Subject Classification codes for orga- radically different definitions, all of which are con- nizing their reviews by topic. troversial. • Philosophy of mathematics – its aim is to provide an account of the nature and methodology of math- 2 Subjects ematics and to understand the place of mathematics in people’s lives. 2.1 Quantity Quantity – 1.1 Mathematics is • an academic discipline – branch of knowledge that • Arithmetic – is taught at all levels of education and researched • Natural numbers – typically at the college or university level. Disci- plines are defined (in part), and recognized by the • Integers – academic journals in which research is published, and the learned societies and academic departments • Rational numbers – or faculties to which their practitioners belong.