Store records in the table at index hk defined by hk k mod n here, h is a hash function, and n is an array size. In this article, i will explain what modular arithmetic is, illustrate why it is of importance for mathematicians, and discuss some recent breakthroughs. In fact, this is the example we shall need for the rsa cryptosystem. Jump to 100 level math and statistics courses 200 level math and statistics courses 300 level math and statistics. Modular arithmetic readings introduction to college mathematics. To show that 10 doesnt have an inverse is harder and is left as an exercise. The second is to have students explain this exploration in mathematics to each other, as if they were true mathematics researchers. Welcome to math all star remainder does not seem to be a big topic in school math. Once you have completed all of the readings and practices, click the next button below to continue the reading for unit 1. It starts with some occurrences of patterns in powers of numbers and in clock arithmetic. Math circles for all ages arizona state university. I multiply out the left side, then get the xs on one side. An exploration of modular arithmetic adding, multiplying, powers, orders, residues. Modular arithmetic connects with primes in an interesting way.
Walter gave an exam in a mathematics class of five students. Students begin with familiar problems involving an analog clock and soon realize they already have some experience with modular arithmetic. View the mathematics upperdivision undergraduate course offering pdf schedule for future semesters. The intersection of the row headed 7 and the column headed 11 gives the number 6. Th e separate teachers edition includes solutions to all questions, mathematics background for each activity, extension, mathematical humor, and more. Mgf 1107 explorations in mathematics this course is designed for students who plan to major in fields that do not require an indepth study of mathematics. The modular arithmetic properties allow me to solve this equation the way i would solve a linear equation, up to a point. The modular arithmetic developed by gauss is a beautiful topic in number. We make the jump from modular arithemetic into rational number arithmetic. Modular arithmetic may 2012 national council of teachers. In the activities of this module prospective teachers. Pdf explorations of students mathematical beliefs and behavior. In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers wrap around when reaching a certain value, called the modulus.
The major topics introduced in this course are financial mathematics, exponential growth and decay, numbers and number systems, and elementary number theory. Math 728 lesson plan modulo arithmetic introduction. Although there will be no output immediately, these definitions are used later in this worksheet. For example, in the exploration of the multiplication table fig.
The authoritative source for course information is the marshall university catalog. It might help us to think about modular arithmetic as the remainder when we divide by the modulus. The author, a former middleschool math teacher, has devoted his entire 35year professional career to creating ways to reach students through engaging, realworld mathematics. We say \a is congruent to b modulo m and write a b mod m if m ja b i. High school mathematics extensions wikimedia commons. High school mathematics extensionsprimesmodular arithmetic. Brilliant guides you through problem solving strategies and challenges you to think outside the box. Ribet for other titles published in this series, go to. Student explorations in mathematics national council of teachers.
Modular arithmetic has been a major concern of mathematicians for at least 250 years, and is still a very active topic of current research. Mathematics and mathematical thinking include basic operations, reading and interpreting data from charts and tables, predicting outcomes based on data, and combinatorics. However, this important subject is not taught in schools. Build quantitative skills in math, science, and computer science with fun and challenging interactive explorations. List of 200 ideastopics for a mathematical exploration. The modulo arithmetic lesson aims to teach students addition and multiplication modulo n. This course provides support and preparation in math for students who suffer from math anxiety, have math sat scores of 450 or below or have not taken a math course in over a year. Originally math circles targeted primarily k12 students. Choose the one alternative that best completes the statement or answers the question. Now that you know fermat s little theorem, solve problem 3 in a. The major topics introduced in this course are financial mathematics, graph theory, numbers and number systems, and elementary number theory. Modular arithmetic may 2012 this issue of students explorations in mathematics introduces students to modular arithmetic and its applications. Math 2001 modular arithmetic exploration april 14, 2020 the later pages of this document show the full additionmultiplication tables for some small.
How do students revisit school mathematics in modular arithmetic. The best way to introduce modular arithmetic is to think of the face of a clock. Firstly, to introduce the basic algorithms for computing exactly with integers, polynomials and vector spaces. Connections will be made between math and music, art, architecture, and nature. For modular arithmetic with a modulus of 12, also known as arithmetic modulo of 12, think of an actual clock with its 12 numbers. We say a b mod m if a and b have the same remainder when divided by m, or there exists an integer k such that a b km. Modular arithmetic motivates many questions that dont arise when studying classic arithmetic. This book aims to provide a complete coverage of this. Our titles are available for most basic curriculum subjects plus many enrichment areas. Gauss wellknown reference to arithmetic as the queen of mathematics.
Therefore 3 doesnt have an inverse in mod 15 arithmetic. The modern approach to modular arithmetic was developed by carl friedrich gauss in his book disquisitiones arithmeticae, published in 1801. Consequently, many students rely on their intuition when attempting to solve such problems. Modular numbers are simply numbers following a different pattern than the typical 0 to 9 we know well. This module focuses on student learning, identity and dispositions of a case study student.
While it is hoped that students will appreciate the richness of opportunities for mathematical exploration, below is a stimulus as a means of helping students to get started on their explorations. Student explorations in mathematics national council of. Below are listings for graduate courses offered through the department of mathematics. I am having trouble understanding modular arithmetic and i have to solve these two problems by hand. Modular arithmetic summary this lesson is an exploration of modular arithmetic. Number theory modular arithmetic stanford university. Jump to 500 level math and statistics courses 600 level math and statistics courses 500 level. Modular arithmetic practice joseph zoller september, 2015 practice problem solutions 1.
For example, in classic arithmetic, adding a positive number a to another number b always produces a number larger than b. This lesson is an exploration of modular arithmetic. A student may be able to verify such a proof, line by line, and at the end will agree. The course requires some exposure to the basic concepts of discrete mathematics, but no computing experience. The modern approach to modular arithmetic was developed by carl friedrich gauss in his book disquisitiones arithmeticae, published in 1801 a familiar use of modular arithmetic is in the 12hour clock, in which the day is divided into two 12. Introductory explorations and the research cycle html pdf definitions html pdf problem posing html pdf examples, patterns, and conjectures html pdf proof html pdf getting information html pdf. Undergraduate courses in mathematics and statistics. View the mathematics graduate course offering and course rotation pdf schedule for future semesters. My class is learning through theorems which isnt helpful at all. In clock arithmetic, students operate in the familiar modulo 12 as this builds upon. Modulararithmetic modular arithmetic is a way of systematically ignoring di. Discrete math for computer science students ken bogart dept. Modular arithmetic, or clock arithmetic, reduces all positive integers to a set of remainders. This study contributes to the realistic mathematics education literature as it documents a student s progression through the definitionof and definitionfor stages of mathematical activity in an.
Graduate texts in mathematics 106 editorial board s. Mathematics mat explorations i 3 credits for humanities majors, the course covers set theory and a number of its applications, topics from logic including propositions, truth tables, number systems, and elementary geometry. We will now state the theorem regarding the existence of inverses in modular arithmetic. What are some good books to learn modular arithmetics. Students collect data and explore a variety of situations that can be modeled using linear, exponential, polynomial, and trigonometric functions. Modular arithmetic may 2012 national council of teachers of. Math for gifted students official site with more information and practice. The major topics introduced in this course are financial mathematics, exponential growth and decay, numbers and. This is a 3unit class on the beauty and utility of mathematics, including patterns and symmetry, voting strategies, and. This lesson plan guides you through teaching modular arithmetic to students using discussion. Many gamification techniques were employed in the beam program which will be discussed in this later.
Other topics may be covered at instructors discretion. This printable worksheet and interactive quiz will provide an assessment of your proficiency at solving math problems using modular arithmetic. How do students revisit school mathematics in modular. Terry is a much loved figure among his students and is. If n is prime then every number except 0 has an inverse in modulo n arithmetic. We are confident that you will enjoy this adventure and find it quite worthwhile. For almost all its history, the study of modular arithmetic has been driven purely by. In some sense, modular arithmetic is easier than integer arithmetic because there are only finitely many elements, so to find a solution to a problem you can always try every possbility. I need this to be broken down like its being taught to a 3. Students sometimes find it difficult to know where to start with a task as openended as the exploration. This is a \partition of the integers into two \classes. Liljedahl 2006 described students explorations and conjecturing in such.
Generalizations of the subject led to important 19thcentury attempts to prove fermats last theorem and the development of significant parts of modern algebra. Magic squares national council of teachers of mathematics. Introductory explorations and the research cycle html pdf definitions html pdf problem posing html. Anybody can master modular arithmetic with a little bit of practice. The systematic study of number theory was initiated around 300b. Below are listings for undergraduate courses offered through the department of mathematics.
Algebra and number theory modular arithmetic goldbachs conjecture probabilistic number theory applications of complex numbers diophantine equations continued fractions general solution of a cubic equation applications of logarithms polar equations patterns in pascals triangle finding prime numbers random numbers. Continuing the example above with modulus 5, we write. Students begin with familiar problems involving an analog clock and soon realize they already have some experience with modular. Sep, 2015 modular arithmetic practice joseph zoller september, 2015 practice problem solutions 1. Learn through interactive problem solving proven to be more effective than lectures. Introductory explorations and the research cycle html pdf definitions html pdf problem posing. If nis an integer, two integers are equal mod nif they di. This course is intended for students not planning to take upper level mathematics. If we use social security number as the key, we will have to deal with an array of size 1010. Modular arithmetic is a branch in mathematics which studies remainders and tackles related problems. In fact, many students apply modular arithmetic every.
Explorations of students mathematical beliefs and behavior article pdf available in journal for research in mathematics education 204. One of the most useful ways of using modular arithmetic is by checking the answers of long additions, subtractions, multiplications, divisions, etc. In fact, the term draws images of a math nerd scribbling foreign symbols on a blackboard, but in reality it isnt that difficult. This means any integer is equivalent to the remainder of that number divided by 5. Modular arithmetic sounds like a mathematical subject that will make your head spin. Student explorations in mathematics is a resource available exclusively to nctm members. A study of gamification techniques in mathematics education.
Divisibility and modular arithmetic are foundational to number theory, which is the theory of the integers. Modular arithmetic is important in number theory, where it is a fundamental tool in the solution of diophantine equations particularly those restricted to integer solutions. Fermat and newton, students begin with a ubiquitous mathematical theorem grade 8 and engage in. Finally, we show that, under given addition and multiplication operations, zmz is a ring for any positive integer m. Modular arithmetic readings for this first reading, you will need to access several external webpages provided by khan academy. Topics include arithmetic functions, multiplicativity, moebius inversion, modular arithmetic, dirichlet characters, gauss sums, primality testing, distribution of primes. She entered the scores in random order into a spreadsheet.
Number theory modular arithmetic and gcd misha lavrov arml practice 92220. Explore magic squares from an historical and mathematical perspective. What are some wellknown uses of modular arithmetic. Mathematics explorations second edition student workbook. Students engage in explorations and lab activities designed to strengthen and expand their knowledge of the topics found in secondary mathematics. Number theory, in turn, is foundational for cryptography, the science of keeping ones communications and data secure from eavesdropping third parties. This elaborate body of instructional materials, and the pedagogy behind it, constitute an approach to earlyage math education that can be described as. This issue of students explorations in mathematics introduces students to modular arithmetic and its applications.
1316 145 431 446 918 202 382 977 1454 346 1023 169 1122 1416 339 93 487 94 435 340 95 1460 1260 1346 703 60 721 819 1126 836 390 1349 1025