A course in discrete structures cornell university. For instance, there are only 12 hours on the face of a clock. The idea of modular arithmetic is essentially very simple and is identical to. Discrete mathematics uses a range of techniques, some of which is seldom found in its continuous counterpart. For the student, my purpose was to present material in a precise, readable manner, with the concepts and techniques of discrete mathematics clearly presented and demonstrated. Modular arithmetic using the extended euclidean algorithm formalizing the backward steps we get this formula. In particular this means that the numbers involved are either integers, or numbers closely related to them, such as fractions or. Information theory also includes continuous topics such as. This books gives an introduction to discrete mathematics for beginning undergraduates. 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. Browse other questions tagged modular arithmetic cryptography or ask your own question. Mathematics for computer science by eric lehman, f. Pseudorandom number generators hash functions cryptology cs 441 discrete mathematics for cs m. Number theory is the branch of mathematics that explores the integers.
Modular arithmetic is often tied to prime numbers, for instance, in wilsons theorem, lucass theorem, and hensels lemma, and generally appears in fields. Exploring more applications of modular arithmetic, examine the chinese remainder theorem, used in ancient china as a fast way to count large numbers of troops. This book consists of six units of study boolean functions and computer arithmetic. Modular arithmetic discrete mathematics number theory. The reason is that computers only have a finite amount of memory. Discrete mathematics and its applications seventh edition kenneth rosen. If the time now is 7 oclock, 20 hours later will be 3 oclock. Discrete mathematics notes pdf in these discrete mathematics notes pdf, we will study the concepts of ordered sets, lattices, sublattices, and homomorphisms between lattices. Pdf discrete mathematicsnumber theory ijesrt journal. Cse 1400 applied discrete mathematics modular numbers. September 29, 2020 fall semester hanufaculty of information technology discrete mathematics divisibility for a, b. Contents introduction integers linear diophantine equations modular arithmetic congruences integers in. Notes for part ia cst 201516 university of cambridge.
Walter gave an exam in a mathematics class of five students. If a divides b we say that a is a factor of b and that b is multiple of a. In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers wrap. One way to think of modular arithmetic is that it limits numbers to a. The formula to find the value of the above number is.
One way to think of modular arithmetic is that it limits numbers to a predefined range 0,1. Discrete structures lecture notes vladlen koltun1 winter 2008 1computer science department, 353 serra mall, gates 374, stanford university, stanford, ca 94305, usa. If and are integers ith m 0 a then diides if there e ists an integer. Modular arithmetic is useful in these settings, since it. The division algorithm modular arithmetic applications of. Free discrete mathematics books download ebooks online. Modular arithmetic connects with primes in an interesting way. Scribd is the worlds largest social reading and publishing site. Using these operations is said to be doing arithmetic modulo m. This text, a short course in discrete mathematics, was developed for the. Biggs discrete mathematics is the branch of mathematics in which we deal with questions involving. Two integers are congruent mod m if and only if they have the same remainder when divided.
Assume a bus arrives at a bus stop every 10 minutes starting at 11. Its elements are congruence classes modulo p, and the group product of two elements may be obtained by ordinary integer multiplication of the elements followed by reduction modulo p. Cs 70 discrete mathematics and probability theory fall 2012 vazirani note 5 modular arithmetic one way to think of modular arithmetic is that it limits numbers to a prede. Discrete mathematics pdf, remember to click the web link beneath and download the ebook or have accessibility to other information. Math 728 lesson plan modulo arithmetic introduction.
Pdf a short course in discrete mathematics semantic scholar. In a perhaps unsympathetic view, the standard presentations and there are many the material in the course is treated as a discrete collection of so many techniques that the students must master for further studies in computer science. One of the simplest settings for discrete logarithms is the group z p this is the group of multiplication modulo the prime p. It takes albert 7 minutes to get to the bus stop while it takes betty minutes. Module ma2c03 discrete mathematics lecture notes for hilary term 2016.
Cs 70 discrete mathematics and probability theory fall 2016 seshia and walrand note 6 1 modular arithmetic suppose you go to bed at 23. Eecs 70 discrete mathematics and probability theory fall 2014 anant sahai note 5 modular arithmetic in several settings, such as errorcorrecting codes and cryptography, we sometimes wish to work over a smaller range of numbers. Mathematics stack exchange is a question and answer site for people studying math at any level and professionals in related fields. Handwritten discrete mathematics notes pdf lecture download. Then if you multiply these together you may be surprised to find the result is negative. The proof of this is just to multiply both sides of the equation. Cs 70 discrete mathematics and probability theory fall 2012.
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 prede. Discrete mathematics divisibility and modular arithmetic definition. Discrete mathematics and its applications, kenneth h. Available on a separate webpage are notes for individual lectures in hilary term 2016 in beamer format. Pdf a short course in discrete mathematics semantic. Review, logic, ppt tautologies offers, logical equivalence ppt predicates and quantifiers. Modular arithmetic is a system of arithmetic for integers, which considers the remainder. Z with a 6 0 we say that a divides b, if there is a q. Indeed, contemporary work in descriptive set theory makes extensive use of traditional continuous mathematics.
Euclids algorithm or the euclidean algorithm is a very efficient and ancient algorithm to find the greatest common divisor gcda, b of two. We claim that the solution is all integers xsuch that xmod b a bc 1, where c 1 is calculated modulo b. Many examples of formal and informal proofs are given. Discrete mathematics is a required course in the undergraduate computer science curriculum.
Also learn about password protection, the mathematics behind the perfect shuffle, and the seed planting technique for raising big numbers to big powers. Discrete mathematics modular arithmetic for any two integers a. In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers wrap around when reaching a certain value, called the modulus. She entered the scores in random order into a spreadsheet, which recalculated the.
Boolean functions and computer arithmetic, logic, number theory and cryptography, sets and functions, equivalence and order, induction, sequences and series. This course does not have a web ct or vista account. Aug 12, 2020 modular arithmetic uses only a fixed number of possible results in all its computation. Primes are an endless source of fascination for mathematicians. Assume 2 integers a and b, such that a 0 a is not equal 0. Related to the magic box method modular arithmetic step 0. In number theory, the more commonly used term is index. Discrete mathematics and its applications seventh edition. Rosen, eighth edition, mcgraw hill, isbn 9780072880083, isbn10 0072880082. Number theory athe math behind the rsa crypto system. Basic facts about numbers 1 rational numbers, irrational numbers, prime, composite, odd, even, n divides m, prime factorization, infinitely many primes, perfect squares, irrationality of integral square roots, residue classes mod d, mod as binary operator, mod as equivalence relation, modular arithmetic, modular addition, modular multiplication. Michelgoemans these notes cover basic notions in algebra which will be needed for discussing several topics.
Principles of discrete applied mathematics, modular arithmetic and. Discrete structures lecture notes stanford university. Cs 70 discrete mathematics and probability theory fall. This example explains why modular arithmetic is referred to by some as clock arithmetic. Rosen discrete mathematics and its applications, tata mcgrawhill. In mathematical terms the question is whether the two fields are. Modular arithmetic and elementary algebra 1 euclids algorithm. Proofs, not computations, form the core of real mathematics. We introduce modular arithmetic, the function that outputs remainders and separates them into equivalence classes. Thus notions and techniques from this base will generally not be justi. Hauskrecht modular arithmetic in cs modular arithmetic and congruencies are used in cs. All information will be made available only on this web page.
Discrete mathematics in computer science october 28, 2020 b11. Browse other questions tagged discrete mathematics modular arithmetic or ask your own question. My goal was to show the relevance and practicality of discrete mathematics to students, who are often skeptical. Michelgoemans these notes cover basic notions in algebra which will. Logic and discrete mathematics for computer scientists. The process is similar when converting from any base to base10.
We say that a divides b if there is an integer c such that b ac. Modular arithmetic, groups, finite fields and probability chapter. It also includes an introduction to modular and distributive lattices along with complemented lattices and boolean algebra. Analogously, in any group g, powers b k can be defined for all integers k, and the discrete logarithm log b a is an integer k such that b k a. For example, if we want the product of two numbers modulo n, then we multiply them normally and the answer is the remainder when the normal product is divided by n. Discrete mathematics divisibility and modular arithmetic. Everything you need to know about modular arithmetic. In mathematics, for given real numbers a and b, the logarithm log b a is a number x such that b x a.
Hauskrecht pseudorandom number generators some problems we want to program need to simulate a random choice. Preclass activities suppose you start programming in java, and declare two variables of type int. So we start from 1 and work backward modular arithmetic. Some operations, like finding a discrete logarithm or a quadratic congruence appear to be as hard as. Number theory has many applications within computer. Number theory is the branch of mathematics that explores the integers and their properties.
This course will roughly cover the following topics and speci c applications in computer science. Inverses in modular arithmetic we have the following rules for modular arithmetic. Modular arithmetic is useful in these settings, since it limits numbers to a prede. In modular arithmetic, the numbers we are dealing with are just integers and the operations used are addition, subtraction, multiplication and division. I am having trouble understanding modular arithmetic and i have to solve these two problems by hand. One of original features of this book is that it begins with a presentation of the rules of logic as used in mathematics. Modular arithmetic applications of modular arithmetic. My goal was to show the relevance and practicality of discrete mathematics to. Modular arithmetic ii discrete math free download as powerpoint presentation. Cs 70 discrete mathematics and probability theory modular. Some sample examination questions annual examination 2016. In modular arithmetic, numbers wrap around upon reaching a given fixed quantity this given quantity is known as the modulus to leave a remainder. High school mathematics extensions wikimedia commons.
1178 669 1268 81 545 1611 1356 1668 87 774 178 754 1031 47 1115 1294 267 531 1242 1674 1443 642 976 1210 1625 758 1429 1544