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# Topic: Modular arithmetic

###### In the News (Wed 13 Feb 19)

 Modular Arithmetic Clock (or modular) arithmetic is arithmetic you do on a clock instead of a number line. Clock arithmetic has negative numbers, but each negative number has a positive number name. In clock arithmetic, you can add, subtract, and multiply; you can divide by some numbers. www.math.csusb.edu /faculty/susan/number_bracelets/mod_arith.html   (585 words)

 Modular arithmetic Modular arithmetic can be handled mathematically by introducing a congruence relation on the integers that is compatible with the operations of the ring of integers: addition, subtraction, and multiplication. The notion of modular arithmetic is related to that of the remainder in division. In music, arithmetic modulo 12 is used in the consideration of the system of twelve-tone equal temperament, where octave and enharmonic equivalency occurs (that is, pitches in a 1â¶2 or 2â¶1 ratio are equivalent, and C-sharp is considered the same as D-flat). www.photonicsknowledge.com /search/Modular_arithmetic   (1339 words)

 Implementing Fast Modular Arithmetic Although signed multiple-precision integer operations would no doubt be useful in some scenarios, it is irrelevant for modular arithmetic, and has been ignored. Modular Multiplication Radix 16: The chart shows that Montgomery performs better than Classical in the case where we operate on two big numbers. Modular Multiplication Radix 256: The chart shows that Montgomery performs better than Classical in all cases, requiring only about 2/3 the time. www.daimi.au.dk /~cmosses/crypt   (1673 words)

 Modular arithmetic - Biocrawler   (Site not responding. Last check: ) Modular arithmetic is a system of arithmetic for integers, sometimes referred to as clock arithmetic, where numbers "wrap around" after they reach a certain value (the modulus). Modular arithmetic is applied in number theory, abstract algebra, cryptography, and visual and musical art. Modular arithmetic is often used as a tool for primality tests and integer factorization. www.biocrawler.com /encyclopedia/Modular_arithmetic   (400 words)

 Modular arithmetic - Wikipedia, the free encyclopedia The notion of modular arithmetic is related to that of the remainder in division. Modular arithmetic is referenced in number theory, group theory, ring theory, abstract algebra, cryptography, computer science, and the visual and musical arts. In music, modular arithmetic is used in the consideration of the twelve tone equally tempered scale, where octave and enharmonic equivalency occurs (that is, pitches in a 1∶2 or 2∶1 ratio are equivalent, and C-sharp is the same as D-flat). en.wikipedia.org /wiki/Modular_arithmetic   (968 words)

 Modular arithmetic Summary One way to understand modular arithmetic is to consider "24-hour clock arithmetic": the arithmetic of time-keeping in which the day runs from midnight to midnight and is divided into 24 hours, numbered from 0 to 23. The notion of modular arithmetic is related to that of the remainder in division. In music, modular arithmetic is used in the consideration of the twelve tone equally tempered scale, where octave and enharmonic equivalency occurs (that is, pitches in a 1∶2 or 2∶1 ratio are equivalent, and C-sharp is the same as D-flat). www.bookrags.com /Modular_arithmetic   (1696 words)

 Modular arithmetic Modular arithmetic is a modified system of arithmetic for integers, sometimes referred to as 'clock arithmetic', where numbers 'wrap around' after they reach a certain value (the modulus). For example, whilst 8 + 6 equals 14 in conventional arithmetic, in modulo 12 arithmetic the answer is 2, as 2 is the remainder after dividing 14 by the modulus 12. Modular arithmetic, first systematically studied by Carl Friedrich Gauss at the end of the eighteenth century, is applied in number theory, abstract algebra and cryptography. www.ebroadcast.com.au /lookup/encyclopedia/mo/Modulo.html   (756 words)

 Modular arithmetic - LearnThis.Info Enclyclopedia   (Site not responding. Last check: ) Modular arithmetic, first systematically studied by Carl Friedrich Gauss at the end of the eighteenth century, is applied in number theory, abstract algebra, cryptography, and visual and musical art. In music, because of octave and enharmonic equivalency (that is, pitches in a 1/2 or 2/1 ratio are equivalent, and C# is the same as Db), modular arithmetic is used in the consideration of the twelve tone equally tempered scale, especially in twelve tone music. In abstract algebra, it is realized that modular arithmetic is a special case of forming the factor ring of a ring modulo an ideal. encyclopedia.learnthis.info /m/mo/modular_arithmetic.html   (1898 words)

 Modular arithmetic: Encyclopedia topic   (Site not responding. Last check: ) Modular arithmetic is applied in number theory (number theory: traditionally, number theory is that branch of pure mathematics concerned with the properties... Modular arithmetic was introduced into mathematics (mathematics: A science (or group of related sciences) dealing with the logic of quantity and shape and arrangement) in the book Disquisitiones Arithmeticae (Disquisitiones Arithmeticae: more facts about this subject) by Carl Friedrich Gauss (Carl Friedrich Gauss: more facts about this subject) in 1801. Modular arithmetic is often used as a tool for primality test (primality test: a primality test is an algorithm for determining whether an input number is prime.... www.absoluteastronomy.com /reference/modular_arithmetic   (798 words)

 Modular arithmetic - Definition up Erdmond.Com Modular arithmetic, first systematically studied by Carl_Friedrich_Gauss at the end of the eighteenth century, is applied in number_theory, abstract_algebra, cryptography, and visual and musical art. The fundamental arithmetic operations performed by most computers are actually modular arithmetic, where the modulus is 2''b'' (''b'' being the number of bits of the values being operated on). In abstract_algebra, it is realized that modular arithmetic is a special case of forming the factor_ring of a ring modulo an ideal. www.erdmond.com /Modular_arithmetic.html   (1937 words)

 Modular Art Modular or "clock" arithmetic is arithmetic on a circle instead of a number line. In ordinary arithmetic, the additive inverse of 4 is -4. In mod 12 arithmetic, the additive inverse of 4 is 8. britton.disted.camosun.bc.ca /modart/jbmodart.htm   (680 words)

 NationMaster - Encyclopedia: Reciprocity law   (Site not responding. Last check: ) In mathematics, in number theory, the law of quadratic reciprocity, conjectured by Euler and Legendre and first satisfactorily proved by Gauss, connects the solvability of two related quadratic equations in modular arithmetic. As a consequence, it allows us to determine the solvability of any quadratic equation in modular arithmetic. There are cubic, quartic (biquadratic) and other higher reciprocity laws; but since two of the cube roots of 1 (root of unity) are not real, cubic reciprocity is outside the arithmetic of the rational numbers (and the same applies to higher laws). www.nationmaster.com /encyclopedia/Reciprocity-law   (549 words)

 CIS 408/608: Cryptography and Security Modular arithmetic is one of the most useful tools in Cryptography. The convenient fact about prime order fields is that the arithmetic is just modular arithmetic which is easy. Exercise: It turns out that the only numbers that give finite fields by simple modular arithmetic are the prime numbers. web.njit.edu /~sraj/modular-arithmetic.htm   (1959 words)

 Dogpile Web Search: Modular Modular arithmetic (sometimes called modulo arithmetic, or clock arithmetic because of its use in the 24-hour clock system) is a system of arithmetic for... Modular engine, or mod motor as it is often referred to, is Ford Motor Company's modern overhead camshaft (OHC) V8 and V10 engine family. Modular homes are houses that are manufactured in a remote facility and then delivered to their intended site of use. www.dogpile.com /info.dogpl/search/web/Modular   (263 words)

 Modular Arithmetic — An Introduction As shown by the preceding examples, one of the powers of modular arithmetic is the ability to show, often very simply, that certain equations and systems of equations have no integer solutions. Without modular arithmetic, we would have to find all of the solutions and then see if any turned out to be integers. Of course, we don't need the formality of modular arithmetic in order to compute this, but when we do this kind of computation in our heads, this is really what we are doing. www.math.rutgers.edu /~erowland/modulararithmetic.html   (1827 words)

 High speed modular arithmetic apparatus having a mask generator and a priority encoder - Patent 4064421 In a high speed arithmetic apparatus, the tally coded output of a modular mask generator addressed by a binary first operand and the tally coded input of a priority encoder are joined together by an interconnecting apparatus. The interconnecting apparatus is responsive in form to a second operand and to a selected arithmetic operation to provide at the output of the priority encoder the binary resultant of the selected operation executed upon the first and second operands. A first example of the arithmetic use of the present invention involves the derivation of the integer K complement of an integer X. The K complement of the number X is given by (K-X). www.freepatentsonline.com /4064421.html   (2896 words)

 Introduction to Modular Arithmetic   (Site not responding. Last check: ) Modular Arithmetic is a form of arithmetic dealing with the remainders after integers are divided by a fixed "modulus" m. Basically, it is a kind of integer arithmetic that reduces all numbers to ones that belongs to a fixed set [0... When the entries of Pascal's Triangle are expressed in terms of modular arithmetic we notice some really interesting patterns. jwilson.coe.uga.edu /EMAT6680/Parsons/MVP6690/Essay1/modular.html   (423 words)

 Modular Equations The laws of modular arithmetic are among the best weapons that we have in our arsenal. However if it requires dealing with equations involving modular arithmetic, many of us may not feel just as comfortable. Fear not; were not going to daunt you with a system of gruesome modular equations we would keep it small and simple. acm.uva.es /p/v107/10787.html   (240 words)

 Basics of Computational Number Theory The plan of the paper is to first give a quick overview of arithmetic in the modular integers. Modular arithmetic is arithmetic using integers modulo some fixed integer N. Modular Inverses - Given a number n and a modulus m, find the inverse of n, ie the number k such that n * k = 1 (mod m). www.math.umbc.edu /~campbell/NumbThy/Class/BasicNumbThy.html   (2263 words)

 Illuminations: Arithme-Tic-Toc Students will be introduced to modular arithmetic by first examining a five-hour analog clock and its mathematical properties. Because of its familiarity to students, one simple way to introduce modular arithmetic is with a 12-hour analog clock. To get students familiar with modular arithmetic, ask, "What time will it be six hours from 5 p.m.?" Students will quickly give the correct answer, 11 p.m. illuminations.nctm.org /LessonDetail.aspx?ID=L671   (1511 words)

 Re: Counterexample to FLT by modular arithmetic Yet it works when a and c are distinct primes, and may work when a and c are coprime; neither, one, or both composite but not sharing any factor other than one, which is a legitimate value for a or both a and b. Counterexample to FLT by modular arithmetic, Doug Goncz Re: Counterexample to FLT by modular arithmetic, Doug Goncz www.usenet.com /newsgroups/sci.math/msg06594.html   (476 words)

 High School Mathematics Extensions/Primes/Modular Arithmetic - Wikibooks, collection of open-content textbooks Modular arithmetic connects with primes in an interesting way. For example, let's start with modulo 7 arithmetic, it's just like ordinary arithmetic except the only numbers we use are 0, 1, 2, 3, 4, 5 and 6. If you enjoyed the material in this chapter, you would also enjoy Further Modular Arithmetic, which is a harder and more rigorous treatment of the subject. en.wikibooks.org /wiki/High_School_Mathematics_Extensions/Primes/Modular_Arithmetic   (2905 words)

 Modular Arithmetic   (Site not responding. Last check: ) Modular arithmetic is arithmetic you do on a clock instead of a number line. This is not the same a clock arithmetic, but it is related. Modular arithmetic can be used in check digit calculations. pages.sbcglobal.net /george.bunson/Programming/modularArithmetic.htm   (515 words)

 Modular arithmetic In mathematics, modular arithmetic is a system of arithmetic for certain equivalence classes of integers, called congruence classes. Sometimes it is suggestively called 'clock arithmetic', where numbers 'wrap around' after they reach a certain value (the modulus). The fundamental arithmetic operations performed by most computers are actually modular arithmetic, where the modulus is 2 www.knowledgefun.com /book/m/mo/modular_arithmetic.html   (1888 words)

 Math Forum - Ask Dr. Math   (Site not responding. Last check: ) Date: 05/07/2003 at 15:50:46 From: Matt Subject: Modular Arithmetic Find an integer n between 0 and 2309 with the property that 34 10 10 10 = n (mod 2310) I do not know how to break the powers down to solve this problem. Date: 05/08/2003 at 22:28:04 From: Doctor Mitteldorf Subject: Re: Modular Arithmetic Dear Matt, The first thing to notice is that 2310 is 2*3*5*7*11. Date: 05/09/2003 at 08:31:46 From: Matt Subject: Thank you (Modular Arithmetic) Thanks for the help on this one. www.mathforum.org /library/drmath/view/62908.html   (374 words)

 goENC.com : Record for ENC#014234 (Secret codes) In this module, the focus is on modular arithmetic and equivalency. In a sample activity on Modular Arithmetic, students assign numerical values to the letters and develop encoding rules for numeric data. They investigate modular arithmetic in depth, including identity elements for addition and multiplication, additive and multiplicative inverses, and communicative and associative properties. www.goenc.com /records/record_generator.asp?encnum=014234   (343 words)

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