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# Topic: Fundamental theorem of arithmetic

 Arithmetic - Wikipedia, the free encyclopedia   (Site not responding. Last check: 2007-10-07) Arithmetic or arithmetics (from the Greek word αριθμός = number) in common usage is a branch of (or the forerunner of) mathematics which records elementary properties of certain operations on numerals, though in usage by professional mathematicians, it often is treated as synonym for number theory. The traditional arithmetic operations are addition, subtraction, multiplication and division, although more advanced operations (such as manipulations of percentages, square root, exponentiation, and logarithmic functions) are also sometimes included in this subject. The arithmetic of natural numbers, integers, rational numbers (in the form of vulgar fractions), and real numbers (using the decimal place-value system known as algorism) is typically studied by schoolchildren, who learn manual algorithms for arithmetic. www.northmiami.us /project/wikipedia/index.php/Arithmetic   (288 words)

 Learn more about Fundamental theorem of arithmetic in the online encyclopedia.   (Site not responding. Last check: 2007-10-07) In mathematics, and in particular number theory, the fundamental theorem of arithmetic is the statement that every positive integer can be written as a product of prime numbers in a unique way. The theorem establishes the importance of prime numbers. The fundamental theorem ensures that additive and multiplicative arithmetic functions are completely determined by their values on the powers of prime numbers. www.onlineencyclopedia.org /f/fu/fundamental_theorem_of_arithmetic.html   (830 words)

 Fundamental theorem of arithmetic - Wikipedia, the free encyclopedia In mathematics, and in particular number theory, the fundamental theorem of arithmetic or unique factorization theorem is the statement that every positive integer greater than 1 is either a prime number or can be written as a product of prime numbers. To make the theorem work even for the number 1, we can think of 1 as being the product of zero prime numbers (see empty product). Fermat's Last Theorem Blog: Unique Factorization, A blog that covers the history of Fermat's Last Theorem from Diophantus of Alexandria to the proof by Andrew Wiles. en.wikipedia.org /wiki/Fundamental_theorem_of_arithmetic   (929 words)

 Arithmetic Arithmetic is a branch of mathematics which records elementary properties of certain arithmetical operations on numbers. The arithmetic of natural numbers, integers, rational numbers (in the form of fractions) and real numbers (in the form of decimal expansions[?]) is typically studied by schoolchildren of the elementary grades. The term "arithmetic" is also sometimes used to refer to number theory; it's in this context that one runs across the fundamental theorem of arithmetic and arithmetical functions. www.wordlookup.net /ar/arithmetic.html   (286 words)

 Encyclopedia: Fundamental theorem of arithmetic   (Site not responding. Last check: 2007-10-07) In arithmetic and number theory the least common multiple or lowest common multiple (lcm) or smallest common multiple of two integers a and b is the smallest positive integer that is a multiple of both a and b. In number theory, an additive function is an arithmetic function f(n) of the positive integer n such that whenever a and b are coprime we have: f(ab) = f(a) + f(b). In number theory, a multiplicative function is an arithmetic function f(n) of the positive integer n with the property that f(1) = 1 and whenever a and b are coprime, then f(ab) = f(a) f(b). www.nationmaster.com /encyclopedia/Fundamental-theorem-of-arithmetic   (1752 words)

 The Fundamental Theorem of Arithmetic In the case of the present theorem, there are three critical phrases, and you should try to explain why each one is there. It would be possible to state the theorem to include this case, but it would distort the main meaning so much that it's not worthwhile. Arithmetic (and, more generally, algebra) are essentially finite mathematics; analysis is essentially the mathematics of the infinite. odin.mdacc.tmc.edu /~krc/numbers/fta.html   (676 words)

 Encyclopedia: Arithmetic   (Site not responding. Last check: 2007-10-07) In mathematics, especially in elementary arithmetic, division is an arithmetic operation which is the reverse operation of multiplication, and sometimes it can be interpreted as repeated subtraction. In mathematics, and in particular number theory, the fundamental theorem of arithmetic or unique factorization theorem is the statement that every positive integer greater than 1 can be written as a product of prime numbers in only one way. Elementary arithmetic is the most basic kind of mathematics: it concerns the operations of addition, subtraction, multiplication, and division. www.nationmaster.com /encyclopedia/Arithmetic   (1062 words)

 Kids.net.au - Encyclopedia Fundamental theorem of arithmetic -   (Site not responding. Last check: 2007-10-07) The fundamental theorem of arithmetic is the statement that every positive integer can be written as a product of prime numbers in a unique way. To make the theorem work even for the number 1, we think of 1 as being the product of zero prime numbers. Essentially, the theorem establishes the importance of prime numbers: they are the "basic building blocks" of the positive integers in that every positive integer can be put together from primes in a unique fashion. www.kids.net.au /encyclopedia-wiki/fu/Fundamental_theorem_of_arithmetic   (562 words)

 The Division Algorithm and the Fundamental Theorem of Arithmetic   (Site not responding. Last check: 2007-10-07) The Division Algorithm and the Fundamental Theorem of Arithmetic Theorem 8.1: (The Division Algorithm) Let a and b be natural numbers with b not zero. Theorem 8.4: If a and b are relatively prime and a divides bc, then a divides c. www.sonoma.edu /users/w/wilsonst/papers/finite/8   (277 words)

 Fundamental theorem of arithmetic - Wikipedia The fundamental theorem of arithmetic is the statement that every positive integer has a unique prime number factorization. However if the prime factorizations are not known, the use of Euclid's algorithm is likely to require less calculation than factorizing the two numbers. The fundamental theorem ensures that multiplicative functions are completely determined by their values on the powers of prime numbers. nostalgia.wikipedia.org /wiki/Fundamental_theorem_of_arithmetic   (211 words)

 Arithmetic article - Arithmetic mathematics numerals addition subtraction multiplication division - What-Means.com   (Site not responding. Last check: 2007-10-07) Arithmetic or arithmetics is a branch of (or the forerunner of) mathematics which records elementary properties of certain operations on numerals. The arithmetic of natural numbers, integers, rational numbers (in the form of fractions), and real numbers (using the decimal place-value system known as algorism) is typically studied by schoolchildren, who learn manual algorithms for arithmetic. Arithmetic article - Arithmetic definition - what means Arithmetic www.what-means.com /encyclopedia/Arithmetic   (209 words)

 Gregory West It is easy to factor a small number like six, but one of the fundamental truths of math, named the fundamental theorem of arithmetic, is that all integers can be expressed as the product of their prime factors. The fundamental theorem of arithmetic implies that prime numbers are important cornerstones in number theory because they are the building blocks from which all other numbers are constructed. Because the fundamental theorem of arithmetic establishes prime numbers as the building blocks of composite numbers, many mathematicians of note have attempted to describe a quick and decisive algorithm that will determine whether a given number is prime or composite. www.cco.caltech.edu /~sciwrite/journal03/west.html   (3312 words)

 PlanetMath: proof of the fundamental theorem of arithmetic   (Site not responding. Last check: 2007-10-07) PlanetMath: proof of the fundamental theorem of arithmetic "proof of the fundamental theorem of arithmetic" is owned by KimJ. This is version 7 of proof of the fundamental theorem of arithmetic, born on 2001-10-18, modified 2005-08-14. planetmath.org /encyclopedia/ProofOfFundamentalTheoremOfArithmetic.html   (122 words)

 HMCo College Algebra Exercises, Chapter 1 The Fundamental Theorem of Arithmetic states that each composite number can be expressed as a product of prime numbers in exactly one way (disregarding the order or the factors). Now we have a contradiction, because by the fundamental theorem of arithmetic, it is impossible for a number to have an odd number of prime factors and an even number of prime factors. The arithmetic mean is often used to find the average of two test scores. college.hmco.com /mathematics/aufmann/collalg/chapter1.html   (625 words)

 Fundamental - Wikipedia, the free encyclopedia If an article link referred you here, you might want to go back and fix it to point directly to the intended page. A fundamental is something that cannot be built out of more basic things, which other things are built upon. Fundamental frequency, a concept in music or phonetics en.wikipedia.org /wiki/Fundamental   (100 words)

 Introduction to Arithmetic: Number Theory; Prime Numbers, Fermat Theorem, Goldbach Conjecture and Diophantine Equations   (Site not responding. Last check: 2007-10-07) The fundamental theorem of arithmetic states that any composite number can be written as the product of prime numbers, called its prime factors, in one and only one way (provided that the order of the factors is not taken into account). Fermat considered Pythagoras' theorem, which states that, for every right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. His proof of Fermat's last theorem is likely to be remembered as one of the greatest ever mathematical achievements. www.geocities.com /mathfair2002/school/arit/arithm3.htm   (2232 words)

 Math 314 The next theorem is sort of a converse of Theorem 18. Theorem 24 (Fundamental Theorem of Arithmetic -- aka Prime Factorization Theorem). Theorem 29 says we only have to check up to the square root. mathserv.monmouth.edu /coursenotes/kuntz/math314/m31406.htm   (583 words)

 Factorization For the natural numbers, this result is often called the Fundamental Theorem of Arithmetic. Theorem: In the natural numbers, each non-zero, non-unit number can be factorized into a product of primes; furthermore, the factorization is unique up to ordering. Theorem: In the integers, each non-zero, non-unit number can be factorized into a product of primes. www.vex.net /~trebla/numbertheory/factor.html   (643 words)

 Prime numbers   (Site not responding. Last check: 2007-10-07) The Fundamental Theorem of Arithmetic says that every positive integer greater than one can be expressed uniquely as a product of primes, apart from the rearrangement of terms. By the fundamental theorem of arithmetic we know that all positive integers factor uniquely into a product of primes. The theorem excludes 1 as a prime number, otherwise the theorem would be false. www.math.utoledo.edu /~dbastos/primes.html   (471 words)

 The Prime Glossary: Fundamental Theorem of Arithmetic We can reword the Fundamental Theorem this way: the canonical factorization of an integer greater than one is unique. This theorem (and indeed any theorem labeled "fundamental") should not be taken too lightly. Basically two properties: first, that every integer can be written as a product of primes (this is a simple consequence of the well ordering principle); and second, if a prime p divides ab, then p divides a or b (this is sometimes used as the definition of prime, see the entry prime number). primes.utm.edu /glossary/page.php?sort=FundamentalTheorem   (318 words)

 3.4 - Fundamental Theorem of Algebra   (Site not responding. Last check: 2007-10-07) If you check out fundamental in the dictionary, you will see that it relates to the foundation or the base or is elementary. Fundamental theorems are important foundations for the rest of the material to follow. Here are some of the fundamental theorems or principles that occur in your text. www.richland.edu /james/lecture/m116/polynomials/theorem.html   (457 words)

 Fundamental Theorem of Arithmetics   (Site not responding. Last check: 2007-10-07) DIMACS Workshop on Feasible Arithmetics and Length of Proofs... The funamental lemma of the Arithmetics - the Division lemma.... WHAT ARE WEAK ARITHMETICS 0 1 2 3 4 5 6 7 8 9... www.scienceoxygen.com /math/302.html   (75 words)

 The Jordan-Hölder Theorem   (Site not responding. Last check: 2007-10-07) Applying Schreier's refinement theorem (Theorem 11.2.2), we get that the two composition series have equivalent refinements. In particular, we use the Jordan-Hölder Theorem to prove the uniqueness part of the Fundamental Theorem of Arithmetic. The Fundamental Theorem of Arithmetic states that every positive integer not equal to a prime can be factored uniquely (up to order) into a product of primes. web.usna.navy.mil /~wdj/tonybook/gpthry/node65.html   (453 words)

 Untitled Document,,, This is called the "fundamental theorem of arithmetic." For example, from the last class, 28=2x2x7. The Fundamental Theorem of Arithmetic may seem obvious to you, based on your experience with integers and how they multiply, but it is actually not easy to prove. One often hears that this material is "Pythagorean," generalizing, as it does, the arithmetic of the integers, by means of geometry. www.mtholyoke.edu /courses/mpeterso/math114/class2.html   (1837 words)

 Fundamental theorem of arithmetic - Internet-Encyclopedia.com   (Site not responding. Last check: 2007-10-07) Find fundamental theorem of arithmetic and more at Lycos Search. Read about fundamental theorem of arithmetic in the free online encyclopedia and dictionary. Find fundamental theorem of arithmetic at one of the best sites the Internet has to offer! www.internet-encyclopedia.com /ie/f/fu/fundamental_theorem_of_arithmetic.html   (909 words)

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