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Numbers, Operations, and Quantitative Reasoning http://online.math.uh.edu/MiddleSchool Basic Definitions And Notation Field Axioms Addition (+): Let a, b, c be real numbers 1. a + b = b + a (commutative) 2. a + (b + c) = (a + b) + c 3. a + 0 = 0 + a = a (associative) (additive identity) 4. There exists a unique number ã such that a + ã = ã +a = 0 (additive inverse) ã is denoted by – a Multiplication (·): Let a, b, c be real numbers: 1. a b = b a (commutative) 2. a (b c) = (a b) c 3. a 1 = 1 a = a (associative) (multiplicative identity) 4. If a 0, then there exists a unique ã such that a ã = ã a = 1 (multiplicative inverse) ã is denoted by a-1 or by 1/a. Distributive Law: Let a, b, c be real numbers. Then a(b+c) = ab +ac The Real Number System Geometric Representation: The Real Line Connection: one-to-one correspondence between real numbers and points on the real line. Important Subsets of 1. N = {1, 2, 3, 4, . . . } – the natural nos. 2. J = {0, 1, 2, 3, . . . } – the integers. 3. Q = {p/q | p, q are integers and q 0} -- the rational numbers. 4. I = the irrational numbers. 5. = Q I Our Primary Focus... S The Natural Numbers: Synonyms 1. The natural numbers 2. The counting numbers 3. The positive integers The Archimedean Principle Another “proof” Suppose there is a largest natural number. That is, suppose there is a natural number K such that nK for n N. What can you say about K + 1 ? 1. Does K + 1 N ? 2. Is K + 1 > K ? Mathematical Induction Suppose S is a subset of N such that 1. 1 S 2. If k S, then k + 1 S. Question: What can you say about S ? Is there a natural number m that does not belong to S? Answer: S = N; there does not exist a natural number m such that m S. Let T be a non-empty subset of N. Then T has a smallest element. Question: Suppose n N. What does it mean to say that d is a divisor of n ? Question: Suppose n N. What does it mean to say that d is a divisor of n ? Answer: There exists a natural number k such that n = kd We get multiple factorizations in terms of primes if we allow 1 to be a prime number. p 2 1, n 0, 1, 2, 2n Fermat primes: 2 1 3, 2 1 5, 2 1 17, . . . 20 21 22 Mersenne primes: 2 1, p a prime p 2 1 3, 2 1 7, 2 1 31, . . 2 3 5 Twin primes: p, p + 2 {3, 5}, {5, 7}, {11,13}, {17,19}, {29, 31}, ... Every even integer n > 2 can be expressed as the sum of two (not necessarily distinct) primes For any natural number n there exist at least n consecutive composite numbers. The prime numbers are “scarce”. Fundamental Theorem of Arithmetic (Prime Factorization Theorem) Each composite number can be written as a product of prime numbers in one and only one way (except for the order of the factors). Some more examples 504 2 3 7 3 2 2,475 3 5 11 2 2 11,250 2 3 5 2 4