Numbers, Operations, and Quantitative Reasoning

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 (associative) 3. a + 0 = 0 + a = a (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 (associative) 3. a  1 = 1  a = a (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...

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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 n  K for n  N. What can you say about K + 1 ? 1. Does K + 1  N ? 2. Is K + 1 > K ?

Mathematical Induction N 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.

Fermat primes: Mersenne primes:

Twin primes: p, p + 2 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