Numbers, Operations, and Quantitative Reasoning

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Numbers, Operations, andQuantitative Reasoning
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http//online.math.uh.edu/MiddleSchool
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Basic Definitions And Notation
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Field Axioms

• Addition () Let a, b, c be real numbers
• a b b a (commutative)
• a (b c) (a b) c (associative)
• a 0 0 a a (additive identity)
• There exists a unique number ã such that
• a ã ã a 0 (additive inverse)
• ã is denoted by a

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• 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)
• If a ? 0, then there exists a unique ã such
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• a? ã ã? a 1 (multiplicative inverse)
• ã is denoted by a-1 or by 1/a.

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Distributive Law Let a, b, c be real numbers.
Then a(bc) ab ac

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The Real Number SystemGeometric Representation
The Real Line
Connection one-to-one correspondence between
real numbers and points on the real line.
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Important Subsets of ?

• 1. N 1, 2, 3, 4, . . . the natural nos.
• J 0, ?1, ?2, ?3, . . . the integers.
• Q p/q p, q are integers and q? 0
• -- the rational numbers.
• 4. I the irrational numbers.
• 5. ? Q ? I

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Our Primary Focus...
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S
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The Natural Numbers Synonyms

• The natural numbers
• The counting numbers
• The positive integers

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The Archimedean Principle
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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 gt K ?
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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?
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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.
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Question Suppose n ? N. What does it mean to
say that d is a divisor of n ?
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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
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We get multiple factorizations in terms of primes
if we allow 1 to be a prime number.
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Fermat primes
Mersenne primes
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Twin primes p, p 2
Every even integer n gt 2 can be expressed as
the sum of two (not necessarily distinct) primes
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For any natural number n there exist at least
n consecutive composite numbers. The prime
numbers are scarce.
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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).
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Some more examples
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