numbers, operations, and quantitative reasoning
TRANSCRIPT
Numbers, Operations, andQuantitative 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 (associative)
3. a + 0 = 0 + a = a (additive identity)
4. There exists a unique number ã such thata + 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 SystemGeometric Representation: The Real Line
-2 -1 0 1 2 3 4 5x
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
n K 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 NN such that
1. 1 S2. 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: ,2,1,0,122 npn
Mersenne primes:
prime a,12 pp
...,1712,512,312210 222
..,3112,712,312 532
Twin primes: p, p + 2
...},31,29{},19,17{},13,11{},7,5{},5,3{
Every even integer n > 2 can be expressed as the sum of two (notnecessarily distinct) primes
For any natural number n there existat 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).
42
22
23
532250,11
1153475,2
732504
Some more examples