Transcript Photo Album - Online Math

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
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 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