numbers, operations, and quantitative reasoning

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