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Chapter 1:
Preliminary Information
Section 1-1: Sets of Numbers
Objectives
Given the name of a set of numbers,
provide an example.
 Given an example, name the sets to which
the number belongs.

Two main sets of numbers

Real Numbers
◦ Used for “real things” such as:
 Measuring
 Counting
◦ Real numbers are those that can be plotted
on a number line

Imaginary Numbers- square roots of
negative numbers
The Real Numbers

Rational Numbers-can be expressed exactly as a
ratio of two integers. This includes fractions,
terminating and repeating decimals.
◦ Integers- whole numbers and their opposites
◦ Natural Numbers- positive integers/counting
numbers
◦ Digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9

Irrational Numbers-Irrational numbers are those
that cannot be expressed exactly as a ratio of
two numbers
◦ Square roots, cube roots, etc. of integers
◦ Transcendental numbers-numbers that cannot be
expressed as roots of integers
Chapter 1:
Preliminary Information
Section 1-2: The Field Axioms
Objective

Given the name of an axiom that applies
to addition or multiplication that shows
you understand the meaning of the
axiom.
The Field Axioms
Closure
 Commutative Property
 Associative Property
 Distributive Property
 Identity Elements
 Inverses

Closure
{Real Numbers} is closed under addition
and under multiplication.
 That is, if x and y are real numbers then:

◦ x + y is a unique real number
◦ xy is a unique real number
More on Closure


Closure under addition means that when
two numbers are chosen from a set, the sum
of those two numbers is also part of that
same set of numbers.
For example, consider the digits.
◦ The digits include 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.
◦ If the digits are closed under addition, it means
you can pick any two digits and their sum is also a
digit.
◦ Consider 8 + 9
 The sum is 17
 Since 17 is not part of the digits, the digits are not
closed under addition.
More on Closure
Closure under multiplication means that
when two numbers are chosen from a set,
the product of those two numbers is also
part of that same set of numbers.
 For example, consider the negative
numbers.

◦ If we choose -6 and -4 we multiply them and
get 24.
◦ Since 24 is not a negative number, the negative
numbers are not closed under multiplication.
The Commutative Property

Addition and Multiplication of real
numbers are commutative operations.
That means:
◦x+y=y+x
◦ xy =yx

Are subtraction and division
commutative?
Associative Property

Addition and Multiplication of real
numbers are associative operations. That
means:
◦ (x + y) + z = x + (y + z)
◦ (xy)z = x(yz)
Distributive Property
Multiplication distributes over addition.
That is, if x, y and z are real numbers,
then:
x (y + z) = xy + xz
 Multiplication does not distribute over
multiplication!

Identity Elements

The real numbers contain unique identity
elements.
◦ For addition, the identity element is 0.
◦ For multiplication, the identity element is 1.
Inverses

The real numbers contain unique inverses
◦ The additive inverse of any number x is the
number – x.
◦ The multiplicative inverse of any number x is
1/x, provided that x is not 0.