chapter 1: preliminary information section 1-1: sets of numbers

Chapter 1:Preliminary InformationSection 1-1: Sets of Numbers

ObjectivesGiven 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 numbersReal 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 NumbersRational 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 InformationSection 1-2: The Field Axioms

ObjectiveGiven the name of an axiom that

applies to addition or multiplication that shows you understand the meaning of the axiom.

The Field AxiomsClosureCommutative PropertyAssociative PropertyDistributive PropertyIdentity ElementsInverses

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 ClosureClosure 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 ClosureClosure 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 PropertyAddition and Multiplication of real

numbers are commutative operations. That means:◦x + y = y + x◦xy =yx

Are subtraction and division commutative?

Associative PropertyAddition and Multiplication of real

numbers are associative operations. That means:◦(x + y) + z = x + (y + z)◦(xy)z = x(yz)

Distributive PropertyMultiplication 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 ElementsThe real numbers contain unique

identity elements.◦For addition, the identity element is

0.◦For multiplication, the identity

element is 1.

InversesThe 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.