chapter 1: preliminary information
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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 - PowerPoint PPT PresentationTRANSCRIPT
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.