Chapter 1: Preliminary Information
description
Transcript of Chapter 1: Preliminary Information
![Page 1: Chapter 1: Preliminary Information](https://reader035.fdocuments.us/reader035/viewer/2022062305/56815fe8550346895dceebbd/html5/thumbnails/1.jpg)
Chapter 1:Preliminary InformationSection 1-1: Sets of Numbers
![Page 2: Chapter 1: Preliminary Information](https://reader035.fdocuments.us/reader035/viewer/2022062305/56815fe8550346895dceebbd/html5/thumbnails/2.jpg)
ObjectivesGiven the name of a set of
numbers, provide an example.Given an example, name the sets
to which the number belongs.
![Page 3: Chapter 1: Preliminary Information](https://reader035.fdocuments.us/reader035/viewer/2022062305/56815fe8550346895dceebbd/html5/thumbnails/3.jpg)
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
![Page 4: Chapter 1: Preliminary Information](https://reader035.fdocuments.us/reader035/viewer/2022062305/56815fe8550346895dceebbd/html5/thumbnails/4.jpg)
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
![Page 5: Chapter 1: Preliminary Information](https://reader035.fdocuments.us/reader035/viewer/2022062305/56815fe8550346895dceebbd/html5/thumbnails/5.jpg)
![Page 6: Chapter 1: Preliminary Information](https://reader035.fdocuments.us/reader035/viewer/2022062305/56815fe8550346895dceebbd/html5/thumbnails/6.jpg)
Chapter 1:Preliminary InformationSection 1-2: The Field Axioms
![Page 7: Chapter 1: Preliminary Information](https://reader035.fdocuments.us/reader035/viewer/2022062305/56815fe8550346895dceebbd/html5/thumbnails/7.jpg)
ObjectiveGiven the name of an axiom that
applies to addition or multiplication that shows you understand the meaning of the axiom.
![Page 8: Chapter 1: Preliminary Information](https://reader035.fdocuments.us/reader035/viewer/2022062305/56815fe8550346895dceebbd/html5/thumbnails/8.jpg)
The Field AxiomsClosureCommutative PropertyAssociative PropertyDistributive PropertyIdentity ElementsInverses
![Page 9: Chapter 1: Preliminary Information](https://reader035.fdocuments.us/reader035/viewer/2022062305/56815fe8550346895dceebbd/html5/thumbnails/9.jpg)
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
![Page 10: Chapter 1: Preliminary Information](https://reader035.fdocuments.us/reader035/viewer/2022062305/56815fe8550346895dceebbd/html5/thumbnails/10.jpg)
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.
![Page 11: Chapter 1: Preliminary Information](https://reader035.fdocuments.us/reader035/viewer/2022062305/56815fe8550346895dceebbd/html5/thumbnails/11.jpg)
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.
![Page 12: Chapter 1: Preliminary Information](https://reader035.fdocuments.us/reader035/viewer/2022062305/56815fe8550346895dceebbd/html5/thumbnails/12.jpg)
The Commutative PropertyAddition and Multiplication of real
numbers are commutative operations. That means:◦x + y = y + x◦xy =yx
Are subtraction and division commutative?
![Page 13: Chapter 1: Preliminary Information](https://reader035.fdocuments.us/reader035/viewer/2022062305/56815fe8550346895dceebbd/html5/thumbnails/13.jpg)
Associative PropertyAddition and Multiplication of real
numbers are associative operations. That means:◦(x + y) + z = x + (y + z)◦(xy)z = x(yz)
![Page 14: Chapter 1: Preliminary Information](https://reader035.fdocuments.us/reader035/viewer/2022062305/56815fe8550346895dceebbd/html5/thumbnails/14.jpg)
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!
![Page 15: Chapter 1: Preliminary Information](https://reader035.fdocuments.us/reader035/viewer/2022062305/56815fe8550346895dceebbd/html5/thumbnails/15.jpg)
Identity ElementsThe real numbers contain unique
identity elements.◦For addition, the identity element is
0.◦For multiplication, the identity
element is 1.
![Page 16: Chapter 1: Preliminary Information](https://reader035.fdocuments.us/reader035/viewer/2022062305/56815fe8550346895dceebbd/html5/thumbnails/16.jpg)
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.