This page includes Lesson 1.1, 1.2 and 1.3 information.  Assignment and video links for each lesson are included at the end of the notes.

Lesson 1.1

In Lesson 1.1 we will be studying/reviewing the real number system.  We will cover sets of number, methods for representing sets of numbers, inequalities, absolute value, union, intersection and graphing.  Students need to read and review all of Lesson 1.1.  To assist you in understanding some of the topics I have included a brief summary of key items in the lesson as well as a video of worked examples.  The link to the video is at the conclusion of the notes for this lesson.  Upon completion of the assignment students should be able to successfully complete a 10 question quiz that will be administered though the quizplace.com website.  Details concerning this are also at the end of the notes.

A set is a collection of objects.  The different objects are called elements of the set.  Sets are denoted by placing braces around the elements in the set.

For example the set of Natural Numbers would look like this {1,2,3,4,5,6,...}.  With the numbers 1, 2, 3 being elements of the set of Natural Numbers.  The repeating dots at the end indicate that the pattern continues into infinite.

Another example of sets would be:

Whole numbers = {0,1,2,3,4,...}  Notice this is the same values as the Natural Numbers with 0 added.

Integers = {...,-4,-3,-2,-1,0,1,2,3,4...}  Notice this set is just the whole numbers and their opposites.

Rational numbers are numbers that can be written as fractions.  So all natural numbers are whole numbers as well as integers as well as rational numbers because they can be written as fractions.  For instance 5 can be written as 5/1.  Other examples of rational numbers would be 1/2, 3/8 and 5/2. 

Irrational numbers are decimal numbers that do not have a repeating pattern and do not terminate.  For instance square root of 7 is an irrational number.  It is a decimal value that does not terminate and has no repeating pattern.  Probably the best known irrational number is pi (π).

A variable is a letter used to represent a value.  For example, x + 7 = 13.  In this example x is representing the value of 6. 

Inequality symbols are as follows:

When you see the symbol < it indicates less than.  For instance 2 < 6.  Two is less than 6.

When you see the symbol > it indicates greater than.  For instance 12 > -2.  Twelve is greater than -2.

When you see the symbol ≤ it indicates less than or equal to and the symbol ≥ indicates greater than or equal to.

Absolute Value is defined as the distance from zero on a number line.  The absolute value of a positive number is the number itself.  The absolute value of a negative number is the opposite of the negative number of positive. and the absolute value of zero is zero.

The textbook demonstrates various methods for representing a set of numbers.  Those methods include the roster method, set-builder notation and interval notation.

The roster method of writing a set encloses the list of the elements of the set in braces.  This was done previously to represent the set of whole numbers at the beginning. Whole numbers = {0,1,2,3,4,...} This is an infinite set since the pattern of numbers continues without end.  An example of a finite set would be {7,8,9,10}.  All elements of this set can be listed.

Different operations are performed on sets of numbers and the resulting answer can be written in interval notation.  We are often asked to calculate the Union or Intersection of two sets of numbers.

The Union of Two Sets is written as A U B.  It is read as A union B.  This symbol indicates the set of all elements that belong to EITHER set A OR set B.  OR is a key word in this instance.  The value can be in either set A or set B or even in both sets. 

The Intersection of Two Sets is written A ∩ B.  It is read as A intersection B.  This symbol indicates the set of all elements that are common to BOTH sets.  Both is a key word in this instance.  The value MUST be in BOTH sets in order for it to be part of the intersection.

Interval Notation is another method for expressing or representing a set of numbers.  For instance, the interval notation (-8,7] indicates the interval of numbers great than -8, and less than or equal to 7.  Notice the ( indicates we start at -8, but do not include it in the set, much like an open circle on a graph used to indicate we started at a number and did not include it in our answer.  The ] indicates that we stop at 7 and we include 7.  Again, this is similar to a closed circle on a number line graph being used to indicate that a number is included as part of the answer.  You will notice as you review the graphing examples that the textbook does not use open and close circles on their graphs.  Instead they have been replaced with parenthesis and brackets.

Lesson 1.2

In Lesson 1.2 we will be reviewing the rules for operations on rational numbers.  We will add, subtract, multiply an divide rational numbers as well as apply the proper order of operations.

Rules for Addition of Real Numbers-In simple terms, when you add to positive, you get a positive answer.  For instance 2 + 3 = 5.  When you add to negative numbers you get a negative answer.  For instance -2 + -3 = -5.  When you add a negative and positive number you can get either a negative or positive result.  You ignore the signs on the numbers and subtract and then you assign a positive or negative sign to your answer depending upon which number was bigger.  For instance 3 + -2 = 1.  When you subtract 3 and 2 you get 1, the answer is positive because 3 is bigger than 2 and it was positive.  Another example would be -3 + 2 = -1.  Again, you subtract 3 and two and get an answer of 1, but this time the answer is negative because the 3 is larger and it was negative.

Rules for Subtraction of Real Numbers-The rule for subtraction is to change it into an addition problem and apply the addition rules.  When you change a problem from subtraction to addition you must change the sign on the number that follows the subtraction sign that was changed.  For instance, 6-2 could be written as 6+(-2).  Another example would be 4-(-2).  This could be rewritten as 4+(+2).  Also, -6-2 could be written as -6 +(-2). 

Rules for Multiplication and Division of Real Numbers-The multiplication and division sign rules are the same.  If you multiply or divide two positive numbers you get a positive answer.  If you multiply or divide one positive and one negative number you get a negative answer.  For instance -3(2) = -6 and 3(-2) also equals -6.  The larger number does not matter.  It is simply 1 negative and 1 positive results in a negative answer.  Two negative multiplied or divided results in a positive answer.  For instance, -5(-4) = 20. 

Special Division rules-You can not divide by zero.  The answer is undefined.  So 6 ÷ 0 is undefined.  Any number other than zero divided by itself is one.  For example 7 ÷ 7 = 1.  Any number divided by 1 is the number itself.  For instance 8 ÷ 1 = 8.

Exponential Expressions-Repeated multiplication of the same factor can be written using an exponent.  For instance, 6 · 6 · 6  = 6³  The number 6 is the base number and the 3 is the exponent.

Order of Operations Agreement-In the mathematical world we must follow the rules and the Order of Operations Agreement is a very important rule to know.  There are four steps and they are as follows:

1.  Perform operations inside the grouping symbols.  Grouping symbols include parentheses, brackets, braces, the absolute value symbol and the fraction bar.

2.  Simplify exponential expressions.

3.  Do multiplication and division as they occur from left to right.  (You can divide before you multiply.)

4.  Do addition and subtraction as they occur from left to right.  (You can subtract before you add.)

Videos for Lesson 1.1 and 1.2

This link will provide you access to a video clip containing some worked examples from your textbook.  This is viewable with Windows Media Player 7 or higher.  You can download Windows Media Player via the web at www.microsoft.com/windows/windowsmedia/download/ .  If you have problems with viewing the lesson please let me know.   Notes and information for 1.2 and 1.3 will be added at a later date.   Lesson 1.1 Lesson 1.2

Assignment

For Lesson 1.1 you need to work the following problems beginning on page 13:  1, 5, 9, 15, 19, 21, 23, 27, 31, 33, 35, 37, 43, 47, 51, 53, 55, 57, 59, 61, 63, 67, 69, 73, 77, 81, 85, 89, 93, and 95. 

For Lesson 1.2 you need to work the following problems beginning on page 25:  3, 7, 11, 15, 19, 23, 27, 31, 35, 43, 47, 51, 55, 59, 63, 67, 71, 75, 78, 81, 53, 89, 90, 95, 97, 103, 106, and 109.

Your textbook comes complete with a solutions manual for all odd problems.  Please utilize your textbook in an appropriate manner to assist you with your assignment.

Weekly Quiz Information

Upon completion of your assignment you will need to complete a 10 question quiz.  You will access your quiz at www.quizplace.com.  You will have to enter Web ID 6686 and then your designated user name and password.  (If you want to access an example please use test1 as the user name and password.)  You will have 20 minutes to complete your quiz.  This quiz must be completed by Sunday night at midnight for you to receive credit.

If you have any questions feel free to contact me at 785-738-9022 or via e-mail at jwidrig@ncktc.edu.