Module 3 The RealThingThismoduleisabout real numbersspecificallytheconceptsofwholenumbers,natural numbers, andintegers. It alsoincludesdiscussionsontheabsolutevalueof anumber as well as basic operations on absolute values of numbers. The ideas that you willencounter in this module will help you understand many things that happen around you. Itwill also prepare you to do greater tasks in mathematicsThe lessons in this module are as follows:Lesson 1 Natural Numbers and Whole NumbersLesson 2 IntegersLesson 3 Ordering Integers Lesson 4Absolute Value of a NumberAfter using this module, you are expected to: grasp the concepts of a natural number, a whole number, and an integer; demonstrate understanding of the concepts of a natural number, a whole number, and an integer; describe and illustrate opposite uantities in real life situations and conditions; visuali!e integers and their order on a number line; understand the absolute value of a number; determine the absolute value of a number; and perform simple addition and multiplication on absolute values of numbers."What this module is all about What you are expected to learnThis is your guide for the proper use of the module:". #ead the items in the module carefully.$. %ollow the directions as you read the materials. &. Answer all the uestions that you encounter. As you go through the module, you will find help to answer these uestions. 'ometimes, the answers are found at theend of the module for immediate feedback.(. To be successful in undertaking this module, you must be patient and industrious in doing the suggested tasks. ). Take your time to study and learn. Happy learning!The following flowchart serves as your uick guide in using this module.$'tartTake the *retest+heck your paper and count your correct answers.Is your score ,-. or above/0es 'can the items you missed.1o'tudy this moduleTake the *osttest*roceed to the next module2'T3*. How to learn from this module4efore you start this module, take the following Pretest.Directions: #ead the following items carefully. Then choose the letter of the best answer from the selections that follow.". The number $,is 55555555.A. a whole number4. a natural number+. an integer6. All of the above$. 7hich of the following statements is 13T true about the number !ero/A. 8ero is a whole number.4. 8ero is a natural number.+. 8ero is an integer.6. 8ero is a real number. &. 7hich of the following statements is T#9: about natural numbers/A. 1atural numbers are counting numbers.4. :verywhole number is a natural number. +. Any real number is a natural number.6. 1atural numbers start with -.(. 7hich of the following statements is T#9: about the numbers -, &, ;, "6. -, ="(, $), =&-, $)-,. The opposite of ?$) is 55555555.A. =)- 4. @$) +. @) 6. -. Turn . 5555555555555555555555B.W#ite the wo#d o""osite o' ea(h o' the wo#ds listed below%The first has beendone for you. What you w#ite:xample: ". Happy ". 'ad $. 1orth $.5555555555555555555555&. Tood &.5555555555555555555555(. 6eposit (.5555555555555555555555). Honest ).5555555555555555555555;. Industrious ;.5555555555555555555555Think about this! ,umbers also have o--osites. 7e will explore this next. +an you think of a way to show numbers andtheir opposites/ 5555555555555555555 ,. . . . . . . . . . . . . . . . .- " $ & ( ) ; > ,. . . . . . . . . . . . . . . . .A 4 + 6 :Answer Keon !age 2,Jook at the number line below. To differentiate the numbers from one another, we saythat The numbers to the right of - are -ositive integers. The numbers to the left of - are negative integers.The integers on a number line are shown as follows:Think about this!- is the only integer that is neither positive nor negative.There are numbers that are less than !ero. These are the numbersyou find at the left of -.The signs @ and ? are the symbols used to indicate addition and subtraction, but in the number line theyindicate the direction of a point from the -=point, not as operations to be performed. In the number line, @ and ? are used as signs of directions. Thus, read @$ as Mpositive $N not Mplus $N. read ?$ as Mnegative $N not Mminus $N.3n the number line where you find points A, 4, +, 6, and :, the point A is paired with ?", point 4 is paired with ?$, and point + is paired with ?&, point 6 is paired with ?;, and point : is paired with ?>. 1ow, look at the following number line. The arrows indicate the opposite numbers.%or example: The opposite of @" is ?", the opposite of ?$ is @$, and the opposite of ?& is @&. It does not really matter which number is given first. 7hat is more important is the idea ofo--osite. 9sing the number line, answer the uestions that follow."". . . . . . . . . . . . . . . . .- " $ & ( ) ; > , ="=$ =& =( =) =; => =,Positi$e Negati$e -ero7hat is the opposite of ?,/ 55557hat is the opposite of >/5555Think about this!7e write a negative integer within parentheses so that we can easily distinguish it from a positive integer. %or example: A=)B.In real life, there are conditions and situations that are opposites. %or example, theopposite of closing a book is opening the book , the opposite of going three steps up thestairs is going three steps down the stairs, the opposite ofa profit of *hp, is a loss of *hp,.7e can express opposites by means of signed numbers. %or example, going threesteps down the stairs could be described as A=& B anda profit of *hp, is described as @,. Aloss of < pesos is expressed as A= pesos is expressed as @>.6o the following activity carefully.It will enhance your understanding of opposites.A#ti$it% 2%2 OPPOSITS )Directions: 7rite the appropriate situation2condition and their corresponding numbers toindicate the exact opposites in each of the given situations2conditions."$. . . . . . . . . . . . . . . . .- " $ & ( ) ; > , ="=$ =& =( =) =; => =,Answer Keon !age 2,:xample: %ive steps forward55@)555 %ive steps backward 55A=)B555Sta#t he#e!". 0ou go north )- kilometers55555 5555555555555555555555555555555555555555555555555555555555555$. 'ixty feet below sea level 55555 5555555555555555555555555555555555555555555555555555555555555&. Toing up two floors of a building. 555555555555555555555555555555555555555555555555555555555555555555(. :arning *" ---.-- 555555555555555555555555555555555555555555555555555555555555555555). 'pending *hp)$).-- 555555555555555555555555555555555555555555555555555555555555555555;. Taining a weight of & pounds 555555555555555555555555555555555555555555555555555555555555555555>. $- +entigrade above free!ing point 55555 55555555555555555555555555555555555555555555555555555555555554efore you continue reading this module, try to do the following exercises to check yourunderstanding of the concepts and processes.A. +rite the o--osite of each of the following on the s-ace -rovie for it."&Selfchec! #Answer Keon !age 2,"B 1egative55555555555555555555555555555555555$B 3pen the door 55555555555555555555555555555555555&B Hop to the right 55555555555555555555555555555555555(B $-- meters forward 55555555555555555555555555555555555)B Turning =,& &'(!lorationAnswer Keon !age 21Think about this! 7e use the symbol n to represent Qthe absolute value of nS. The absolute value ofa number is either positive or negative. In our number line, 3 3 = and 3 3 = . Iore formally, we state the following definitionThink about this! To put it simply, if you ignore the sign of a number, the result is called theabsolutevalue of that number. Thus, 8 8 =, and 150 150 = . 0ou can do simple operations withtheabsolutevaluesof numbersbysimplyignoringthesignsandthenperformingtheindicated operations. %or example: If you evaluate14 10 + , then you write26 14 10 14 10 = + = + . If youwantto evaluate 15 28 , then youhave15 28 W420 15 28 = . 4efore you continue reading this module, try to do the following exercises to check your understanding of the concepts and processes.$-Selfchec! %Answer Keon !age 21 The absolute value of a number is its distance from - on the number line.%or any number , ifa positive number or -. ifis a negative number.A. Tive the absolute value of each of the following:"B 19 5555555555$B 35 5555555555&B 295555555555(B 56 5555555555)B 91 55555555554. :valuate each of the following:"B 13 25 + 555555555$B 4 15 =555555555&B 72 22 + 555555555(B ( )( ) 6 12 555555555)B 45 82 555555555 The numbers that are used for counting are called natural numbers% These are C", $, &, LDThe natural numbers and 'ero are called whole numbers% These are C-, ", $, &, LDThe numbers that consists of !ero, the positive, and the negative numbers are called integers.$"&et's summari(eAnswer Keon !age 22 These are CL=&, =$, =", -, ", $, &,LD The absolute value of a number is the number regardless of its sign.0ou have Kustlearned what is necessaryto hurdle this Posttest%Directions: #ead the following items carefully. Then choose the letter of the best answerfrom the selections that follow.". The number &,is 55555555.A. a whole number.4. a natural number.+. an integer.6. All of the above$. 7hich of the following statements is true about the number !ero/i 8ero is a whole number.ii 8ero is a natural number.iii 8ero is an integer.A. i only4. i and ii only+. i and iii only6. All of the above&. 7hich of the following statements is true about natural numbers/A. 1atural numbers are counting numbers.4. A whole number is a natural number. +. Any number is a natural number.6. 1atural numbers start with -.(. 7hat is true about the numbers ?", -, &, =;, , "--, =$)-6. &--, $>, . Turn A. ".V $.V &.Y (.V ).V ;.Y >.Y ,.Y , ="), ="-, -, ",, $$ $. ($, &,, &-, =&), =(-, =)>&. = (),=&-, =";, ="$, =>, -, ( &.(), "(, -, ="$, =">, =$;, =&-(.=;-, =&&, ="&, =", -, ", $$, )" (. )", $$, "", -, =", ="&, =$&, =&&, ). =$,, =") ). )$, &", "$, (, =$, =";, =$-, =$>Lesson 3Self1Check $ -age ",.A. ".V $.V &.Y (.V4. ". =),, ="(, =,, -, ", &, $; +. ". $(, &, ", =",, =$;, =),$. =(), =",, =&, -, &;, "-- $. "--, &;, &&, -, =",, =$&&.=$--, ="$-, -, ", ,, ")- &. "--, ")-, ,, ", -, =$--(.=>,, =$;, "), ),, ;(, ,, ;(, ),, "), ="), =$; ). =. + "$. 6&. A ,. + "&. 4(. +