Real Numbers



1080135-78005What Is This Module About?Numbers are always a part of our everyday life. When we count our money, we deal with numbers. When we reckon how old we are, we also deal with numbers. Many of our day-to-day activities deal with numbers. But are you aware what real numbers are ?In this module, you will learn more about numbers, its definitions and classifications. You will learn more about numbers by reading the following lessons in this module. This module has 4 lessons, namely :Lesson 1–Real NumbersLesson 2–Number LineLesson 3–Absolute Value of a Number and Addition of IntegersLesson 4–Properties of Equality and Properties of Real Numbers What Can You Learn From This Module?At the end of this module, you should be able to:define and classify real numbers;draw number line and graph integers on a number line;find the absolute value of a number;perform operations dealing with absolute values;add integers using the number line;determine the properties of equality; anddetermine the properties of real numbers.Before reading this module, answer first some questions to determine what you already know about the topics in this module. Let’s See What You Already KnowDirections: Encircle the letter of the correct answer.It comprises the set of positive numbers, negative numbers and zero.integersc.rational numbersirrational numbersd.real numbersIt comprises of the set of counting numbers and 0.integersc.real numbersrational numbersd.whole numbersttHow do you graph 5 on the number line? a.–3 –2 –10123456ttb.–3 –2 –10123456ttc.–3 –2 –10123456ttd.–3 –2 –10123456ttWhich graph represents numbers less than 6 ? a.–3 –2 –10123456ttb.–3 –2 –10123456ttc.–3 –2 –10123456ttd.–3 –2 –10123456ttWhich graph represent numbers greater than or equal to 4 ? a.–3 –2 –10123456ttb.–3 –2 –10123456ttc.–3 –2 –10123456ttd.–3 –2 –101234566.? 8 ? ? 3 ? ?.a. –11b.–5c.5d.117.? 4 ? ? 5 ? 6 ? ?.a. –15b.–5c.5d.15What is the value of?11 ? 3 ? 20 ? .a. –28b.–6c.6d.28What property is illustrated by (1+ 2) + 3= 3 + (1+2)?AssociativeCommutativeIdentityInverseWhat property is illustrated by 7 + 0 = 7?AssociativeCommutativeIdentityInverseWell, how was it? Do you think you fared well? Compare your answers with those in theAnswer Key on page 30 to find out.If all your answers are correct, very good! You may still study the module to review what you already know. Who knows, you might learn a few more new things as well.If you got a low score, don’t feel bad. This only goes to show that this module is for you. It will help you to understand some important concepts that you can apply in your daily life. If you study this module carefully, you would learn the answers to all the items in the test and a lot more! Are you ready?You may now go to the next page to begin Lesson 1.LESSON 1Real NumbersWhat is a real number? What number consists the set of rational numbers? of irrational numbers?How can you differentiate counting numbers from whole numbers?These are all discussed in this lesson.484314597311310801352656498Every thing that we do deals with numbers. When we wake up, we look at the clock, we deal with numbers. Before leaving our house, we count our money, while riding the bus, jeepney or taxi, we count our change. When buying our lunch or merienda, we count the money that we give to the cashier and count our change. All of these deal with numbers. But what kind of numbers are we referring to?You were starting to talk when your parents taught you the alphabet and the numbers 1,2,3 and so on. Hence you were introduced to the concept of counting numbers or natural numbers. Thus counting numbers or natural numbers are numbers that start with 1 followed by 2, followed by 3, and so on. So we say that the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. . . are natural or counting numbers.When you include 0 with the set of counting numbers or natural numbers, you will have the set of whole numbers. Hence whole numbers consist of 0 and the set of counting or natural numbers. So, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 . . . are whole numbers.When you include negative numbers with the set of whole numbers you will produce the set of integers. Thus the set of integers consists of the set of positive and negative numbers including 0. Thus . . . –5,–4,–3,–2,–1,0,1, 2, 3 . . . compose the set of integers.There are also numbers that are expressed in fraction or decimal forms. As we know, fractions can be expressed in decimal forms. To express a fraction in decimal form, divide the numerator by the denominator, then the quotients are in decimal forms.Example: 0.51? 2 1.02 1 001 0.25? 4 1.004 8 202001 0.2? 5 1.05 1 00Notice that in expressing the fractions112 , 4and1into decimal forms the division stopsbecause the remainders are zero. The decimals that we get as quotients are called terminating decimals.1Now, notice if we express the following fractions into decimal form: 3 ,11and 91 0.33333 96 6991040109369104010 9 36 9 141091? 3 1.00001 ? 6 0.1666 1.00001 ? 9 0.1111 1.0000The remainders are not zero, right? This means, that we could continue dividing, as we like because there is always a remainder. We also noticed that a digit or some digits in the decimal form, keep on repeating. These decimals are called repeating decimals. A bar is written over a digit or over several digits to indicate that they repeat.Thus, in our examples:1 ? 0.33331 ? 0.166661 ? 0.11119There is a bar over the digits 33, 66 and 11 because these digits keep on repeating.These decimal forms whether terminating or repeating decimals when combined with the set of integers make up the set of rational numbers.There are numbers which do not correspond to any rational numbers. For purposes of naming them, they are called irrational numbers.Greek mathematicians found several examples of irrational numbers. These are the following:n7π,where n is not a perfect square; exampleWhen you combine the set of rational and irrational numbers you will have the set of real numbers. Hence the set of real numbers consists of the set of rational and irrational numbers. Let’s ReviewBased on the descriptions you have read, answer the following:Can you consider all counting numbers as integers? Why? Can you consider all positive numbers as whole numbers? Why? Can you consider all integers as counting numbers? Why? Are all integers, rational? Why? Are all real numbers, rational? Why? Compare your answers with mine.Can you consider all counting numbers as integers? YesWhy? Because integers consist of 0, positive integers and negative integers. Counting numbers are positive integers.Can you consider all positive numbers as whole numbers? YesWhy? Because whole numbers consist of 0 and positive numbers or the set of counting or natural numbers.Can you consider all integers as counting numbers? NoWhy? Because integers also include negative integers which are not used as counting numbers.Are all integers, rational? YesWhy? Because rational numbers consist of the set of integers, decimals that terminate, and decimals that do not terminate but repeat.Are all real numbers, rational? NoWhy? Because real numbers also include the set of irrational numbers. Let’s RememberCounting or natural numbers are number that starts with 1,2,3, . . .Whole numbers consist of the set of counting or natural numbers and 0.Integers consist of 0, positive numbers and negative numbers.If you combine negative numbers with the set of whole numbers, you produce the set of integers.When you add integers with decimals that terminate or decimals that do not terminate but repeat, then you produce the set of rational numbers.When you combine the set of rational1numbers with the set of irrational numbers, youproduce the set of real numbers.4979805-192559Let’s See What You Have LearnedTell whether the following number is rational or irrational: 1.2where ? = 3.1415. . .2.53.4.0.1255.0. 66666.– 12.757.08.0.3759.– 6710.Rearrange the jumbled words/phrases on the right hand column to identify the following:the union of the set of rationalLAER and irrational numbers1,2,3,BERSNUM TCOUNING0 and the set of counting numbersLEWHO BERSNUMwhole numbers and negativeSITENGER numbersCompare your answers with those in the Answer Key on page 30.LESSON 2Number LineIn lesson 1, you learned how to define and classify real numbers. In this lesson you will learn to draw the number line and graph the integers on the number line.How can I graph these points on the number line?A number line is a visual representation of the set of real numbers. Each point on the line can represent a number in the set of real numbers. On this line, you can choose any point O with coordinate 0 and call it the origin. On the number line, equal divisions could be placed on the left and on the right of O. To each point on the right of O, you associate positive integers and to each point on the left of O, you associate the negative numbers.Below is an example of a number line.tt–6 –5 –4 –3 –2 –10123456This is also a number line.tt–3–2–10123 Let’s LearnLet us consider the following graphs.ttT–6 –5 –4 –3 –2 –10123456Graph 1Graph 1 shows a point T whose coordinate is –4.ttE–6 –5 –4 –3 –2 –10123456Graph 2 Graph 2 shows a point E whose coordinate is 0.tCan you draw a graph showing a point O with coordinate 3?tCompare your answers with the graph below.ttO–6 –5 –4 –3 –2 –10123456This graph shows a point O whose coordinate is 3. Did you get the right answer? If you did, good!The graph below shows the points whose coordinates are less than 5. Notice that the line started at 5, represented by a dot, and was drawn to the left with an arrow at its end point to indicate the points to the left of 5.t tt–6 –5 –4 –3 –2 –10123456The graph below shows the points whose coordinates are less than 0. Notice that the line started from 0 point and was drawn to the left to indicate the points to the left of 0.t tt–6 –5 –4 –3 –2 –10123456ttCan you graph a line whose coordinates are less than –2 ?Compare your answers with mine.t tt–6 –5 –4 –3 –2 –10123456The graph above, shows the points whose coordinates are less than –2.The graph below shows the points whose coordinates are greater than 5. Notice that the line started at 5 as designated by a dot and an arrow was drawn to the right to indicate the points to the right of 5.ttt–6 –5 –4 –3 –2 –10123456Similarly,tThe graph below shows the points whose coordinates are greater than 0. Notice that the line started at 0 as designated by a dot and an arrow was drawn to the right to indicate the points to the right of 0.tt–6 –5 –4 –3 –2 –10123456ttCan you locate points whose coordinates are greater than –3 ?Compare your answer with mine.tt t–6 –5 –4 –3 –2 –10123456This shows the points whose coordinates are greater than –3.Now, suppose you are asked to draw a graph showing the points whose coordinates are between –2 and 2, inclusive. How are you going to do it?First, let us define what inclusive is.Inclusive means that you include the endpoints. Then we follow the following steps:STEP 1.Graph the points whose coordinates are greater than or equal to –2. This is shown below.ttt–6 –5 –4 –3 –2 –10123456graph 1STEP 2.Graph the points whose coordinates are less than or equal to 2.ttt–6 –5 –4 –3 –2 –10123456tSTEP 3.Find the common points of the two graphs.tt–6 –5 –4 –3 –2·–10123456graph 1···········tttgraph 2–6 –5 –4 –3 –2 –10123456··The above illustration shows that the common points are presented using the dotted lines. This means that the common points are –2,–1,0,1 and 2.Step 4.Having determined the common points, show this in a graph. Graph 3 then, shows the points whose coordinates are between –2 and 2, inclusive.tt–6 –5 –4 –3 –2 –10123456graph 3How can you graph the points whose coordinates are between –3 and 5, inclusive?ttSTEP 1ttSTEP 2ttSTEP 3ttSTEP 4Compare your answers with mine.tSTEP 1.STEP 2.–6 –5 –4 –3 –2 –10123456t–6 –5 –4 –3 –2 –10123456graph 1ttgraph 2STEP 3.Find the common points of the two graphs.ttt–6 –5 –4 –3··tt··–2 –10123456····graph 1·····tgraph 2–6 –5 –4 –3 –2 –10123456··STEP 4.Having determined the common points, show this in a graph. Graph 3 shows the points whose coordinates are between –3 and 5, inclusive.tt–6 –5 –4 –3 –2 –10123456graph 3To show the points whose coordinates are between –2 and 2, exclusive, follow these steps.Exclusive means that you do not include the endpoints.STEP 1.Graph the points whose coordinates are greater than –2.tt–6 –5 –4 –3 –2 –10123456STEP 2.Graph the points whose coordinates are less than 2.tt–6 –5 –4 –3 –2 –10123456STEP 3.Find the common points of the two graphs. The graph below shows the points whose coordinates are between –2 and 2, exclusive.tt–6 –5 –4 –3 –2 –10123456Notice that we did not include –2 and 2, as these are the endpoints. So the points exclusive of –2 and 2 are –1, 0 and 1.How can you graph the points whose coordinates are between –3 and 5, exclusive?ttSTEP 1ttSTEP 2ttSTEP 3Compare your answers with mine.STEP 1.Graph the points whose coordinates are greater than –3.tt–6 –5 –4 –3 –2 –10123456STEP 2.Graph the points whose coordinates are less than 5.tt–6 –5 –4 –3 –2 –10123456STEP 3.Find the common points of the two graphs. The graph below shows the points whose coordinates are between –3 and 5, exclusive.1080135255436tt–6 –5 –4 –3 –2 –10123456Let’s RememberA number line is a visual representation of the set of real numbers.Each point on the line can represent a number in the set of real numbers.1002030-100611Let’s ReviewGraph the point A whose coordinate is 6.Graph the points whose coordinate is less than 1.Graph the points whose coordinate is greater than 2.Graph the points whose coordinate is greater than or equal to 7.Graph the points whose coordinate is between –3 and 3 exclusive.Graph the points whose coordinate is between –2 and 5 pare your answers with those in the Answer Key on pages 30–31.LESSON 3Absolute Value of a Number and Addition of IntegersIn lesson 1, you learned how to define and classify real numbers. In Lesson 2, you learned to draw the number line and graph the integers on the number line.At the end of this lesson you should be able to:determine the absolute value of a number;perform operations dealing with absolute values; and1019810412044add integers using the number line.Let’s LearnThe absolute value of a number, denoted by origin on the number line.Let us look at the following examples:x , is the distance of that number from theLet us determine the absolute value of 3 or 3 .From the definition, we know that an absolute value of a number is the distance of that number from the origin on the number line. Illustrating this using the number line, we count 3 units to the right from 0, which is the origin. This is shown in the graph below.tt123Therefore,–6 –5 –4 –3 –2 –101234563 ? 3 since you move 3 units to the right to obtain 3 from 0.Let us have another example.Determine the absolute value of 5 or what is 5 ?Using the number line, we count 5 units to the right from 0, which is the origin. This is shown in the graph below.tt12345Therefore5 ? 5–6 –5 –4 –3 –2 –101234561019810-86006Let’s Try ThisWhat is 7 ? What is 9 ? Compare your answers with mine.7 ? 79 ? 9Similarly,5 ? 5 since you also move 5 units to the left to obtain –5 from 0.tt54321For the–6 –5 –4 –3 –2 –101234562 ? 2 , you move 2 units to the left to obtain 2 from 0.tt21–6 –5 –4 –3 –2 –10123456The absolute value of 0 or 0? 0 since you move 0 unit to obtain 0.1026160368466tt–6 –5 –4 –3 –2 –10123456Let’s Try ThisWhat is? 7 ? What isWhat is? 9 ? ? 7 14 ? Compare your answers with mine.? 7 ? 7? 9 ? 9? 7 14 ? 714Notice that to obtain the absolute value of a number, you simply copy the number regardless of sign. Hence,?17 ? 1725 ? 25? 7 ? 25 ? 7 ? 25? 32?10 ? 21 ? 10 ? 21? – 11What is15 ? ?13 ? What is 15 ? ? 23 ? Compare your answer with mine.15 ? ?13 ? 15 ?13? 215 ? ? 23 ? 15 ? 23? ?8Did you get the right answers ?One way to add integers is using the number line. Let us look at the examples below. Note if there is no sign before the number, the number is said to be positive. If there is a – sign before the number, the number is negative.Example 1.How can we show the sum of 3 and 2 using the number line?In order to get the sum, follow these steps.STEP 1. Plot the point 3 in the number line. This is shown in the graph below.tt–6 –5 –4 –3 –2 –10123456tSTEP 2. From 3, count 2 units to the right.12t–6 –5 –4 –3 –2 –10123456Notice that you end up with 5. Hence 2 + 3 = 5.Example 2.How can we show the sum of 3 and 3 using the number line?In order to get the sum, follow these steps.STEP 1. Graph 3 on the number line.tt–6 –5 –4 –3 –2 –10123456STEP 2. From 3, count 3 units to the right.tt123–5 –4 –3 –2 –10123456Notice that you end up with 6. Hence 3 + 3 = 6. What is the sum of 3 and 7? Compare your answers with mine.3 + 7 = 10Notice that the sum of two positive numbers is positive.Example 3.How can we show the sum of –1 and –3 using the number line?In order to get the sum, follow these steps.STEP 1. Graph –1 on the number line.tt–6 –5 –4 –3 –2 –10123456STEP 2. From –1, count 3 units to the left since the sign of –3 is negative.tt321–6 –5 –4 –3 –2 –10123456Notice that you end up with –4. Hence –1 + –3 = –4 using the number line. Example 4.How can we show the sum of –2 and –4 using the number line?In order to get the sum, follow these steps.STEP 1. Graph –2 on the number line.tt–6 –5 –4 –3 –2 –10123456STEP 2. From –2, count 4 units to the left since the sign of –4 is negative.tt4321–6 –5 –4 –3 –2 –10123456Notice that you end up with –6. Hence –2 + – 4 = –6. What is the sum of –3 and –7? Compare your answer with mine.–3 + –7 = –10Notice that the sum of two negative numbers is negative.Example 5.How can we show the sum of –3 and 5 using the number line?In order to get the sum, follow these steps.STEP 1. Graph –3 on the number line.tt–6 –5 –4 –3 –2 –10123456STEP 2. From –3, count 5 units to the right since the sign of 5 is positive.tt12345–6 –5 –4 –3 –2 –10123456Notice that you end up with 2. Hence –3 + 5 = 2.Example 6.How can we show the sum of 8 and –4 using the number line?In order to get the sum, follow these steps.STEP 1. Graph 8 on the number line.tt–6 –5 –4 –3 –2 –1012345678STEP 2. From 8, count 4 units to the left since the sign of –4 is negative.tt4321–6 –5 –4 –3 –2 –1012345678Notice that you end up with 4. Hence 8 + –4 = 4Example 7.How can we show the sum of –6 and 4 using the number line?In order to get the sum, follow these steps.STEP 1. Graph –6 on the number line.tt–6 –5 –4 –3 –2 –1012345678STEP 2. From –6, count 4 units to the right.tt1234–6 –5 –4 –3 –2 –1012345678Notice that you end up with –2. Hence –6 + 4 = –2Example 8.How can we show the sum of –6 and 5 using the number line?In order to get the sum, follow these steps.STEP 1. Graph –6 on the number line.tt–6 –5 –4 –3 –2 –1012345678STEP 2. From –6, count 5 units to the right.tt12345–6 –5 –4 –3 –2 –1012345678Notice that you end up with –1. Hence –6 + 5 = –1There is a module on Positive and Negative Integers which you may want to read to learn more about integers and the four fundamental operations involving integers.The absolute value of a number is the distance of that number from the origin on the number line.You can get the absolute value of a number using the number line.You can add integers, using the number line. Let’s Remember1080135-212879Let’s See What You Have LearnedEvaluate: 1.2.3.? 29= 15 13= 35 ? ? 8= 4.?19 ? 31= 5.?18 ? ?15= 6.7 + (–23)= 7.–28 + 35= 8.–100 + 90= 9.–8 + 13 + 2= 10.–6 + –6 +5= Compare your answers with those in the Answer Key on page 31.LESSON 4Properties of Equality and Properties of Real NumbersIn lesson 1, you learned how to define and classify real numbers. In Lesson 2, you learned to draw the number line and graph the integers on the number line. In Lesson 3, you learned to determine the absolute value of a number, perform operations dealing with absolute value and add integers using the number line.At the end of this lesson you should be able to:determine the properties of equality; anddetermine the properties of real numbers. Let’s LearnThere are several properties of equality. These properties are as follows:Reflexive Property of Equality: x = x2696845634531We can associate this idea with a mirror . When you look at a mirror, what can you see? If we are in our normal state of mind, what we see is simply the physical aspect of us. Nothing more.Symmetric Property of Equality: If x = y then y = x .What we can see at the right side, we can also see at the left side. This illustrates symmetric property.For example , if 3+ 5 = 8 then 8 = 3 + 5Transitive Property of Equality: If x = y and y = z , then x = z.The transitive property can be illustrated in this example. You are a girl, and your name is Rosie. You have a sister named Maxima. But Maxima has a sister named Grace. Then Rosie has a sister named Grace, too.Similarly, if 3 + 5 = 8 and 8 = 1 + 7 then 3 + 5 = 1+ 7.Addition Property of Equality (APE): If x = y then x + z = y + z.This means that adding the same number to both sides of the equality results in an equality.For example, if 5 = 3 + 2,Then6 + 5 = 6 + 3 + 211 =9 +211 = 11Multiplication Property of Equality (MPE): If x = y then ax = ay.This means that multiplying both sides of an equality by the same number results in an equality.For example, if 6 = 3 (2),Then2(6) = 2 (3) (2)12 = 2 (6)12 = 12If we have properties of equality, we also have properties of real numbers. The following are the different properties of real numbers.Closure PropertyFor any x, y which belongs to the set of real numbers then,x + y belongs to the set of real numbersThis means that when you get the sum of two real numbers the result is also a real number.x(y) belongs also to the set of real numbersThis means that the product of two real numbers is also a real number.Associative PropertyFor any x, y and z which belongs to the set of real numbers, then a.( x + y ) + z = x + ( y + z )This means that even if we add the first two numbers then add the third number, the result will be the same if we add the second and the third number then add the first number.Hence (3 + 6) +7 = 3 + (6 + 7) by associative propertyIs (10 + 5) + 7 = 10 + (5 + 7) ? What property justifies it? Compare your answers with mine .YesAssociative Property of Equality b.[ (x)(y) ] (z) = (x) [ (y)(z) ]This means that even if we multiply the first two numbers, then multiply the third number, the result will be the same if we multiply the second and the third number, then multiply with the first number.Hence [(3)(4)] (5 ) = (3) [ ( 4) (5) ]Commutative PropertyFor any x, y which belongs to the set of real numbers then,x + y = y + xThis means that the change of places does not affect the sum.Hence 3 + 5 = 5 + 3x (y) = y ( x)This means that the change of places does not affect the product. Hence 3 (5 ) = 5 (3)Is (3) (7) = (7) (3) ? What property justifies it ? Compare your answers with mine.YesCommutative PropertyExistence of an Identity Element for Addition and MultiplicationFor any x which belongs to the set of real numbers, there exists 0 and 1 which belongs to the same set such that,a. x + 0 = 0 + x = xThis means that if we add 0 to any number the value will not be affected. Hence 5+ 0 = 5–10 + 0 = –10 3/2 + 0 = 3/2b. x (1) = 1 (x) = xThis means that any number multiplied by 1 will give us the same number.Hence 3 (1) = 3–8 (1) = –8–1/4 (1) = –1/4Existence of an Inverse Element for Addition and MultiplicationFor any x which belongs to the set of real numbers, there exists –x and x ? ? (is not equal to) 0 such that,x + (– x) = 0 , here –x is called the additive inverseHence –7 is the additive inverse of 7.1 wherexHence5 is the additive inverse of – 5 .77What is the additive inverse of –6? 2What is the additive inverse of? What is the additive inverse of – 7 ?9Compare your answers with mine 62–79x ( 1 ) = 1 herexis called the multiplicative inversexHence 1 is the multiplicative inverse of 5, as 1 (5) = 1Hence5is the multiplicative inverse of5??3 , as 2 ? 3 ? ? 1323 ? 2 ?Distributive Property of Multiplication over AdditionFor any x, y, z which belongs to the set of whole numbers,( x + y ) z = xz + yzz ( x + y ) = zx + zyFor example x = 3, y = 2 z = 5 then Substituting the values, we have:a. ( x + y ) z = ( 3 + 2 ) ( 5 )= 5 ( 5 )= 25b. xz + yz = 3 ( 5 ) + 2 ( 5 )= 15 + 10= 25The following properties are considered properties of equality:ReflexiveSymmetricTransitiveAddition Property of EqualityMultiplication Property of EqualityThe following properties are properties of real numbers:ClosureAssociativeExistence of an Identity elementExistence of an Inverse elementCommutativeDistributive Property of Multiplication over Addition. Let’s Remember Let’s See What You Have LearnedIdentify the property illustrated:1.If 10 = 8 + 2, then 8 + 2 = 102.5 + 0 = 53.4 + 7 = 114.6(3) = 185.–6 + 6 = 06.4y = 4y7.if 3 + 7 = 10 and 10 = 5 + 5 then 3 + 7 = 5 + 58.7 (1 + 9) = 7 (1) + 7 (9)9.9 ( 1 ) = 1910.(3 + 7) + 6 = 6 + (3 + 7)Compare your answers with those in the Answer Key on pages 31–32. Let’s Sum UpIn Lesson 1, the following are the important learnings:Counting or natural numbers are numbers that start with 1, 2, 3, …Whole numbers consist of the set of counting or natural numbers and 0.Integers consist of 0, positive numbers and negative numbers.If you combine negative numbers with the set of whole numbers you produce the set of integers.When you add integers with decimals that terminate or decimals that do not terminate but repeats, then you produce the set of rational numbers.When you combine the set of rational numbers with the set of irrational numbers, you produce the set of real numbers.In Lesson 2, the following are the important learnings:A number line is a visual representation of the set of real numbers.Each point on the line can represent a number in the set of real numbers. In Lesson 3, the following are the important learnings:The absolute value of a number is the distance of that number from the origin on the number line.You can get the absolute value of a number using the number line.You can add integers using the number line. In Lesson 4, the following are the important learnings:The following properties are considered properties of equality:ReflexiveSymmetricTransitiveAddition Property of EqualityMultiplication Property of EqualityThe following properties are properties of real numbers:ClosureAssociativeExistence of an Identity elementExistence of an Inverse elementCommutativeDistributive Property of Multiplication over Addition What Have You Learned?Directions: Answer the following:Union of nonnegative and negative numbers .integersc.rational numbersirrational numbersd.real numbersIt starts with 0, 1, 2, 3, etc. .integersc.real numbersrational numbersd.whole numbersttHow do you graph 2 on the number line? a.–6 –5 –4 –3 –2 –10123456ttb.–6 –5 –4 –3 –2 –10123456ttc.–6 –5 –4 –3 –2 –10123456ttd.–6 –5 –4 –3 –2 –10123456tttWhich graph represents numbers greater than 6 ? a.–6 –5 –4 –3 –2 –10123456tttb.–6 –5 –4 –3 –2 –10123456tttc.–6 –5 –4 –3 –2 –10123456tttd.–6 –5 –4 –3 –2 –10123456ttWhich graph represent numbers less than or equal to 4 ? a.–6 –5 –4 –3 –2 –10123456tttb.–6 –5 –4 –3 –2 –10123456tttc.–6 –5 –4 –3 –2 –10123456ttd.–6 –5 –4 –3 –2 –101234566.|8| – |–3| = .a. –11b. –5c. 5d. 117.– 4 + |–5| – |–6| = .a. –15b. –5c. 5d. 15What is the value of |10| + 2 – |–18| ?a. –28b. –6c. 6d. 28What property is illustrated by (1· 2) · 3 = 3 · (1 · 2)?AssociativeCommutativeIdentityInverseWhat property is illustrated by 7 + (–7) = 0?AssociativeCommutativeIdentityInverseCheck your answers with those given in the Answer Key on page 32 of this module.If you got a score of :0 – 3Study the whole module again.4 – 6Go back to the parts of the module which you did not understand very well. 7 – 8Good! Just review the items which you did not get right.9 – 10Very good! You learned a lot from this module.You are now ready to go on to the next one.1080135-114581Answer KeyLet’s See What You Already Know (pages 1–3)1.a6.cd7.ba8.bb9.b5.d10.cLesson 1Let’s See What You Have Learned (pages 7–8)1.irrationalrationalrationalrationalrationalrationalrationalrationalrationalirrational1.real numberscounting numberswhole numbersintegersLesson 2Let’s Review (page 14)Graph of point A whose coordinate is 6.tt–6 –5 –4 –3 –2 –10123456Graph of numbers whose coordinate is less than 1.t tt–6 –5 –4 –3 –2 –10123456Graph of numbers whose coordinate is greater than 2.ttt–6 –5 –4 –3 –2 –10123456Graph of numbers whose coordinate is greater than or equal to 7.tt t–6 –5 –4 –3 –2 –1012345678Graph of numbers whose coordinate is between –3 and 3 exclusive.tt–6 –5 –4 –3 –2 –10123456Graph the numbers whose coordinate is between –2 and 5 inclusive.tt–6 –5 –4 –3 –2 –10123456Lesson 3Let’s See What You Have Learned (page 20)1.? 29 ? 292.15 13 ? 15 133.35 ? ? 8 ? 35 ? 8 ? 434.?19 ? 31 ? 19 ? 31 ? ?125.?18 ? ?15 ? 18 ?15 ? 36. 7 + (–23) = –167. –28 + 35 = 78. –100 + 90 = –109. –8 + 13 + 2 = 710. –6 + –6 +15 = 3Lesson 4Let’s See What You Have Learned (pages 25–26)SymmetryIdentity for additionClosure for additionClosure for multiplicationInverse for additionReflexiveTransitiveDistributive property of multiplication over additionInverse for multiplicationCommutativeWhat Have You Learned? (pages 27–28)1. (a)since the set of integers includes the set of nonnegative numbers and negative numbers.(d)since whole numbers comprises of counting numbers and 0.(a)since the rest represents numbers less than 2, or numbers greater than 2 or numbers greater than or equal to 2.(c)since the rest represents numbers less than 6, or numbers less than or equal to 6 or numbers greater than or equal to 6.(b)since the graph shows numbers less than 4 and equal to 4. 6. (c)since |8| – |–3| = 8 – 3 = 5.7. (b)since? 4 ? ? 5 ? ? 6 ? ?4 ? 5 ? 6 ? ?5 .8. (b)since 10 ? 2 ? ?18 ? 10 ? 2 ?18 ? ?6 .(b)since it involves only the change of places.(d)since the sum is 0. ReferencesSia, Lucy O., et al. 21st Century Mathematics, Second Year. Quezon City: Phoenix Publishing House, Inc. Reprinted 2000.Capitulo, F. M. Algebra, a Simplified Approach. Manila: National Bookstore, 1989. ................
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