NUMBER SYSTEMS
MODULE - 1Number SystemsAlgebra1NotesNUMBER SYSTEMSFrom time immemorial human beings have been trying to have a count of theirbelongings- goods, ornaments, jewels, animals, trees, sheeps/goats, etc. by using varioustechniques- putting scratches on the ground/stones- by storing stones - one for each commodity kept/taken out.This was the way of having a count of their belongings without having any knowledge ofcounting.One of the greatest inventions in the history of civilization is the creation of numbers. Youcan imagine the confusion when there were no answers to questions of the type “Howmany?”, “How much?” and the like in the absence of the knowledge of numbers. Theinvention of number system including zero and the rules for combining them helped peopleto reply questions of the type:(i) How many apples are there in the basket?(ii) How many speakers have been invited for addressing the meeting?(iii) What is the number of toys on the table?(iv) How many bags of wheat have been the yield from the field?The answers to all these situations and many more involve the knowledge of numbers andoperations on them. This points out to the need of study of number system and its extensionsin the curriculum. In this lesson, we will present a brief review of natural numbers, wholenumbers and integers. We shall then introduce you about rational and irrational numbers indetail. We shall end the lesson after discussing about real numbers.OBJECTIVESAfter studying this lesson, you will be able to illustrate the extension of system of numbers from natural numbers to real(rationals and irrational) numbersMathematics Secondary Course3
MODULE - 1Number SystemsAlgebraNotes identify different types of numbers; express an integer as a rational number; express a rational number as a terminating or non-terminating repeating decimal,and vice-versa; find rational numbers between any two rationals; represent a rational number on the number line; cites examples of irrational numbers; represent find irrational numbers betwen any two given numbers; round off rational and irrational numbers to a given number of decimal places; perform the four fundamental operations of addition, subtraction, multiplicationand division on real numbers.2, 3, 5 on the number line;1.1 EXPECTED BACKGROUND KNOWLEDGEBasic knowledge about counting numbers and their use in day-to-day life.1.2 RECALL OF NATURAL NUMBERS, WHOLE NUMBERSAND INTEGERS1.2.1 Natural NumbersRecall that the counting numbers 1, 2, 3, . constitute the system of natural numbers.These are the numbers which we use in our day-to-day life.Recall that there is no greatest natural number, for if 1 is added to any natural number, weget the next higher natural number, called its successor.We have also studied about four-fundamental operations on natural numbers. For, example,4 2 6, again a natural number;6 21 27, again a natural number;22 – 6 16, again a natural number, but2 – 6 is not defined in natural numbers.Similarly, 4 3 12, again a natural number12 3 36, again a natural number4Mathematics Secondary Course
MODULE - 1Number SystemsAlgebra126 6 is a natural number butis not defined in natural numbers. Thus, we can say that24i) a) addition and multiplication of natural numbers again yield a natural number butNotesb) subtraction and division of two natural numbers may or may not yield a naturalnumberii) The natural numbers can be represented on a number line as shown below. 1 2 3 4 5 6 7 8 9 .iii) Two natural numbers can be added and multiplied in any order and the result obtainedis always same. This does not hold for subtraction and division of natural numbers.1.2.2 Whole Numbers(i) When a natural number is subtracted from itself we can not say what is the left outnumber. To remove this difficulty, the natural numbers were extended by the numberzero (0), to get what is called the system of whole numbersThus, the whole numbers are0, 1, 2, 3, .Again, like before, there is no greatest whole number.(ii) The number 0 has the following properties:a 0 a 0 aa – 0 a but (0 – a) is not defined in whole numbersa 0 0 0 aDivision by zero (0) is not defined.(iii) Four fundamental operations can be performed on whole numbers also as in the caseof natural numbers (with restrictions for subtraction and division).(iv) Whole numbers can also be represented on the number line as follows: 0 1 2 3 4 5 6 7 8 9 .1.2.3 IntegersWhile dealing with natural numbers and whole numbers we found that it is not alwayspossible to subtract a number from another.Mathematics Secondary Course5
MODULE - 1Number SystemsAlgebraNotesFor example, (2 – 3), (3 – 7), (9 – 20) etc. are all not possible in the system of naturalnumbers and whole numbers. Thus, it needed another extension of numbers which allowsuch subtractions.Thus, we extend whole numbers by such numbers as –1 (called negative 1), – 2 (negative2) and so on such that1 (–1) 0, 2 (–2) 0, 3 (–3) 0., 99 (– 99) 0, .Thus, we have extended the whole numbers to another system of numbers, called integers.The integers therefore are., – 7, – 6, – 5, – 4, – 3, – 2, – 1, 0, 1, 2, 3, 4, 5, 6, 7, .1.2.4 Representing Integers on the Number LineWe extend the number line used for representing whole numbers to the left of zero andmark points – 1, – 2, – 3, – 4, . such that 1 and – 1, 2 and – 2, 3 and – 3 are equidistantfrom zero and are in opposite directions of zero. Thus, we have the integer number line asfollows: . –4 –3 –2 –1 0 1 2 3 4.We can now easily represent integers on the number line. For example, let us represent– 5, 7, – 2, – 3, 4 on the number line. In the figure, the points A, B, C, D and E respectivelyrepresent – 5, 7, – 2, – 3 and 4. –7 –6A –5 –4CD –3 –2 –1 0 1 2 3E 4 5 6B 7 8We note here that if an integer a b, then ‘a’ will always be to the right of ‘b’, otherwisevise-versa.For example, in the above figure 7 4, therefore B lies to the right of E. Similarly,– 2 – 5, therefore C (– 2) lies to the right of A (–5).Conversely, as 4 7, therefore 4 lies to the left of 7 which is shown in the figure as E is tothe left of B For finding the greater (or smaller) of the two integers a and b, we follow the followingrule:i) a b, if a is to the right of bii) a b, if a is to the left of bExample 1.1: Identify natural numbers, whole numbers and integers from the following:Solution:15, 22, – 6, 7, – 13, 0, 12, – 12, 13, – 31Natural numbers are: 7, 12, 13, 15 and 22whole numbers are: 0, 7, 12, 13, 15 and 22Integers are: – 31, – 13, – 12, – 6, 0, 7, 12, 13, 15 and 226Mathematics Secondary Course
MODULE - 1Number SystemsAlgebraExample 1.2: From the following, identify those which are (i) not natural numbers (ii) notwhole numbers– 17, 15, 23, – 6, – 4, 0, 16, 18, 22, 31Solution:i) – 17, – 6, – 4 and 0 are not natural numbersNotesii) – 17, – 6, – 4 are not whole numbersNote: From the above examples, we can say thati)all natural numbers are whole numbers and integers also but the vice-versa isnot trueii)all whole numbers are integers alsoYou have studied four fundamental operations on integers in earlier classes.Without repeating them here, we will take some examples and illustrate themhereExample 1.3: Simplify the following and state whether the result is an integer or not12 4, 7 3, 18 3, 36 7, 14 2, 18 36, 13 (–3)Solution:12 4 48; it is an integer7 3 7; It is not an integer318 3 6; It is an integer36 7 36; It is not an integer.714 2 28, It is an integer18 36 18; It is not an integer3613 (–3) – 39; It is an integerExample 1.4: Using number line, add the following integers:(i) 9 (– 5)(ii) (– 3) (– 7)B9Solution:(i) –7 –6 –5 –4 –3 –2 –1 0 1 2 3 45A 5 6 7 8 9A represents 9 on the number line. Going 5 units to the left of A, we reach the pointB, which represents 4. 9 (–5) 4Mathematics Secondary Course7
MODULE - 1Number SystemsAlgebra(ii)NotesA –37 – 10 –9 –8 –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5B 6 7 8 9 10Starting from zero (0) and going three units to the left of zero, we reach the pointA, which represents – 3. From A going 7 units to the left of A, we reach the pointB which represents – 10. (–3) (–7) – 101.3 RATIONAL NUMBERSConsider the situation, when an integer a is divided by another non-zero integer b. Thefollowing cases arise:(i) When ‘a’ is a multiple of ‘b’Suppose a mb, where m is a natural number or integer, thena mb(ii) When a is not a multiple of bais not an integer, and hence is a new type of number. Such a number isbcalled a rational number.In this casepThus, a number which can be put in the form q , where p and q are integers and q 0, iscalled a rational number2 5 6 11, , are all rational numbers.Thus, ,3 8 2 71.3.1 Positive and Negative Rational Numbersp(i) A rational number q is said to be a positive rational number if p and q are bothpositive or both negative integersThus3 5 3 8 12,,, ,are all positive rationals.4 6 2 6 57p(ii) If the integes p and q are of different signs, then q is said to be a negaive rationalnumber.8Mathematics Secondary Course
MODULE - 1Number SystemsAlgebraThus, 7 6 12 16,,,are all negaive rationals.2 5 4 31.3.2 Standard form of a Rational NumberNotesWe know that numbers of the form p p pp,,andq q qqare all rational numbers, where p and q are positive integersWe can see that p p ( p ) p p( p ) p , p , , q ( q )q q q ( q ) q qIn each of the above cases, we have made the denominator q as positive.p, where p and q are integers and q 0, in which q is positive (orqmade positive) and p and q are co-prime (i.e. when they do not have a common factorother than 1 and –1) is said to be in standard form.A rational numberThus the standard form of the rational number2 2 5 3is. Similarly,andare 3365rational numbers in standard form.Note: “A rational number in standard form is also referred to as “a rational number in itslowest form”. In this lesson, we will be using these two terms interchangably.For example, rational number182can be written as in the standard form (or the lowest273form) .25 5, in standard form (or in lowest form) can be written as(cancelling out 3575 from both numerator and denominator).Similarly,Some Important Results(i) Every natural number is a rational number but the vice-versa is not always true.(ii) Every whole number and integer is a rational number but vice-versa is not always true.Mathematics Secondary Course9
MODULE - 1Number SystemsAlgebraExample 1.5: Which of the following are rational numbers and which are not?Notes15 18 75 2, , 17, , , 7 5 63Solution:(i) –2 can be written asp 2, which is of the form q , q 0. Therefore, –2 is a rational1number.(ii)p5is a rational number, as it is of the form q , q 03(iii) –17 is also a rational number as it is of the form(iv) Similarly, 17115 18 7, andare all rational numbers according to the same argument67 5Example 1.6: Write the following rational numbers in their lowest terms:(i) 24192(ii)12168(iii) 2149Solution:(i) 24192 (ii)(iii)10 121 12 14 14121is the lowest form of the rational number14168 2149 1 3 8 3 8 881 24is the lowest form of the rational number819212168 3 7 3 7 77 3 21is the lowest form of the rational number749Mathematics Secondary Course
MODULE - 1Number SystemsAlgebra1.4 EQUIVALENT FORMS OF A RATIONAL NUMBERA rational number can be written in an equivalent form by multiplying/dividing the numeratorand denominator of the given rational number by the same number.NotesFor example2 2 2 4 , 3 3 2 64 ,68,122 2 4 8 , 3 3 4 122 8 16 3 8 24162etc. are equivalent forms of the rational number243Similarly3 6 21 27 .8 16 56 72and4 8 12 28 .7 14 21 49are equivalent forms of43and respectively.87Example 1.7: Write five equivalent forms of the following rational numbers:(i)317(ii) 59Solution:(i)33 2633 4 12 3 ( 3) 9 , , 17 17 2 34 17 17 4 68 17 ( 3) 513 83 7 2124 , 17 8 136 17 7 119 Five equivalent forms of3are176 12 9 24 21,,, ,34 68 51 136 119Mathematics Secondary Course11
MODULE - 1Number SystemsAlgebra(ii)As in part (i), five equivalent forms of 5are9 10 15 20 60 35,,,,18 27 36 108 63Notes1.5 RATIONAL NUMBERS ON THE NUMBER LINEWe know how to represent integers on the number line. Let us try to representnumber line. The rational numberAs 0 1on the21is positive and will be represented to the right of zero.211 1, lies between 0 and 1. Divide the distance OA in two equal parts. This221. Similarly R, the mid-point of OA’,2can be done by bisecting OA at P. Let P represent1represents the rational number .2A’–1–2Similarly,P12R O0A2134can be represented on the number line as below:3C’B’A’–3–2–1OA01PB2CD344/344 2, therefore lies between 1 and 2. Divide the distance AB in three equal33parts. Let one of this part be APAs 1 Now1241 1 OA AP OP33Mathematics Secondary Course
MODULE - 1Number SystemsAlgebraThe point P represents4on the number line.3Notes1.6 COMPARISON OF RATIONAL NUMBERSIn order to compare two rational numbers, we follow any of the following methods:(i) If two rational numbers, to be compared, have the same denominator, compare theirnumerators. The number having the greater numerator is the greater rational number.Thus for the two rational numbers59and , with the same positive denominator171795 as 9 517 1795 17 1717,(ii) If two rational numbers are having different denominators, make their denominatorsequal by taking their equivalent form and then compare the numerators of the resultingrational numbers. The number having a greater numerator is greater rational number.For example, to compare two rational numbers63and, we first make their117denominators same in the following manner:9 7 423 11 33and 11 7 777 11 77As 42 33,6 342 33or 77 77 11 7(iii) By plotting two given rational numbers on the number line we see that the rationalnumber to the right of the other rational number is greater.For example, take23and, we plot these numbers on the number line as below:432–2–1Mathematics Secondary Course0A (3)B(3)4123413
MODULE - 1Number SystemsAlgebra0 Notes2323 1 and 0 1. It means and both lie between 0 and 1. By the method3434of dividing a line into equal number of parts, A representsAs B is to the right of A, Out of23and B represents343 223 or 4 33423 3and , is the greater number.34 4CHECK YOUR PROGRESS 1.11. Identify rational numbers and integers from the following:4, 3 512 3 15, , 6, , 36, ,7 8 74 62. From the following identify those which are not :(i) natural numbers(ii) whole numbers(iii) integers(iv) rational numbers45 37 3, , 15,0, , ,16,17 4 3743. By making the following rational numbers with same denominator, simplify the followingand specify whether the result in each case is a natural number, whole number, integeror a rational number:739 1(v) 2 2(i) 3 1045(vi) 2 7(ii) 3 (iii) 8 13(iv) 12 12(vii) 8 34. Use the number line to add the following:(i) 9 (–7)(ii) (–5) (–3)(iii) (–3) (4)5. Which of the following are rational numbers in lowest term?14Mathematics Secondary Course
MODULE - 1Number SystemsAlgebra8 5 3 6 2 3 15, ,,,,12 7 12 727 246. Which of the following rational numbers are integers?15 5 13 27 7 3 6 10,,,,,,5 15 5 914 27. Write 3 rational numbers equivalent to given rational numbers:Notes2 5 17,,35 68. Represent the following rational numbers on the number line.2 3 1, ,5 4 29. Compare the following rational numbers by (i) changing them to rational numbers inequivalent forms (ii) using number line:23and3435(d) and711(a)37–2 1(b) and(c)and5932 73(e)and621.7 FOUR FUNDAMENTAL OPERATIONS ON RATIONALNUMBERS1.7.1 Addition of Rational Numbers(a) Consider the addition of rational numbersp r,q qp rp r q qqFor example2 5 2 5 7 3 3333 9 3 9 12(ii) 17 17171714 5 14 5 9 3(iii) 3 3 33(i)andMathematics Secondary Course15
MODULE - 1Number SystemsAlgebra(b) Consider the two rational numbersNotesprand .qsp r ps rq ps rq q s qs sqqsFor example,3 23 3 4 2 9 8 17 4 34 312124 7 4 8 5 7 35 32 3(ii) 5 85 84040(i)From the above two cases, we generalise the following rule:(a) The addition of two rational numbers with common denominator is the rational numberwith common denominator and numerator as the sum of the numerators of the tworational numbers.(b) The sum of two rational numbers with different denominators is a rational number withthe denominator equal to the product of the denominators of two rational numbers andthe numerator equal to sum of the product of the numerator of first rational numberwith the denominator of second and the product of numerator of second rationalnumber and the denominator of the first rational number.Let us take sone examples:Example 1.8: Add the following rational numbers:(i)Solution:62and77(i)(ii) 34and1717(iii) 35and11112 6 2 6 8 7 7772 6 8 7 7 7(ii)4 ( 3) 4 ( 3) 4 3 1 17 17171717 164 ( 3) 1 17 1717Mathematics Secondary Course
MODULE - 1Number SystemsAlgebra 5 3 ( 5) ( 3) 5 3 8 (iii) 111111 11 11 8 5 3 11 11 11 NotesExample 1.9: Add each of the following rational numbers:(i)Solution:13and74(ii)(i) We have32and57(iii)45and 9153 1 4 73 7 1 4 4 7 7 421 4 21 4 2828 2825 28 3 1 25 3 7 4 1 21 4 25 or2828 4 7 28 4 7(ii)2 3 7 52 5 3 7 7 5 5 710 21 35 3510 21 31 35352 3 31 2 5 3 7 10 21 31 or 7 5 35 353535 (iii)5 ( 4 ) 9 155 15 ( 4 ) 9 9 15 15 975 ( 36 ) 135 135 Mathematics Secondary Course17
MODULE - 1Number SystemsAlgebra 75 36 39 3 13 13 135135 3 45 45 5 15 9 ( 4 ) 75 36 39 13 5 ( 4 ) 13 or 9 15135135 45 9 1545 Notes1.7.2 Subtraction of Rational Numbersp r p r q qqp r ps qr(b) q sqs(a)Example 1.10: Simplify the following:(i)7 1 4 4Solution: (i)(ii)(ii)3 2 5 127 1 7 1 6 2 3 3 4 444 2 2 23 2 3 12 2 5 5 12 5 12 12 5 36 10 36 10 60 6060 26 13 2 13 60 30 2 301.7.3 Multiplication and Division of Rational Numbers p r (i) Multiplication of two rational number and , q 0, s 0 is the rational s q numberprwhere qs 0ps product of numeratorsproduct of denominators(ii) Division of two rational numbersnumber18prand , such that q 0, s 0, is the rationalqsps, where qr 0qrMathematics Secondary Course
MODULE - 1Number SystemsAlgebra p r p s In other words q s q r Or (First rational number) (Reciprocal of the second rational number)NotesLet us consider some examples.Example 1.11: Multiply the following rational numbers:(i)32and79Solution:(i)(ii)5 2 and 6 19 (iii)7 2 and 13 5 3 2 3 23 22 7 9 7 9 7 3 3 21 3 2 2 7 9 21(ii)5 2 5 ( 2 ) 6 19 6 19 2 55 2 3 19575 5 2 57 6 19 (iii)7 2 7 ( 2 ) 13 5 13 5 7 2 7 2 14 13 5 13 5 65 7 2 14 13 5 65Example 1.12: Simply the following: 3 7 (i) 4 12 (ii)Mathematics Secondary Course9 105 16 12 87 29 (iii) 27 18 19
MODULE - 1Number SystemsAlgebraSolution:(i)Notes 3 7 4 12 3 12 4 7 7 12 Reciprocal of 12 is 7 3 12 3 3 4 9 4 77 47 3 7 9 4 12 7(ii) 9 105 16 2 9 2 16 105 - 1052 Reciprocal of 2 is - 105 9 23 3 2 2 8 3 352 8 3 35 3 3 8 35280 9 105 3 16 2 280(iii) 87 29 27 18 87 18 87 18 29 3 2 9 2 9 3 291 27 29 27 29 87 29 2 27 18 120Mathematics Secondary Course
MODULE - 1Number SystemsAlg
Notes Mathematics Secondary Course MODULE - 1 Algebra 5 2 12 6 is a natural number but 4 6 is not defined in natural numbers. Thus, we can say that i) a) addition and multiplication of natural numbers again yield a natural number but b) subtraction and division of two natural numbers may or may not yield a natural number
on systems science and engineering within the IEEE SMC So-ciety. They include autonomous and bio-inspired robotic and unmanned systems, blockchain, conflict resolution and group decision making, enterprise systems, infrastructure systems, intelligent systems, model-based systems engineering, service systems, system of systems, and system biology.
Number Bonds to 10 Number Line Challenge Cards Number Bonds to 10 Number Line Challenge Cards . Worksheet 1 9 6 6 2 4 2 6 4 0 5 1 10 8 6 3 3 3 7 10 1 5 5 8 1 7 6 8 . Number Shape Number Bonds to 10 Missing Numbers Use the number shapes to work out the missing number in each ques
Vocabulary bookmark General same, identical, different missing number/s number facts, number pairs number bonds greatest value least value number line number track number square hundred square number cards, number grid abacus counters, cubes, blocks, rods die, dice, spinner dominoes pegs, peg board pin board geo-strips same way,
2. REPORT TYPE Final 3. DATES COVERED - 4. TITLE AND SUBTITLE Electronic Warfare and Radar Systems Engineering Handbook 5a. CONTRACT NUMBER 5b. GRANT NUMBER 5c. PROGRAM ELEMENT NUMBER 6. AUTHOR(S) NAVAIR Electronic Warfare/Combat Systems 5d. PROJECT NUMBER 5e. TASK NUMBER 5f. WORK UNIT NUMBER 7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES)
Operating Systems, Embedded Systems, and Real-Time Systems Janez Puhan Ljubljana, 2015. CIP-CatalogingInPublication NationalandUniversityLibrary,Ljubljana 004.451(078.5)(0.034.2) PUHAN,Janez,1969-Operating Systems, Embedded Systems, and Real-Time Systems [Electronic
DIGGING SYSTEMS PARTS CATALOG SPRING 2017. . 08 The DuraTooth Advantage 10 19K Digging Systems 12 33K Digging Systems 16 35K Digging Systems 20 33K/35K Digging Systems 24 50K Digging Systems 36 70K Digging Systems 40 70K/110K Digging Systems . It's no different with digging chain. Ditch Witch .
Business Intelligence Systems KID Systems Application Systems Technical Systems Communications Systems Legal Systems Quality Systems Security Systems Business Domain Strategies . L2. Emergent System Models PIMs,
Number Problems Here is a number line. The number 14 is shown. Mark the number 9 on the number line. 0 14 Here are some digit cards. Meg and Sam each use two of the cards to make a number. What is the difference between their two numbers? I have I have made the largest number you can make. I have made the smallest number you can make.
