NUMBER SYSTEM NATURAL NUMBERS As we start from 1 and move along this number line from left to right we find numbers and numbers. The simplest numbers are 1, 2, 3, 4 the numbers being used in counting. These are called natural numbers and denoted by N Set of all non-fractional number from 1 to + 00 The smallest natural number is 1 but there is no largest number as it goes upto infinity (00) Prime numbers: All natural numbers that have one and itself only as their factors are called prime numbers i.e. prime numbers are exactly divisible by 1 and themselves e.g. 2,3,5,7,11,13,17,19,23....etc and they are denoted by P Composite numbers: All natural number, which are not prime are composite numbers. If C is the set of composite number then C = {4,6,8,9,10,12 1 is neither prime nor composite number Co-prime numbers: If the H.C.F. of the given numbers (not necessarily prime) is 1 then they are known as co-prime numbers. e.g. 4, 9, are co-prime as H.C.F. of (4, Any two consecutive numbers will always be co-prime o 1 2 3 415161718 9 10
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NUMBER SYSTEM WHOLE NUMBERS v/ The natural numbers along with the number zero form the set of whole numbers i.e. numbers 0, 1, 2, 3, 4 are whole numbers. v/ These numbers are denoted by W The smallest whole number is 0 (zero) v/ Set of numbers from 0 to +00 , W = {0 1 2 34 INTEGERS All natural numbers, 0 and the negatives of natural numbers form the collection of integers. These are denoted by Z or I Z comes from the German word "zahlen", which means "to count" Set of all-non fractional numbers from - 00 to + 00, | or Z = ( ...e, -3,-2,- 0 2 3 456 7 Natural numbers numbers 8 9 10 The Number Line
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NUMBER SYSTEM FRACTIONS -J On the number line there are numbers like -1/2, 5/2, 10/3 which are called Fractions Common fraction : Fractions whose denominator is not 10 Decimal fraction : Fractions whose denominator is 10 or any power of 10 Proper fraction : Numerator < Denominator e.g. 3/5, 6/11 Improper fraction : Numerator > Denominator e.g. 5/2, 10/3 2 Mixed fraction : Consists of integral as well as fractional part e.g. 3— 3 Compound fraction : Fraction whose numerator and denominator themselves are fractions Improper fraction can be written in the form of mixed fractions 1/2 .3 .2 3 2 3 4
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NUMBER SYSTEM RATIONAL NUMBERS -J Definition - A number of the form p/q where p and q are both integers having no common factor i.e., p and q are co-primes and q * 0 is called a rational number (Division by zero is not permissible) The collection of rational numbers is denoted by Q 'Rational' comes from the word 'ratio', and Q comes from the word 'quotient' All natural numbers, whole numbers, integers and fractions are rational numbers For example, 6 can be written as 6/1. Here, p = 6 and q = 1. Similarly— 20 can also be written as 20/1. Here, p = — 20 and q = 1. So, 6 and — 20 are also rational numbers therefore, the rational numbers also include the natural numbers, whole numbers and integers You will notice that fractions do not have a unique representation in the form p/q, where p and q are integers and q O. For example, h = 2/4 = 10/20 = 25/50 = 47/94 and so on. These are equivalent rational numbers (or fractions). However, when we say that p/q is a rational number, or when we represent p/q on the number line, we assume that q O and that p and q have no common factors other than 1 (that is, p and q are co-prime). So, on the number line, among the infinitely many fractions equivalent to lh , we will choose lh to represent all of them 1/2 .3 .2 3 2 3 4
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NUMBER SYSTEM Question: Are the following statements true or false? Give reasons for your answers — 1. Every whole number is a natural number 2. Every integer is a rational number 3. Every rational number is an integer Answer: 1. False, because zero is a whole number but not a natural number 2. True, because every integer m can be expressed in the form m/ 1, and so it is a rational number 3. False, because h, 3/5 or 4/9 are not an integers Question: Is zero a rational number? Can you write it in the form p/q , where p and q are integers and q O? Answer: Yes zero is a rational number because it can be written in the form p/q such as 0/1, 0/1999 etc and where q O Question: Are the following statements true or false? Give reasons for your answers — Every natural number is a whole number. 1. 2. Every integer is a whole number. 3. Every rational number is a whole number Answer: 1. True 2. False, because -5, -6 are integers but not whole numbers 3. False, because h, 3/5 or -4/9 are not whole numbers
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NUMBER SYSTEM TO FIND OUT RATIONAL NUMBERS BETWEEN TWO RATIONALS Mid Value Method: If r and s be two rational numbers and r < s then — Since r < s 2 Again since r < s 2 Therefore 2 i.e. lies between r & s. 2 Hence first rational number between r & s is — 2
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NUMBER SYSTEM TO FIND OUT RATIONAL NUMBERS BETWEEN TWO RATIONALS Mid Value Method: v/ If r and s be two rational numbers then — is between r and s 2 v/ If r < s then r < 2 Example. Find a rational number between 3 and 4. SOLUTION. we know that is between r and s. Therefore, a rational number between 3 and 4 is — 2 2 7 2 Hence — is a rational number between 3 & 4 2
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NUMBER SYSTEM Exercise 1.1 — Note: Use Mid Value method only QI. Find a rational number between -1/3 & 1/4 Q2. Find six rational numbers between 3 and 4 Q3. Find 3 rational numbers between 1/3 & 1/2 Q4. Find five rational numbers between 3/5 and 4/5 Q5. Find five rational numbers between 2/3 and 8 Solutions given in separate annexure
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NUMBER SYSTEM TO FIND OUT RATIONAL NUMBERS BETWEEN TWO RATIONALS Denominator Method: Note: Use this method when you have to find rational numbers between two natural numbers or two positive fractions with common denominator Find n rational numbers between a and b (a < b) b—a 1st Step: find d = 71+1 2nd Step: find numbers — 1st rational number will be a + d 2nd rational number will be a + 2d 3rd rational number will be a + 3d and so on nth rational number is a + nd
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NUMBER SYSTEM TO FIND OUT RATIONAL NUMBERS BETWEEN TWO RATIONALS Denominator Method: Example 1. Insert 3 rational numbers between 2 and 3 Solution. Here a = 2, b=3&n=3 d = = 1/4 4 Therefore 1st rational number = 2nd rational number will be = a + 2d = 2+ 3rd rational number will be = 1 4 2 4 3 4 9 4 10 4 4
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NUMBER SYSTEM TO FIND OUT RATIONAL NUMBERS BETWEEN TWO RATIONALS Example 2. Find five rational numbers between 3/5 and 4/5 Here a = 3/5, b = 4/5 & = 5 4/5-3/5 1 1 6 30 Therefore 1st rational number = a + d = -+ — = — 3 1 5 30 30 21 30 22 30 23 30 19 30 2nd rational number = a + 2d = - + — 3 2 5 30 3 3 3rd rational number = 5 30 4th rational number = 5 30 5 3 5th rational number = a + 5d 5 30
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NUMBER SYSTEM TO FIND OUT RATIONAL NUMBERS BETWEEN TWO RATIONALS Numerator Method: Note: Use this method to find rational numbers between — ' Two negative fractions with common denominator One negative & one positive fractions with common denominator ' Two positive fractions where denominators are not equal Example 1. Find 10 rational numbers between - 1/9 and 4/9 Solution: We have -1/9 and 4/9 . The numerators of the both the fractions are multiplied by such a number so that the difference between the new numerator be at least 10 + 1 i.e., 11. On multiplying the numerators -1 and 4 by 3, we get — 3 and 12 Difference between new numerators is 15; which is greater than 11 27' 27' 27' 27' 27' 27' Thus, the given fractions become 3 12 —&— 27 27 2 1 1 27' 2 3 4 5 6 On increasing the numerators by 1 of -3/27 successively, we get the required 10 numbers as - 27' 7 27
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NUMBER SYSTEM Example 2. Find 7 rational numbers between 1/6 and 5/21 Solution: We have 1/6 and 5/21. L.C.M. of 6 and 21 is 42 To make denominators equal i.e. 42 we multiply numerator and denominator of 1/6 by 7 and that of 5/21 by 2, we get 7/42 and 10/42 To insert seven rational numbers we multiply the numerators and denominators of both the fractions by such a number 2 or 3 so that the difference between the numerators is at least 7+1 = 8. Multiplying the numerators and denominators of or 4 both fractions by 3, we get — = Zand 42 30 126 126 On increasing the numerators by 1 of 21/126 successively, we get the required seven rational numbers between 1/6 and 5/21 22 23 24 25, 26 27 28 126 126' 126' 126' 126' 126' 126
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NUMBER SYSTEM Exercise 1.2 — Note: Use Denominator or Numerator method only QI. Find a rational number between -1/3 & 1/4 Q2. Find six rational numbers between 3 and 4 Q3. Find 3 rational numbers between 1/3 & 1/2 Q4. Find five rational numbers between 3/5 and 4/5 Q5. Find five rational numbers between 2/3 and 8 Solutions given in separate annexure
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NUMBER SYSTEM Exercise 1.3 — Note: Use any method as appropriate QI. Find five rational numbers between 1 and 2 Q2. Find six rational numbers between -1 and 3 Q3. Find four rational numbers between 2/3 and 5/3 Q4. Find six numbers between 3/4 and 7/4 Q5. Insert six numbers between 1/2 and 2/5 Solutions given in separate annexure
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NUMBER SYSTEM CONVERSION OF A RATIONAL NUMBER INTO DECIMAL DIVISION METHOD Case 1. The remainder becomes zero To convert a rational number into decimal, we divide numerator by denominator. For example - In case of 7/8 we divide 7 by 8 and after a finite number of steps, the remainder becomes zero. Such a decimal expansion is known as terminating decimals (Finite Decimals) 0.875 64 60 56 40 40 7 Therefore - = . 0 875 8
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NUMBER SYSTEM CONVERSION OF A RATIONAL NUMBER INTO DECIMAL DIVISION METHOD Case 2. The remainder never becomes zero For example - In case of 2/3 & 11/7 we notice that the remainders repeat after a certain stage. In other words we have a repeating block of digits in the quotient, we say that this expansion is non-terminating or recurring decimals (Infinite Decimals) 2 Therefore - = 3 0.666... or 0.6 11 Therefore — = 1.571428571428.... Or 1.571428 7 Period of decimals for 11/7 = 571428 Length of the period for 11/7 = 6 Note: Both terminating & recurring decimals are rational numbers 2 — 0.666 11 1.571428 ... 7 2 0666 ... 3120 18 20 20 18 2 7 t.57t428.„ 7 40 50 40 to 7 30 20 4
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NUMBER SYSTEM Exercise 1.4 — Write the following in decimal form and say what kind of decimal expansion each has: (ii) (iv) (v) (vi) 36/100 1/11 3/13 2/11 329/400 8 Solutions given in separate annexure
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NUMBER SYSTEM CONVERSION OF DECIMALS Conversion of terminating decimals into the form m/n (i) Remove decimal point from the numerator and write 1 in the denominator and put as many zeros on the right hand side of 1 as the number of digits after the decimal point (ii) Convert the rational number obtained into its lowest term by dividing the numerator and denominator by the common factor EXAMPLE. Convert the decimal numbers into the form m/n (i) 0.35 (ii) 0.175 (iii) 0.0025 SOLUTION: 35 +5 (i) 0.35 = = (Dividing by the common factor 5) = 7/20 100 +5 175 +25 (ii) 0.175 = 175/1000 = (Dividing by the common factor 25) = 7/40 1000 +25 25 +25 (iii) 0.0025 = 25/10000 = (Dividing by the common factor 25) = 1/400 10000 +25
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NUMBER SYSTEM CONVERSION OF DECIMALS Types of non terminating recurring decimals Pure recurring decimals: A decimal number in which all the digits after the decimal point are repeated. For example, 0.4, 0.32 0.675 etc Mixed recurring decimals: A decimal number in which at least one digit after the decimal point is not repeated and others are repeated. For example, 0.72, 0.645 , 2.1275, etc
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NUMBER SYSTEM CONVERSION OF DECIMALS Conversion of non terminating pure recurring decimals (i) Put the given decimal number equal to x. (ii) Remove the bar if any and write the repeating digits at least twice. (iii) If the repeating decimal has 1 place repetition, multiply by 10, if there is 2 place repetition, multiply by 100 and so on (iv) Subtract the number in step (ii) from the number obtained in step (iii). (v) Divide both sides of the equation by the coefficient of x EXAMPLE 1. Express 0.7 in the form m/n . SOLUTION: (1) Let x = 0.7 then x = 0.777 There is only one repeating digit (7) after decimal point so we multiply both sides of (1) by 10. lox = 777 (2) Subtracting (1) from (2), we get 10x — x = (7.77....) - (0.77......)
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NUMBER SYSTEM CONVERSION OF DECIMALS Conversion of non terminating pure recurring decimals — Continued: EXAMPLE 2. Represent 0.23 in the form m/n . SOLUTION: (1) Let x = 0.23 then x = 0.232323 We have two repeating digits after decimal point, so we multiply both sides of (1) by 100. (2) 100 x: 23.23 23 Subtracting (1) from (2), we get 100 x — x = (23.2323....) - (0.2323.....) = 23 X = 23/99 o.n = 23/99
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NUMBER SYSTEM CONVERSION OF DECIMALS Conversion of non terminating pure recurring decimals — Short Cut Method: Remove the decimal & write down the given number as numerator In the denominator write down as many 9's equal to the number of repeating digits EXAMPLE: Represent 0.23 in the form m/n SOLUTION: Write 23 as numerator and 99 in the denominator (as two repeating digits) 0.23 = 23/99
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NUMBER SYSTEM CONVERSION OF DECIMALS Conversion of non terminating mixed recurring decimals Put the given decimal number equal to x 1. 2. Count the number of digits without bar after the decimal point. Let the number of digits without bar be n 3. Multiply both sides of the equation obtained in (1) by Ion EXAMPLE: Express 0.2562 in the form m/n SOLUTION. Let x = 0.2562... (1) There are two digits (25) without bar after decimal point in the given number. So we multiply both sides of (1) by 102 100x — 25.62 100x — 25+0.62 —25+92 1 oox +62 1 oox 99 2475+62 99 2537 1 oox 2537 X 9900 2537 0.2562 9900 = 100
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NUMBER SYSTEM REPRESENTING DECIMALS ON THE NUMBER LINE To represent a decimal on a number line, divide each segment of the number line into ten equal parts. E.g. To represent 8.4 on a number line, divide the segment between 8 and 9 into ten equal parts. 8.4 represented on a number line 8.0 The arrow is four parts to the right of 8 where it points at 8.4. 8.4 Likewise, to represent 8.45 on a number line, divide the segment between 8.4 and 8.5 into ten equal parts. 8.45 represented on a number line The arrow is five parts to the right of 8.4 where it points at 8.45. 8.43 8.3 Similarly, we can represent 8.456 on a number line by dividing the segment between 8.45 and 8.46 into ten equal parts. 8.456 represented on a number line The arrow is six parts to the right of 8.45 where it points at 8.456. 8.43 8.456 8.46
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NUMBER SYSTEM Classification of Numbers Rational Integers Real Numbers Irrational WChole } Natural
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