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NUMBER SYSTEM Real Number Rational or irrational numbers are called real number. Since all rational and irrational numbers can be represented on number line, that’s why number line is called real number line. The set of real numbers is denoted by R. Rational Numbers Rational numbers are those numbers, which can be expressed p , where p and q are integers and q ≠ 0 in the form q (Division by 0 is not defined). Every integer p is a rational p number because it can be written as . 1 If rational number is converted into decimal form, then (i) Either they are terminating decimal. (ii) Or they are non-terminating, but repeating decimals. Example 13 (i) 4 7 (ii) 3 = = Example (i) Convert 12 8 1.5 = (ii) Convert 17 4 2. 4.25 = Terminating Decimal To Rational Number 2.33333…… Example Rational number is said to be in its simplest form if its numerator and denominator have no common factor other p is a rational number, then it is in simplest that one. So, if q form if p and q have no common divisor other than 1. (i) Convert 6.4 into the form 6.4 = 64 in its simplest form. 80 4 16 = 5 20 8.25 (i) Convert Number 64 10 p . q 32 5 = (ii) Convert 8.25 into the form Example Rational p . q Steps 1. Remove decimal point from numerator. 2. Write 1 in denominator and put as many Zeros (0) as decimal places in the number. 3. Reduce the fraction to its simplest form by dividing the numerator and denominator by their common divisor. 3.25 Conversion Between Decimal Number 17 into decimal form. 4 We can convert a number in decimal form to the form Simplest Form Of A Rational Number 64 = 80 12 into decimal form. 8 And Rational number can be converted to decimal form. For this, we divide the numerator of rational number with its denominator till the quotient is up to desired decimal places. The conversion of rational number to decimal form can be categorized in the following ways: 1. Rational Number To Finite/Terminating Decimals The rational numbers for which the division terminates after a finite number of steps or with finite decimal part are known as finite/terminating decimal. = 825 100 = p . q 33 4 3. Rational Number To Repeating/Recurring Decimal Some times on dividing the numerator of a rational number with its denominator, the quotient starts repeating itself. Such decimals are called Repeating/Recurring decimal. Steps 1. Take the numerator of rational number as dividend and denominator as divisor. 2. Divide the numerator by denominator. 3. Divide the numerator by denominator till quotient starts repeating itself. Example (i) Convert 8 into decimal form. 3 8 = = 2.6666...... 2.6 Steps 3 1. Take the numerator of rational number as dividend and denominator as divisor. 16 2. Divide the numerator by denominator till the remainder is (ii) Convert into decimal form. 45 0 (Zero). 16 = = 0.3555..... 0 .3 5 45 4. Repeating Decimal To Rational Number To convert a terminating decimal to a rational number, the following steps are used. p and name the equation (i). q 1. Put the number equal to 2. Multiply the equation with that power of 10, so only repeating decimal remain on the left side of decimal. That means, if only 1 digit is repeating then multiply the equation by 10, if 2 digits are repeating then multiply the equation by 100, if three 3 digits are repeating then multiply the equation by 1000 and so on. After multiplying, name the equation (ii). Subtract equation (i) from equation (ii). Write the decimal in the simplest form. Example (i) Convert 0.35 into the form Let p . q ……………(i) Multiplying both sides by 100 p = 35.353535…… 100 q ……………(ii) Subtracting (i) from (ii) p p – = 35.353535…… – 0.353535…… 100 q q ⇒ Example (ii) Convert 36.5424242…… into the form Multiplying both sides by 10 p = 365.424242…… 10 q Multiplying (i) by 100 p = 36542.424242…… 1000 q ⇒ 990 p 35 = q 99 ⇒ Let p = 23.434343…… q Multiplying both sides by 100 p = 2343.434343…… 100 q p . q ⇒ ……………(i) ……………(ii) Subtracting (i) from (ii) p p – = 2343.434343…… – 23.434343…… 100 q q ⇒ 99 ⇒ p = 36.5424242…… q p . q ……………(i) ……………(ii) ……………(ii) Subtracting (ii) from (iii) p p – 10 = 36542.424242…… – 365.424242…… 1000 q q p = 35 q (ii) Convert 23.43 into the form Determine the number of digits after the decimal point which do not have bar on them. Let there be n digits without bar just after the decimal point. Multiply both sides of x by such power of 10, so that only repeating decimals are remaining on the right side of the decimal point. Use the method of converting pure recurring decimal to the form p/q and obtain the value of x. Note In case of repeating non-terminating with whole part zero and decimal part only with repeating digits only, simply remove the decimal and in denominator put as many 9 (nine) as there are repeating decimal digit. Let p = 0.353535…… q ⇒ 99 3. 4. Steps 3. 4. 5. 2. p = 2320 q p 2320 = q 99 p = 36177 q p 36177 = q 990 p 12059 = q 330 Irrational Number The number, which can not be written in the form p , where q p and q are integers and q ≠ 0 , is called irrational number. If irrational number is converted into decimal form, then it is neither terminating and nor repeating. Example (i) 2 (ii) 3 = 1.732050807........ = 1.414215........ p q While converting a recurring decimal that has one or more digit before the repeating digits, it is necessary to isolate the repeating digits. In order to convert a mixed recurring decimal to the form p/q, we follow the following steps: Surd Steps 1. Obtain the mixed recurring decimal and write it equal to any variable, say x . is a rational number and a1 / n is an irrational number. Here a is called radicand and n is called order of the surd. Conversion Of Mixed Recurring Decimal To From Note Because irrational numbers are non-terminating and nonrepeating in the decimal form, that is why some times we take approximate value of irrational number to solve the problem. If a is a rational number and n is a positive, so n a is surd, if a Every surd is a radical, but every radical is not a surd. In other word, if a is a real number and n is a positive integer then a is not a surd, if a is irrational or a1 / n is rational. Every surd is made up of two parts, i.e. rational part and irrational part. The part outside the radical sign is called rational part and the part under the radical sign is called irrational part. A surd which has rational factor other than one (unit) is called mixed surd. n Example (i) 6 3 5 is a surd with of order 3. Its rational part is 6 and its irrational part is 3 5 5 (ii) 20 12 is a surd with of order 5. Its rational part is 20 5 12 . Note 1. If the rational part of surd is 1, then it need not be mentioned. Example = 1 2 2. 2 If the order of surd is 2, then it need not be mentioned. Example 2 = 3 3 Types Of Surd Surds are divided into various categories depending upon their characteristics. 1. Like Surds Surds with same irrational factor and of same order are called like surds. Example (i) 2 (ii) 7 3 , 82 3 7 10 , 5 10 2. Unlike Surds Surds with different irrational factor or of different order are called unlike surds. Example (i) 62 2 , 62 5 (ii) 72 3 , 73 3 3. Pure Surd A surd which has one (unit) as its rational factor is called pure surd. Example (i) (ii) (iii) (i) 2 3 (ii) 53 11 (iii) 207 3 5. Depending Upon Number Of Terms . and its irrational part is Example Depending upon the terms, the surds are divided into various categories. (i) Monomial Surd A surd consisting of only one term is called monomial surd. Example (i) 2 3 (ii) 5 7 (iii) 2 + 5−2 6 = 2 + 3+2−2 6 = 2 + = 2 + = 2+ 3− 2 = 3 ( 3) − ( 2) ( 3 − 2) 2 2 − 2( 3 )( 2 ) 2 (ii) Binomial Surd An expression consisting of two terms, where both terms are monomial surds or one term is monomial surd and the other is a rational number is called monomial. In other words, either it is sum/difference of two monomial surds or sums/difference of a surd and rational number. Example (i) 5 3 − 2 7 (ii) 2 + 3 (iii) 4 3 − 6 5 (iii) Trinomial Surd An expression consisting of three terms, where all the tree terms are monomial surds or one term is monomial and other term is binomial surd is called trinomial surd. Example (i) 2+ 3+ 5 2 3 (ii) 3 4 7 4. Mixed Surd 7+ 2 + 3 (iv) Conjugate Surd Two binomial surds which differ only in the sign (+ or − ) between the terms are called conjugate surd. Example (i) 3 − 2 and 3+ 2 (ii) 4 2 + 3 5 If surds are unlike surds, then first convert them to like surds. and 4 2 − 3 5 Laws Of Surd Example There are few law which are used to simplify the mathematical operations, where surds are involved. These laws are as follows : 1. n n a = a n× n an = (an )1/n = a 2. 3. 4. n a ×nb n a n b mn = n = a ab a b mn a = n = 1 n mn a = a m 1/n a = (a1/n )1/m = a1/n × 1/m = a1/mn = mn a 7 Operation Between Surds While performing mathematical operations on surds, we have to follow various rules. 1. Surd Addition 1 3 3 1 3 6 − 5 − 3 (i) 6 3 − 5 3 − = 18 − 15 − 1 3 3 = = 2 3 3 (ii) 5 2 − 4 64 − 6 8 = = = = = = 5.21/2 − 26 ×1/4 − 23/6 5.21/2 − 23/2 − 21/2 5.21/2 − 23×1/2 − 21/2 5 2 − 23 − 2 5 2 −2 2 − 2 2 2 3. Multiplication Only same order surds can be multiplied. In such case, we multiply rational part with rational part and irrational part with irrational part. Only like surds can be added. In such case, we add rational part with rational part and write the irrational part as it is. If surds are of different order, first convert them to same order surds. If surds are unlike surds, then first convert them to like surds Example Example (i) = 1 6 3 +5 3 + 3 3 1 3 6 + 5 + 3 (i) 23 7 × 33 5 = 63 35 (ii) 32 2 × 53 3 × 6 4 4 12 6 = 3×5×6 34 3 3 = 90 12 26 × 26 × 34 = 9012 212 × 34 (ii) 5 2 + 4 64 + 6 8 = 90 × 2 = 5 2 + 26 + 23 = 1 5.2 2 + = 1 5.2 2 3 2 +2 = 18 + 15 + 1 3 3 6 4 1 ×6 24 1 = = = = 5.2 2 + 2 + 2 3× 1 6 1 + 22 3× 1 2 1 + 22 5 2 + 23 + 2 5 2 +2 2 + 2 = = = 12 43 ( ) 26 × 3 4 × 22 3 12 4 180 × 3 3 4× 1 12 1 180 × 3 3 180 3 3 Note Like surds and same order surds are different things. 1. In case of like surd, both the radicand and order of the surds must be same. Example 8 2 37 10 2. Surd Subtraction Only like surds can be subtracted. In such case, we subtract the rational part with rational part and write the irrational part as it is. × 5 12 34 × 6 3 = 2 12 = 2. and 87 10 . In case of same order surd, only the order of the surd must be same, but radicand may be different. Example 37 4 and (i) Rationalising factor of 87 6 4. Division Only same order surds can be divided. In such case, we divide rational part with rational part and irrational part with irrational part. 1 53 × = × If surds are of different order, first convert them to same order surds. = 1 53 × 53 = 5(1/3 + 2/3) Example = = 53/3 5 62 8 ÷ 22 4 = 32 2 5 is 3 52 , i.e 3 25 . 1 1− 5 3 1 53 (i) 3 3−1 5 3 2 2. Rationalising Factor Of Binomial Surd If binomial surd is the form a ± b (ii) 106 162 ÷ 23 3 factor is a m b . 6 = 106 162 ÷ 2 32 = 56 18 then its rationalizing (i) If binomial surd is in the form a + rationalising factor is a − b , then its b. (ii) If binomial surd is in the form a − Rationalization rationalising factor is a + b , then its b. Process of converting an irrational numerator/denominator into rational numerator/denominator is called rationalization. Surd in denominator of a fraction makes the fraction complicated. With rationalization, the surd in denominator is changed to a rational number which makes the term simpler. Note Simplest rationalising factor of binomial surd is its conjugate surd. Rationalizing Factor Example If two surds on multiplying with each other result into a rational number, then each one of them is called rationalizing factor of the other. Example (i) 2 and 2 × = = 8 are rationalizing factor of each other. ( 2 )2 − ( 3 )2 2 − 3 −1 7 − 4 is 7 + 4. ( 7 − 4) × ( 7 + 4) (ii) 3 5 and 5 are ratioanalizing factor of each other. 3 5 × 5 = 3× 5 = 15 3 ×3 9 = 3 = = = ( 7 )2 − ( 4 )2 7 − 4 3 3. Rationalising Factor Of Trinomial Surd 3 3 and 3 = = = = (ii) Rationalising factor of 4 3 ( 2 + 3) × ( 2 − 3) 8 16 (iii) (i) Rationalising factor of ( 2 + 3 ) is ( 2 − 3 ) . 9 are rationalizing factor of each other. 3 Treat the trinomial surd as binomial surds. For this, group any two terms and treat the grouped terms as one term and the remaining surd as second term. 3 × 32 If the trinomial surd is a ± b ± c , then after grouping the 1/3 terms ( a ± b ) ± 3 2×1/3 ×3 1/3 = 3 = 1 2 + 33 3 = 1+ 2 3 3 = 3 33 c. 2/3 ×3 So, rationalise factor of ( a m b) m 1. Rationalising Factor Of Monomial Surd Rationalising factor of monomial surd Example n 1− a is a 1 n . c a± b± c and m ab . Note In case of trinomial surd, even after rationalization once, a surd still exists. To remove that surd, we need to rationalize the expression again. So, in case of trinomial surd, we have to do the rationalization two times. Example (i) Rationalising factor of (4 + 2 6 ) . 3 − 2 − 1 are ( 3 − 2 ) + 1 and After grouping the terms ( 3 − 2 ) − 1 Now, [( ] [ ] 3 − 2) −1 × ( 3 − 2) + 1 2 ( 3 − 2 ) − (1) = = = (i) Remove irrational number from the denominator 4 2 2+ 3 3 + 2 − 2 6 −1 = 4−2 6 Rationalising again, (4 − 2 6 ) × (4 + 2 6 ) = (4)2 − (2 6 )2 = = 16 − 24 −8 (ii) Rationalizing factor of 7 + 6 + 13 are ( 7 + 6 ) - 13 and ] × [( 7 + 6 ) − 13 = ( 7 + 6 )2 − ( 13 )2 = = 7 + 6 + 2 42 − 13 42 = −4 2 − 3 (i) Remove the irrational number from the denominator 3 3 = 3( 7 + 4 ) ( 7 )2 − ( 4 )2 7+ 4 1 1 3 . = = 6 . 6 2 6 6 6 3 2. Binomial Surd As Denominator ( 3 − 2) + 1 ( 3 − 2 )2 − (1)2 3 − 2 +1 3 + 2 − 2 6 −1 3 − 2 +1 4-2 6 3 − 2 +1 ( ) 22- 6 Rationalising again, = 6 ( 3 − 2) + 1 ( 3 − 2) + 1 3 − 2 +1 2 × ( 3 − 2) − 1 2(2 − 6 ) (ii) Simplify = 7+ 4 3( 7 + 4) 3 3 3 × . 7+ 4 × = = 6 7− 4 3( 7 + 4) 7−4 1. Monomial Surd As Denominator 2 ) 3 = = = ( (ii) Simplify Note Same non-zero number can be multiplied by numerator and denominator of a term. This property is used in rationalization. × ) (i) Remove irrational number from denominator Irrational number as the denominator of a term/expression makes the term/expression very complicated. Using rationalization, we can convert the irrational number in the denominator to a rational number. Rationalization is used to simplify the term/expression, thus simplify whole calculation. 3 ( 3. Trinomial Surd As Denominator Advantage Of Rationalization 1 ) 4 2− 3 −1 = 84 ) = = 2 × 42 = = ( 3 2 42 Rationalising again, 2 42 × 4 2− 3 ( 2 )2 − ( 3 )2 7− 4 ] 2− 3 4 2− 3 2−3 42 . 7 + 6 ) + 13 ( . 2− 3 = After grouping the terms ( 7 + 6 ) + 13 Now, [( × 4 2+ 3 × (2 + 6 ) (2 + 6 ) 2 3 +3 2 −2 2 −2 3 +2+ 6 [ 2 (2)2 − ( 6 )2 = 2+ 2 + 6 2(4 − 6) = 2+ 2 + 6 −4 = −2− 2 − 6 4 (ii) Simplify 1 7 + 6 + 13 . ] 1 3 − 2 −1 . 1 ( 7 + 6 ) + 13 = = × ( 7 + 6 ) − 13 ( 7 + 6 ) − 13 ( 7 + 6 ) − 13 ( 7 + 6 )2 − ( 13 )2 7 + 6 − 13 2 42 Rationalizing again, 2 42 = ( × 7 + 6 − 13 ( 2 ) ( 42 ) = 7 6 + 6 7 − 546 84 Here are few laws of surds. These laws are used to simplify the mathematical operations, where real numbers are involved. 7 + 6 + 2 42 − 13 7 + 6 − 13 7 6 + 6 7 − 546 84 Laws Of Real Numbers 7 + 6 − 13 = = 42 If a is a real number such that a > 0 and p, q are rational numbers, then 1. ap .aq = ap + q 42 2. ap ÷ aq = ap − q 3. (a ) 4. ap .bp = ( ab ) )( 42 ) p q = apq p Operations Between Rational And Irrational Numbers The result of a mathematical operation may be rational or irrational depending on the type of values used in the operation. Operand Operand Rational Rational Irrational Irrational Rational Irrational + - Operation × Rational 7 × 4 = 28 2 × 11 = 22 Rational/Irrational ÷ Rational 7 + 5 = 12 20 + 14 = 34 Rational/Irrational Rational 5-3=2 9-3=6 Rational/Irrational (2 + 3 ) + (2 − 3 ) = 4 (6 + 3 ) − (2 + 3 ) = 4 2 × 8 =4 8 × 2 =2 (2 + 3 ) + ( −2 + 3 ) = 2 3 (9 + 3 ) − (9 − 3 ) = 2 3 2 × 3= 6 10 × 2 = 5 Irrational Irrational Irrational Rational 8÷2=4 42 ÷ 14 = 3 Rational/Irrational Irrational 4 + 6 =4 + 6 6 − 2 =6 − 2 4 × 3 =4 3 10 ÷ 3 = 10 ÷ 3 5 + 2 =5 + 2 10 − 4 3 = 10 − 4 3 5 × 2 7 = 10 7 Note Rational ≠ 0 8 ÷ 4= 8 ÷ 4 Note Rational ≠ 0