NUMBER SYSTEMS CHAPTER 1. (A) Main Concepts and Results


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1 CHAPTER NUMBER SYSTEMS Min Concepts nd Results Rtionl numbers Irrtionl numbers Locting irrtionl numbers on the number line Rel numbers nd their deciml expnsions Representing rel numbers on the number line Opertions on rel numbers Rtionlistion of denomintor Lws of exponents for rel numbers A number is clled rtionl number, if it cn be written in the form p q, where p nd q re integers nd q 0. A number which cnnot be expressed in the form p q (where p nd q re integers nd q 0) is clled n irrtionl number. All rtionl numbers nd ll irrtionl numbers together mke the collection of rel numbers. Deciml expnsion of rtionl number is either terminting or nonterminting recurring, while the deciml expnsion of n irrtionl number is nonterminting nonrecurring
2 EXEMPLAR PROBLEMS If r is rtionl number nd s is n irrtionl number, then r+s nd rs re irrtionls. Further, if r is nonzero rtionl, then rs nd r re irrtionls. s For positive rel numbers nd b : (i) b = b = b ( + b )( b ) = b ( + )( ) (v) ( ) + b = + b + b b b b = b If p nd q re rtionl numbers nd is positive rel number, then (i) p. q = p + q ( p ) q = pq p q p q = p b p = (b) p Multiple Choice Questions Write the correct nswer: Smple Question : Which of the following is not equl to 5 5 Solution : Answer 5 5 (C) (D)? 5 0 EXERCISE. Write the correct nswer in ech of the following:. Every rtionl number is nturl number n integer (C) rel number (D) whole number 90504
3 NUMBER SYSTEMS. Between two rtionl numbers there is no rtionl number there is exctly one rtionl number (C) there re infinitely mny rtionl numbers (D) there re only rtionl numbers nd no irrtionl numbers. Deciml representtion of rtionl number cnnot be terminting nonterminting (C) nonterminting repeting (D) nonterminting nonrepeting 4. The product of ny two irrtionl numbers is lwys n irrtionl number lwys rtionl number (C) lwys n integer (D) sometimes rtionl, sometimes irrtionl 5. The deciml expnsion of the number is finite deciml.44 (C) nonterminting recurring (D) nonterminting nonrecurring. Which of the following is irrtionl? Which of the following is irrtionl? (C) 7 (D) (C) 0.4 (D) A rtionl number between nd is + (C).5 (D)
4 4 EXEMPLAR PROBLEMS 9. The vlue of in the form p q, where p nd q re integers nd q 0, is (C) (D) is equl to (C) (D) is equl to 5 5 (C) 5 (D) 0 5. The number obtined on rtionlising the denomintor of 7 is (C) (D) is equl to ( ) + (C) (D) + 4. After rtionlising the denomintor of 7, we get the denomintor s 9 (C) 5 (D) 5 5. The vlue of is equl to (C) 4 (D) 8. If =.44, then + is equl to 90504
5 NUMBER SYSTEMS (C) 0.44 (D) equls (C) 8. The product 4 equls (D) (C) (D) 9. Vlue of ( ) 4 8 is 9 0. Vlue of (5) 0. (5) 0.09 is (C) 9 (D) 4 (C) 4 (D) 5.5. Which of the following is equl to x? 8 x x ( ) 4 x (C) ( ) x (D) x 7 7 x (C) Short Answer Questions with Resoning Smple Question : Are there two irrtionl numbers whose sum nd product both re rtionls? Justify. Solution : Yes. + nd re two irrtionl numbers. ( ) ( ) + + =, rtionl number. ( + ) ( ) = 7, rtionl number. So, we hve two irrtionl numbers whose sum nd product both re rtionls. Smple Question : Stte whether the following sttement is true: There is number x such tht x is irrtionl but x 4 is rtionl. Justify your nswer by n exmple
6 EXEMPLAR PROBLEMS Solution : True. Let us tke x = 4 Now, x = ( 4 ) =, n irrtionl number. x 4 = ( ) 4 4 =, rtionl number. So, we hve number x such tht x is irrtionl but x 4 is rtionl. EXERCISE.. Let x nd y be rtionl nd irrtionl numbers, respectively. Is x + y necessrily n irrtionl number? Give n exmple in support of your nswer.. Let x be rtionl nd y be irrtionl. Is xy necessrily irrtionl? Justify your nswer by n exmple.. Stte whether the following sttements re true or flse? Justify your nswer. (i) is rtionl number. There re infinitely mny integers between ny two integers. Number of rtionl numbers between 5 nd 8 is finite. There re numbers which cnnot be written in the form p q, q 0, p, q both (v) (vi) re integers. The squre of n irrtionl number is lwys rtionl. is not rtionl number s nd re not integers. 5 (vii) is written in the form p, q 0 nd so it is rtionl number. q 4. Clssify the following numbers s rtionl or irrtionl with justifiction : (i)
7 NUMBER SYSTEMS 7 (v) 0.4 (vi) 75 (vii) (viii) ( + 5 ) ( ) (ix) (x) (D) Short Answer Questions Smple Question : Locte on the number line. Solution : We write s the sum of the squres of two nturl numbers : = = + On the number line, tke OA = units. Drw BA = units, perpendiculr to OA. Join OB (see Fig..). By Pythgors theorem, OB = Using compss with centre O nd rdius OB, drw n rc which intersects the number line t the point C. Then, C corresponds to. Remrk : We cn lso tke OA = units nd AB = units. Fig.. Smple Question : Express 0. p in the form, where p nd q re integers nd q q 0. Solution : Let x = 0. so, 0x =. or 0x x =. 0. = or 9x =. or x =. =
8 8 EXEMPLAR PROBLEMS Therefore, 0. 7 = = Smple Question : Simplify : ( 5 5 ) ( ). Solution : ( 5 5 )( ) = = = 0 0 Smple Question 4 : Find the vlue of in the following : = Solution : + = + = ( ) ( ) ( ) ( ) ( ) = = 8 = + Therefore, + = or = Smple Question 5: Simplify : 5 ( ) Solution : ( ) = ( 5 ( ) + ( ) )
9 NUMBER SYSTEMS 9 = 5( + ) 4 = 5 ( 5) = [ ] = 5 4 EXERCISE.. Find which of the vribles x, y, z nd u represent rtionl numbers nd which irrtionl numbers: (i) x = 5 y = 9 z =.04. Find three rtionl numbers between (i) nd 0. nd 0. 5 nd 7 7 nd 4 5. Insert rtionl number nd n irrtionl number between the following : (i) nd 0 nd 0. nd 5 nd (v) 0.5 nd 0. (vi) nd 7 u = 4 (vii).57 nd. (viii).000 nd.00 (ix). nd (x).7589 nd Represent the following numbers on the number line : 7, 7.,, 5 5. Locte 5, 0 nd 7 on the number line.. Represent geometriclly the following numbers on the number line : (i)
10 0 EXEMPLAR PROBLEMS 7. Express the following in the form p q, where p nd q re integers nd q 0 : (i) (v) (vi) 0.4 (vii) (viii) Show tht = 7 9. Simplify the following: (i) (v) (vi) ( ) (vii) (viii) + 8 (ix) 0. Rtionlise the denomintor of the following: (i) (v) + (vi) (vii) (viii) 5. Find the vlues of nd b in ech of the following: (ix) (i) 5+ =
11 NUMBER SYSTEMS 5 9 = = b = + 5b If = +, then find the vlue of.. Rtionlise the denomintor in ech of the following nd hence evlute by tking =.44, =.7 nd 5 =., upto three plces of deciml. (i) Simplify : + (v) + (i) ( ) ( 5) 4 (v) 9 7 (vi) (vii)
12 EXEMPLAR PROBLEMS (E) Long Answer Questions Smple Question : If = 5 + nd + b? b =, then wht will be the vlue of Solution : = 5 + b = = 5 + Therefore, 5 5 = ( ) = + b = ( + b) b Here, + b = (5 + ) + (5 ) = 0 = 5 = b = (5 + ) (5 ) = 5 ( ) = 5 4 = Therefore, + b = 0 = 00 = 98 EXERCISE.4. Express in the form p q, where p nd q re integers nd q 0.. Simplify : If =.44, =.7, then find the vlue of 4. If = If x = nd y =. Simplify : ( ) ( ) 4 5, then find the vlue of , then find the vlue of x + y Find the vlue of ( ) ( 5 ) ( 4 )
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2 Rtionl Numbers Integers such s 5 were importnt when solving the eqution x+5 = 0. In similr wy, frctions re importnt for solving equtions like 2x = 1. Wht bout equtions like 2x + 1 = 0? Equtions of this
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