5. Every rational number have either terminating or repeating (recurring) decimal representation.

Transcription

1 CHAPTER NUMBER SYSTEMS Points to Rememer :. Numer used for ounting,,,,... re known s Nturl numers.. All nturl numers together with zero i.e. 0,,,,,... re known s whole numers.. All nturl numers, zero nd negtive numers together i.e....,,,,, 0,,,,,... re known s Integers. p. Rtionl Numers : Numers of the form where p, q oth re integers nd q 0. For e.g.,, q et.. Every rtionl numer hve either terminting or repeting (reurring) deiml representtion. Terminting Repeting (Reurring) For eg. 0.,. et. For e.g here, prime ftors of denomintor re = nd only.. 8 et. 6. There re infinitely mny rtionl numers etween ny two given rtionl numers.. Irrtionl Numers : Numers whih nnot e written in the form of q p, where p, q re integers nd q 0. For e.g.,,,, , 9 et. 8. Rel numers : Colletion of oth rtionl nd irrtionl numers. For e.g.,, 0,,, et. 9. Every rel numer is represented y unique point on the numer line. Also, every point on the numer line represents unique rel numer. 0. For every given positive rel numer, we n find geometrilly.. Identities relted to squre root : Let p, q e positive rel numers. Then, (i) p p pq p. q ; q 0 q q (iii) ( p q ) ( p q ) p q (iv) ( p q ) p pq q. Lws of Rdils : Let, y > 0 e rel numers nd p, q e rtionls. then (i) p q = p+q p q pq p q pq (iii) ( ) p p p (iv).y ( y) MATHEMATICS IX NUMBER SYSTEMS

2 ILLUSTRATIVE EXAMPLES Emple. Find si rtionl numers etween nd. NCERT. Solution. We know tht etween two rtionl numers nd, suh tht <, there is rtionl numer. A rtionl numer etween nd is ( ). 6 Now, rtionl numer etween nd is. Emple. Solution. 8 A rtionl numer etween nd is. Also, rtionl numer etween nd is 8 6 A rtionl numer etween nd is 8 A rtionl numer etween nd is This n e represented on numer line s follows : (without using formul) NUMBER SYSTEMS MATHEMATICS IX OR We hve, nd We need to find si rtionl numers etween nd i.e nd, whih re,,,,,. Ans Find four rtionl numers etween 6 nd nd. 6 6 rtionl numers etween nd i.e. nd re,,,.

3 p Emple. Epress 0. in the form of rtionl numer,. q Solution. Let () multiplying oth sides y 00, we get 00 =......() Sutrting () from (), 00 = = 99 Ans. Emple. Represent on the numer line. Solution. Emple. Let XOX e numer line with O s origin. Let OA = unit. Drw AB OA suh tht AB = unit. Join OB. Then using Pythgors theorem, In OAB OB OA AB units Agin, drw DC OB suh tht BC = unit. Join OC. then, OC OB BC ( ) () units. With O s entre nd OC s rdius, drw on r, meeting OX t P. Then OC = OP = units. Stte whether the following sttements re true or flse. Justify your nswers. (i) Every irrtionl numer is rel numer. Every point on the numer line is of the form m, where m is nturl numer. (iii) Every rel numer is n irrtionl numer NCERT Solution. (i) True, sine olletion of rel numers is mde up of rtionl nd irrtionl numers. Flse, no negtive numer n e the squre root of ny nturl numer. (iii) Flse, for emple is rel ut not irrtionl. Emple 6. Write the following in deiml form nd sy wht kind of deiml epnsion eh s: 6 9 (i) (iii) (iv) (v) (vi) NCERT Solution. (i) 0. 6 Terminting deiml epnsion. 00 Consider, y long division, we hve : MATHEMATICS IX NUMBER SYSTEMS

4 , whih is non-terminting nd repeting deiml epnsion. 8 (iii). By long division, we hve , 8 terminting deiml epnsion. (iv) Consider, y long division, we hve , whih is non-terminting nd repeting deiml epnsion. NUMBER SYSTEMS MATHEMATICS IX

5 Emple. (v) Consider,, y long division, we hve , whih is non-terminting nd repeting deiml epnsion. 9 (vi) Consider,, y long division, we hve , 00 whih is terminting deiml epnsion. Wht n e the mimum numer of digits e in the repeting lok of digits in the deiml epnsion of? Perform the division to hek your nswer. NCERT MATHEMATICS IX NUMBER SYSTEMS

6 Solution Thus, The mimum numer of digits in the quotient while omputing re. p Emple 8. Look t severl emples of rtionl numers in the form ( q 0), where p nd q re integers q with no ommon ftors other thn nd hving terminting deiml representtions (epnsions). Cn you guess wht property q must stisfy? NCERT Solution. Let us onsider vrious suh rottionl numers hving terminting deiml representtion. 6 NUMBER SYSTEMS MATHEMATICS IX

7 0. ; 0. ; ; 0.088; 0. et. 00 from the emples shown ove, it n e esily oserve tht, If the denomintor of rtionl numer in stndrd form hs no prime ftors other thn or or oth, then nd the only then it n e represented s terminting deiml. Emple 9. Visulise.6 on the numer line, using suessive mgnifition. NCERT Solution. We know tht.6 lies etween nd. We divide portion of numer line etween nd in 0 equl prts i.e..,.,...,.9 nd then look t the intervl [.,.8] through mgnifying glss nd oserve tht.6 lies etween. nd.8 (see figure). Now, we imgine tht eh new intervls [.,.], [.,.],..., [.9, ] hve een su-divided into 0 equl prts. As efore, we n now visulize through the mgnifying glss tht.6 lies in the intervl [.6,.]. (see figure). Agin,.6 lies etween.6 nd.. So, let us fous on this portion of the numer line, nd imgine to divide it gin into 0 equl prts. The first mrk represents.6, seond mrk represents.6, nd so on. So,.6 is the th mrk in these sudivisions. Emple 0. Rell, is defined s the rtio of the irumferene (sy ) of irle to its dimeter (sy d). Tht is,. This seem to ontrdit the ft tht is irrtionl. How will you resolve this d ontrdition? NCERT Solution. There is no ontrdition. Rememer tht when you mesure length with sle or ny other devie, you only get on pproimte rtionl vlue. So, you my not relise tht either or d is irrtionl. Emple. Simplify the following : (i) ( ) ( ) (iii) ( ) (iv) 8 0 MATHEMATICS IX NUMBER SYSTEMS

8 Solution. (i) ( ) 8 ( ) ( ) (iii) ( ) ( ) ( )( ) (iv) 6 Emple. Find. geometrilly. Solution. Drw AB =. units nd etend it to C suh tht BC = unit. Find mid-point O of AC. With O s entre, nd OA s rdius, drw semiirle. Drw BD AC, interesting semiirle t D. Then BD =. units. With B s entre nd BD s rdius, drw n r, interseting AC produed t E. Then, BE BD. units. Emple. Find vlue of nd, where. Solution. We hve, ( ) () ( ) 6 6 nd Emple. Simplify the following : (i) (iii) ( / (iv) () Solution. (i) (iii) (iv) ( ) ( ) NUMBER SYSTEMS MATHEMATICS IX )

9 Emple. Simplify : Solution. Given epression PRACTICE EXERCISE.. Represent eh of the following rtionl numers on the numer line : (i). (i) Find rtionl numers etween nd. Find rtionl numers etween nd.. (i) Represent nd on the some numer line (iii).6 (iv). Represent on the numer line.. Without tul division, find whih of the following rtionls re terminting deimls. (i) 9. Find the deiml epnsions of the following : 6 (i) 9 6 (iii) (iii) 0 (iv) 80 (iv) (v) Find the deiml representtion of. Dedue from the deiml representtion of, without tul lultions, the deiml representtion of,, nd. p. Epress eh of the following reurring deimls in the form of rtionl numer, : q (i) (iii) 0. 6 (iv).6 MATHEMATICS IX NUMBER SYSTEMS 9

10 8. (i) Find three irrtionl numers etween nd. Insert three irrtionl numers etween nd. 9. Give n emple of two irrtionl numers whose : (i) differene is rtionl numer. differene is n irrtionl numer. (iii) sum is rtionl numer. (iv) sum is n irrtionl numer. (v) quotient is rtionl numer. (vi) quotient is n irrtionl numer. (vii) Produt is rtionl numer. (viii) Produt is n irrtionl numer. 0. Simplify eh of the following : (i) ( ) ( ) ( ) (iii) 0 (iv) ( ) (v) ( ) ( ) (vii) 0 8. Represent the following on the numer line : (i).. (iii) 6. 8 (iv) 9.. Rtionlize the denomintor : (i). Rtionlize the denomintor: (i). Simplify the following : (i) 6 0 NUMBER SYSTEMS MATHEMATICS IX (iii) (iii). If nd re rtionl numers nd if, find nd. 6. If nd y re the rtionl numers nd y, find nd y.. Evlute the following : (iv) (iv) (i) 8 (iii) / ( ) (iv) (6) / (v) ( 8) (vi)

11 8. Find the vlue of if : (i) (iii) 6 6 (iv) 9 9. Find whether the produt of irrtionl numers ( ), ( ), ( ) nd ( ) is rtionl or irrtionl numer. 0. (i) Given., find vlue of., find vlue of.. (i) Prove tht is not rtionl numer. Prove tht is n irrtionl numer.. Simplify the following : (i) (iii).. 6 (iv). 6.. Simplify the following : n n (i) n n. Assuming tht is positive rel numer nd,, re rtionl numers, show tht : (i). If y z 6, show tht 0 y z 6. (i) If, find the vlue of. If 6, find the vlue of. Simplify the following : (i) 8. Rtionlize the denomintor of following :. y y z z 6 (i) 6 6 MATHEMATICS IX NUMBER SYSTEMS

12 9. Prove tht : (i) Represent the following on the numer line : M.M : 0 (i) (iii) (iv) Generl Instrutions : PRACTICE TEST Q. - rry mrks, Q. -8 rry mrks nd Q. 9-0 rry mrks eh.. Find three rtionl numers etween. Represent on the numer line.. Rtionlise the denomintor : 6. Find deiml representtion of.. Simplify the following : nd. (i) ( ) ( )( ) 6. If nd re rtionl numers nd, find nd.. Evlute the following : (i) 8. Epress 0. 6 s rtionl numer in the simplest form. Time : hour NUMBER SYSTEMS MATHEMATICS IX / 9. Represent. on the numer line. Also, give step of onstrutions Simplify : 6 6

13 . (i) ANSWERS OF PRACTICE EXERCISE 6 0,,,,,,,., (iv) nd (v). (i).. 8 (iii) 0. 6 (iv) , 0. 8, 0. 8, 0. 8, (i) (iii) (iv) (i) , nd nd (i) nd nd (iii) nd (iv) nd (v) 0 nd (vi) 0 nd 6 (vii) nd (viii) nd 0. (i) 6 8 (iii) 8 (iv) 0 6 (v) (vi) 0. (i) ( ) (iii) (iv) ( ). (i) ( ) ( 6 ) (iii) ( ) (iv) ( ) (i)., 6. =, y = 0. (i) (iii) (iv) (v) 8 6 (vi) 8. (i) 8 (iii) (iv) 9. Rtionl 0. (i) (i) 90. (i) 6. (i) 6. (i) (i) ( 6) ( ). 6. ANSWERS OF PRACTICE TEST 0,, (i) ,. (i) (iii) (iv) MATHEMATICS IX NUMBER SYSTEMS