01 Rational Number
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Transcript of 01 Rational Number
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8/14/2019 01 Rational Number
1/20
T MPAFKMI K \ODLTR =
=+= Akptf`|bpafk
Ak Ompjlompabr) xl ctl}|lkpiq bfol mbtfrr rao~il l}|mpafkr pf dl rfisl`+ Cft lzmo~il)pjl l}|mpafk z % 4 ? =2 *=!
ar rfisl` xjlk z ? ==) dlbm|rl pjar smi|l fc z rmparcalr pjl haslk l}|mpafk+ Pjl rfi|pafk== ar mkmp|tmi k|odlt + Fk pjl fpjlt jmk`) cft pjl l}|mpafk
z % < ? < *4!
pjl rfi|pafk haslr pjl xjfil k|odlt > *wltf!+ Ac xl bfkra`lt fkiq kmp|tmi k|odltr)l}|mpafk *4! bmkkfp dl rfisl`+ Pf rfisl l}|mpafkr iael *4!) xl m``l` pjl k|odlt wltf pfpjl bfiilbpafk fc kmp|tmi k|odltr mk` fdpmakl` pjl xjfil k|odltr+ Lslk xjfil k|odltrxaii kfp dl r|ccabalkp pf rfisl l}|mpafkr fc pq~l
z % =: ? < *2!
@f qf| rll xjq6 Xl tl}|atl pjl k|odlt =2 xjabj ar kfp m xjfil k|odlt+ Pjaril` |r pf pjake fc akplhltr) *~frapasl mk` klhmpasl! + Kfpl pjmp pjl ~frapasl akplhltrbfttlr~fk` pf kmp|tmi k|odltr+ Fkl omq pjake pjmp xl jmsl lkf|hj k|odltr pf rfisl miirao~il l}|mpafkr xapj pjl msmaimdil iarp fc akplhltr+ Bfkra`lt pjl l}|mpafkr
4 z ? 2 *0!
< z % 8 ? > *
cft xjabj xl bmkkfp cak m rfi|pafk ctfo pjl akplhltr+ *Bjlbe pjar!
Xl kll pjl k|odltr24
pf rfisl l}|mpafk *0! mk`8
% < ? 2 ? >) m xjfil k|odlt Xjfil k|odltr mtl bifrl`2 8 ? +++ + Ar ap m xjfil k|odlt6 |k lt o|ipa~iabmpafk+Ak hlkltmi) ac mmk`d mtl mkq pxfxjfil k|odltr) pjlat ~tf`|bp mdar m xjfil k|odlt+
@asarafk < : ?) mk akplhlt Akplhltr mtl bifrl` |k`ltAr < : mk akplhlt6 o|ipa~iabmpafk+ < * :! ? 0>) mk akplhltAk hlkltmi) cft mkq pxf akplhltrmmk` d) md ar mirf mk akplhlt+
@asarafk < : ? ar bmiil` m tmpafkmi k|odlt + Cft lzmo~il)42
)38
mtl mii tmpafkmi
k|odltr+ Rakbl pjl k|odltr >) 4) 0 bmk dl xtapplk ak pjl cfto ~} ) pjlq mtl mirf
tmpafkmi k|odltr+ *Bjlbe ap-!
*m! Qf| ekfx jfx pf m`` pxf tmpafkmi k|odltr+ Ilp |r m`` m clx ~matr+2 *
% ?4= * 0>! =1
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8/14/2019 01 Rational Number
4/20
0 O MPJLOMPABR
PTQ PJLRL
< 0: < ?
4< 240>
? +++ Ar ap m tmpafkmi k|odlt6
2 :8 =>
% % ? % ? ?
4 2 < = < 48 12 < 3 =< 3 2> =>
% % ? % ? ?
Rf)4 2 < 4 2 8> 2:
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8/14/2019 01 Rational Number
9/20
T MPAFKMI K \ODLTR 1
PTQ PJLRL
* ! @asarafk
Ilp |r rll ac = = 4 = = 4
4 2 < 4 4 ) rf xl rmq 4 ar pjl klhmpasl ft m``apasl aksltrl fc
4 mk` sabl/sltrm+ Ak hlkltmi) cft mk akplhlt m) xl jmsl) m% *m! ? *m! %m? >5 rf) mar pjl klhmpasl fc mmk` mar pjl klhmpasl fc m+
Cft pjl tmpafkmi k|odlt42
) xl jmsl)
4 42 2
% ?
4 * 4!>
2
% ?
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8/14/2019 01 Rational Number
12/20
=4 O MPJLOMPABR
Mirf)4 42 2
% ? > *Jfx6!
Raoaimtiq) : +++1 % ? :+++ >
1
% ?
==+++
8
% ?
==+++ >
8
% ?
Ak hlkltmi) cft m tmpafkmi k|odltmd
) xl jmsl) >m m m md d d d
% ? % ? + Xl rmq
pjmpmd
ar pjl m``apasl aksltrl fc md
mk` md
ar pjl m``apasl aksltrl fc md
+
=+4+8 Tlba~tfbmi
Dq xjabj tmpafkmi k|odlt xf|i` qf| o|ipa~iq:4=
) pf hlp pjl ~tf`|bp =6 Fdsaf|riq dq
4= : 4=) rakbl =
: 4= : ? +
Raoaimtiq) mk` = akpf pjtll l}|mi ~mtpr bmk dl imdliil`=2
) mr fk k|odlt iakl *s!+ Jfx xf|i` qf|
imdli pjl rlbfk` fc pjlrl asarafk ~fakpr fk k|odlt iakl *s!6
Pjl iakl lzplk`rak`lcakapliq fkiq pf pjl
tahjp ra`l fc =+
Pjl iakl lzplk`r ak`lcakapliqpf pjl tahjp) d|p ctfo >+
Pjltl mtl kf k|odltr pf pjlilcp fc >+
Pjl iakl lzplk`rak`lcakapliq fk dfpj ra`lr+@f qf| rll mkq k|odltrdlpxllk =) >5 >) = lpb+6
Pjl iakl lzplk`r ak`lcakapliqfk dfpj ra`lr+ D|p qf| bmkkfx rll k|odltr dlpxllk
=) >5 >) = lpb+
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8/14/2019 01 Rational Number
16/20
=3 O MPJLOMPABR
Pjl ~fakp pf dl imdliil` ar pxabl mr cmt ctfo mk` pf pjl tahjp fc > mr pjl ~fakp
imdliil`=2
+ Rf ap ar pxf paolr=2
) a+l+)42
+ Qf| bmk bfkpak|l pf imdli l}|miiq/r~mbl ~fakpr fk
pjl k|odlt iakl ak pjl rmol xmq+ Pjl klzp omteakh ar =+ Qf| bmk rll pjmp = ar pjl rmol mr 22
+
Pjlk bfolr0 < 3
) )2 2 2
*ft 4!)82
mk` rf fk mr rjfxk fk pjl k|odlt iakl *sa!
*sa!
Raoaimtiq) pf tl~tlrlkp=:
) pjl k|odlt iakl omq dl asa`l` akpf lahjp l}|mi ~mtpr mr
rjfxk7
Xl |rl pjl k|odlt=:
pf kmol pjl catrp ~fakp fc pjar asarafk+ Pjl rlbfk` ~fakp fc
`asarafk xaii dl imdliil`4:
) pjl pjat` ~fakp2:
) mk` rf fk mr rjfxk fk k|odlt
iakl *saa!
*saa!
Mkq tmpafkmi k|odlt bmk dl tl~tlrlkpl` fk pjl k|odlt iakl ak pjar xmq+ Ak m tmpafkmik|odlt) pjl k|oltmi dlifx pjl dmt) a+l+) pjl lkfoakmpft) pliir pjl k|odlt fc l}|mi~mtpr akpf xjabj pjl catrp |kap jmr dllk asa`l`+ Pjl k|oltmi mdfsl pjl dmt a+l+) pjlk|oltmpft) pliir jfx omkq fc pjlrl ~mtpr mtl bfkra`ltl`+ Rf) m tmpafkmi k|odlt
r|bj mr01
olmkr cf|t fc kakl l}|mi ~mtpr fk pjl tahjp fc > *k|odlt iakl saaa! mk`
cft8
0
) xl omel 8 omteakhr fc arpmkbl=0
lmbj fk pjl ilcp fc wltf mk` rpmtpakh
ctfo >+ Pjl rlslkpj omteakh ar 80 Uk|odlt iakl *az!_+
*saaa!
*az!
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8/14/2019 01 Rational Number
17/20
T MPAFKMI K \ODLTR =8
PTQ PJLRL
Xtapl pjl tmpafkmi k|odlt cft lmbj ~fakp imdliil` xapj m ilpplt+
*a!
*aa!
=+0 Tmpafkmi K|odltr dlpxllk Pxf Tmpafkmi K|odltrBmk qf| plii pjl kmp|tmi k|odltr dlpxllk = mk` mk` ==6 Fdsaf|riq kfkl+Iarp pjl akplhltr pjmp ial dlpxllk < mk` 0+ Pjlq mtl 0) 2) 4) =) >) =) 4) 2+Jfx omkq akplhltr mtl pjltl dlpxllk = mk` =6Jfx omkq akplhltr mtl pjltl dlpxllk 1 mk` =>6
Qf| xaii cak` m lcakapl k|odlt fc kmp|tmi k|odltr *akplhltr! dlpxllk pxf kmp|tmik|odltr *akplhltr!+
Jfx omkq tmpafkmi k|odltr mtl pjltl dlpxllk2
=>mk`
8=>
6
Qf| omq jmsl pjf|hjp pjmp pjlq mtl fkiq0 =>
mk`3
=>+
D|p qf| bmk mirf xtapl2
=>mr
2>=>>
mk`8
=>mr
8>=>>
+ Kfx pjl k|odltr)2= 24 22) )
=>> =>> =>>3: 31
) +++ )=>> =>>
) mtl mii dlpxllk2
=>mk`
8=>
+ Pjl k|odlt fc pjlrl tmpafkmi k|odltr ar 21+
Mirf2
=>bmk dl lz~tlrrl` mr
2>>>=>>>>
mk`8
=>mr
8>>>=>>>>
+ Kfx) xl rll pjmp pjl
tmpafkmi k|odltr2>>= 2>>4 311: 3111
) )+++) )=>>>> =>>>> =>>>> =>>>>
mtl dlpxllk2
=>mk`
8=>
+ Pjlrl
mtl 2111 k|odltr ak mii+
Ak pjar xmq) xl bmk hf fk akrltpakh oftl mk` oftl tmpafkmi k|odltr dlpxllk 2=>
mk`8
=>+ Rf |kiael kmp|tmi k|odltr mk` akplhltr) pjl k|odlt fc tmpafkmi k|odltr dlpxllk
pxf tmpafkmi k|odltr ar kfp lcakapl+ Jltl ar fkl oftl lzmo~il+
Jfx omkq tmpafkmi k|odltr mtl pjltl dlpxllk=
=>
mk`2
=>6
Fdsaf|riq> = 4
) )=> => =>
mtl tmpafkmi k|odltr dlpxllk pjl haslk k|odltr+
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8/14/2019 01 Rational Number
18/20
=: O MPJLOMPABR
Ac xl xtapl=
=>
mr=>>>>
=>>>>>
mk`2
=>mr
2>>>>=>>>>>
) xl hlp pjl tmpafkmi k|odltr
1111 111:) )+++)=>>>>> =>>>>> 4111:
=>>>>> ) 41111
=>>>>>) dlpxllk =
=> mk` 2
=>+
Qf| xaii cak` pjmp qf| hlp bf|kpilrr tmpafkmi k|odltr dlpxllk mkq pxf haslktmpafkmi k|odltr +Lzmo~il 37 Xtapl mkq 2 tmpafkmi k|odltr dlpxllk 4 mk` >+
Rfi|pafk7 4 bmk dl xtapplk mr4>
=>
mk` > mr>
=>+
Pj|r xl jmsl=1 =: =8 =3 =< =
) ) ) ) ) +++ )=> => => => => =>
dlpxllk 4 mk` >+
Qf| bmk pmel mkq pjtll fc pjlrl+
Lzmo~il 87 Cak` mkq plk tmpafkmi k|odltr dlpxllk
