01. Rational Numbers

8/12/2019 01. Rational Numbers

1/26

hmijsgkuhmrpgysjls.ji

=LGMSQE

^ QJBI @ IUH E^\

^mtjbim` IuhnersAedjijtjbi 6

M iuhnern

mjs m rmtjbim` iuhner jd m mia n

mre jiteoers mia n js ibt equm` tb zerb. nlmiibt ne equm` tb zerb nelmuse ajvjsjbi ny zerbjs ibt m``bwea. Durtger, m rmtjbim` iuhner js smjatb ne ji tge stmiamra dbrh br sjhp`est dbrh wgeitge iuhermtbr mia aeibhjimtbr gmve ib lbhhbidmltbr btger tgmi =.

Braer bd ^mtjbim` IuhnersEx.= Mrrmioe tge db``bwjio drmltjbis ji msleiajio

braer.

7

1,

=4

?,

.

\b`. @LH bd aeibhjimtbrs

7, =>, =4, mia < 8 > > > > < 8 17.

Qgei7

> > < 8 >1

Ibw we wj`` djia equjvm`eit drmltjbis bd tge

ojvei iuhners gmvjio >1 ji tge aeibhjimtbr.

Geile,=>

58

>=>

>5

8

>1

=3

mia7

?8


3/26


Ex.:4

5+

4

58 3.

4

5js tge maajtjve jiverse bd

4

5.

\untrmltjbi bd ^mtjbim` Iuhners

_gei we gmve tb suntrmlt m rmtjbim` iuhner, smy:

5

drbh:

7, we maa tge maajtjve jiverse bd

:

5, j.e.,

:

5tb

:

7. Qgus,

:

7

:

58

:

7+

:

58

:

578

:

:1

=mia

>

=9

1

=, sb

1

=8

>3

=3.

(m`` mre equjvm`eit).

@mw bd Maajtjbi 6

Ojvein

m,

a

l, mia

d

emre tgree rmtjbim` iuhners.

(j) Jdn

m9

a

ltgei

n

m+

d

e9

a

l+

d

e.

(jj) Jdn

m8

a

ltgei

n

m+

d

e8

a

l+

d

e.

(jjj) Jdn

m0

a

ltgei

n

m+

d

e0

a

l+

d

e.

Ex. Lbisjaer tge tgree rmtjbim` iuhners

>3

=3

3

1.

Geile,>3

=3

3

=3

19

>3

3

1

>3

=?9

>3

?.

_gjlg js true.

Ex., mia

? 8

=,

_gjlg js true.

Srbperty bd Hu`tjp`jlmtjbi 6

@etn

m,

a

lmia

d

ene tgree rmtjbim` iuhners.

Jdd

ejs m pbsjtjve iuhner, tgei

(j) Jdn

m9

a

l, tgei

n

m

d

e9

a

l

d

e.

(jj) Jdn

m0

a

l, tgei

n

m

d

e0

a

l

d

e.

(jjj) Jdn

m8

a

l, tgei

n

m

d

e8

a

l

d

e.


6/26


Ex.

=9

7

=

7

=

?

=8

?

=(Qrue)

_bra Srbnèhs

Ex.:1 15

=1

8 ?>:1 =1

15

8 >bd m`` tge nbbks

Sbwers

E xp bi ei tj m` I bt mt jb i m ia ^m tj bi mÌuhners 6

Expbieitjm` ibtmtjbi lmi ne exteiaea tb rmtjbim`

iuhners. Dbr exmhpè6

5

1

5

1

5

1lmi ne

wrjttei ms

4

5

8

4

5

4

5

>

>

4

)5(8

5

(jj)

?

7ms tge pbwers bd rmtjbim`

iuhners.


7/26


\b`.41

>?8

111

(

8

< js

=.

^eljprblm` bd

1

8

1

=

8

1

1

=8

1

1

>

=. Qgeredbre, tge reljprblm` bd

> js >=. Qge reljprblm` bd 8 >

5

1

8

>

5

1

^eljprblm` bd

=

>=

>=

>=

5

58 5.

M`sb, sjile (5)> 8 >5, sb >58 5.

Qgus >5 gms twb rbbts, 5 mia 5.

\b >58 + 5.

\jhj`mr`y, =48 + 1, 1:8 + ?, mia sb bi.


9/26


Qmn`e

Iuhner \qumre Lune

= = =

> 1 7

< : >?

1 =4 41

5 >5 =>5

4 =4

? 1:

: 7= ?>:

=3 =33 =333

Qb djia tge squmre rbbt bd m perdelt squmre

ny prjhe dmltbrjsmtjbi hetgba 6

\teps 6

=- Express tge ojvei iuhner ms m prbault bd prjhe

dmltbrs.

>- Hmke pmjrs bd sjhj`mr dmltbrs.

> > > < < 5 5.

\b`. )55()>>(

8 > > < 5 8 43

Ex.14 Djia tge squmre rbbt bd =?41 ny djiajio prjhe

dmltbrs.

=?41 8 > > < < ? ?

\b`. =?418 ??

8 > < ?

8 1>

==

??

1:?

=1?

=?41>

Ibte tgmt jd ieomtjve dmltbrs mre lbisjaerea, we

sgm`` gmve =?418 1>

Ex.1? Djia tge sjhp`est miswer dbr 71nm: .

\b`. 71nm: 8 11>> nnmm1> nm

>

n

m8

mtbrhjiaeibtgebdsqumre

iuhermtbrtgebdsqumre

Ex.17 Djia tge squmre bd

1

1

>

7

58

77

55

8

41

>5

Ex.53 Djia tge squmre rbbt bd=11

41.

\b`.=11

418

>>>>>>>>>

81>

4

>

8

>

8

=>

78

Lunes mia Lune ^bbtsJd x js m rmtjbim` iuhner, hu`tjp`yjio x wjtg jtse`d< tjhes ojves x x x mia js aeibtea ms x > ? ? ?.

\b`. < )???>>>(

8 < )?>()?>()?>(

8 > ? 8 =1


10/26


Ex.5> Djia tge lune rbbt bd : mia =3< 8 =333,

j.e., tge lune rbbt bd miy perdelt lune ne`bw tgbusmia

js m sjioè ajojt.

Ex.5< Djia < ?>:.

\b`. _e kibw tgmt jts lune rbbt js m sjioè ajojt ms

tgere mre bi`y tgree ajojts. Qgeredbre ?>: js tge

lune bd :, ms tge uijts ajojt bd m lune wj`` ne :

bi`y dbr tge iuhner :.

Ex.51 Djia tge lune rbbt bd =3. @emve tge tgree ajojts bi tge rjogt mia `bbk mt

=3>

[(5) + (=?) 8 >>X

Ex.5: Maa==

1

mia

==

?.

\b`. _e djrst express==

1

ms m rmtjbim` wjtg pbsjtjve

aeibhjimtbr.

_e gmve,==

1

8

)=()==(

)=(1

8

==

1

==

1

+

==

?8

==

1+

==

?8

==

?)1( 8

==

1.

Ibw, we express=>

5mia

7

1.

_e gmve,

=>5 8

>=>>5 8

>1=3 mia,

7< 8

5+

7

1

=3+

>1

:8

>1

:=38

>1

=:

Ex.4= Maa:

?mia 1.

\b`. _e gmve, 1 8=

1.

Lèmr`y, aeibhjimtbrs bd tge twb rmtjbim` iuhners

mre pbsjtjve. _e ibw rewrjte tgeh sb tgmt tgey

mgve m lbhhbi aeibhjimtbr js >1.


11/26


Ibw we re-wrjte tge ojvei rmtjibm` iuhners jitb

dbrhs ji wgjlg nbtg bd tgeh gmve tge smhe

aibhjimtbr.

7

=>

>5

8

>1

=3

7

58

>1

:+

>1

)=3(8

>1

=3:8

>1

=

Ex.4< \jhp`jdy 6=5

7

+

7

<

>

5

?

\b`.(j) _e gmve,

>

7

<

7

7

5


12/26


(jj) _e gmve,

1

?

>

>

5

?

8 1

?

>=

>

5?

8

1

1

?

)>(=

>

>

5)?(

8 1

1

)>(

?

>

>

5

)?(8

=4

????

>5

??

8>5=4

?4

8

>5=4

?4

8

133

?4

8133

==?41:

Ex.44 Express emlg bd tge db``bwjio ms pbwer bd m

rmtjbim` iuhner wjtg pbsjtjve expbieit 6

(j)

5

5

7

>

5>

>>

8 =3=

>

=

5

(jj) _e gmve,

(4= 7=)= + (>=

=

7

=

4

=

8

==

4

>1

1=

=

4

=

>1

= ==

m

=m =

(jjj) _e gmve,

(5= )71(

8

>

=

7

=

1

=

81

=

7

(?))< 8 (=1)5 >>

(jjj) 1 (jv)

>

=

(v) 5=3?

>

>>>

ihi

hm

m

m

8 ??

>

=>

i

i

m

=m

8

?

?

?

>

=

>

=

(jjj) _e gmve,

1 8 (< >)1 [mi ni 8 (mn)iX

8 41

814

=

i

i

m

=m

81

1

4

=X==[ 1

8

1

4

=

i

i

i

n

m

n

m

(jv) _e gmve,

4>

=

>

=

>

=

>

=

>

=

(v) _e gmve,5=3?


15/26


\b`. (j) _e gmve,

5575 >)>>(

855755

5

7

5

>)>(>

>

>

8 55>>)>(

8 >=5 >5 8 >=55 8 >>3

(jj) _e gmve,

(1)< 5< (5) < 8 =34

(jjj) _e gmve,

1

=

>

=

>

5

=

8

=>>

5

=

hiih m)m(

8

>>

5

=

8

)>()>(

5

=

84>5

=

5

)=(

5

=1

11

(jj) _e gmve,

>

1

=

>

=

=

1

=

>

=

>

=

1

=

>

=

8>5

41

5

7>

>

(jj) _e gmve,

5

1

(

5

1

8 41

=>5

1

:

1

5

)>(

54

==>5

411

=>5:

(jjj) _e gmve,

5 8 >53

Ex.?? \jhp`jdy6

(j)75

(jj)

5?

55

45

=>5=3(5

m5

m=35

m>5

8>

m5

>

m5

m>5

m5 1171)>(>

7>

1>

111

m>

4>5m

>

5

(jj) _e gmve,

5?

(5=3 2

\b`. @et (1)> 8 ne hu`tjp jea ny x tb oet =3> .

Qgei, x (1) > 8 =3>

>> )1(=3x

>

>

)1(==3x

>

>

)1(

=3x

>5

1

=33

=4

=3

)1(x

>

>

Geile, requjrea iuhner js

>5

1

Ex.?: Ny wgmt iuhner sgbu`a (=>)= ne ajvjaea sb

tgmt tge qubtjeit hmy ne

=

2

\b`. @et tge requjrea iuhner ne x. Qgei,

==

x)=>(

==

x

)=>(

==

)=>(x

>

=x

m

n

n

m

miam

=

m

==

=7

=

=7

=

=>

=x

Ex.73 Ny wgmt iuhner sgbu`a

<

ne ajvjaea sb

tgmt tge qubtjeit hmy ne

>

>?

1

2


17/26


\b`. @et tge requjrea iuhner ne x. Qgei,

>?

1x

>

<

>?

1

x

=

>

<

>?

1

>

?

x

ii

m

n

n

m

>

?

=

x

>

>

>

>?

1>

>?

1

>?

7

>?

1

>?

7x

?1>

Ex.7= Djia x sb tgmt

x7==5

>

=h>

m

>

>h = 8 h 8 < + =

>h 8 >

h 8 =

Ex.7< Jd

1>

>

\b`. _e gmve,

1>

>

>

>

=>>4>

4>

>

=

(v) >=4333333 (vj) 3.33335>:=31

(vjj) :5?


18/26


(jjj) 3.333333541 8 5.41 =3?

[ Qge aeljhm` pbjit js hbvea ? p`mles tb tge

rjogtX

(jv) 3.33333>= 8 >.= =34

[

Qge aeljhm` pbjit js hbvea 4 p`mle tb tgerjogtX

(v) >=433333 8 >.=4 =37

[Qge aeljhm` pbjit js hbvea 7 p`mles tb tge

`edtX

(vj) 3.33335>: =31 8 5.>: =35 =31

8 5.>: =35+1 8 5.>: =3=

(vjj) :5?< =31 8 :.5?< =3< =31

8 :.5?< =3?5 hetre

8 =.>?5 =35 hetre

(v) Qge tgjlkiess bd m ibrhm` pmper js 3.3? hh

8 ? =3> hh

Ex.7? Jd tge ajmheters bd tge \ui mia tge Emrtg mre

=.1 =3: hetres mia =.>?5 =3? hetre respeltjve`y.

Lbhpmre tgese twb.

\b`. _e gmve,

?

?>

?

:

=3>?5.=

=3=31.=

=3>?5.=

=31.=

Emrtgtgebd.Ajm

suitgebd.Ajm

8 =33~=33?5.=

1.=

=~

?5 h. Lbhpmre

tgese twb.

\b`. _e gmve,

\jze bd rea n`bba le`` 8 3.33333? h 8 ?=34 h

\jze bd p`mit le`` 8 3.3333=>?5 8 =.>?5=35 h

5

4

=3>?5.=

=3?

le``p`mitbd\jze

le``n`bbareabd\jze

8>?5.=

=3?

>?5.=

=3? =54

8

>

=~

?5.=

?.3

\b, m rea n`bba le`` js mpprbxjhmte`y gm`d bd m

p`mit le`` ji sjze.


19/26


].= Express emlg bd tge db``bwjio ms m rmtjbim`

iuhner bd tge dbrhqp , wgere p mia q mre jiteroers

mia 3q 6

(j) >< (jj) (1)>

(jjj) >

=

(v)

>

].> Djia tge vmùe bd emlg bd tge db``bwjio6

(j) == 1< (jj) >=3 >)1>>

1

=

=

(jjj) >== >)1>(

(jv) === 4)>5(

].1 \jhp`jdy6

(j) >== )

=)>>

)>>>

>

=

ne hu`tjp`jea

sb tgmt tge prbault hmy ne equm` tb

=

?

1

2

].7 Ny wgmt iuhner sgbuà (=5)= ne ajvjaea sb

tgmt tge qubtjeit hmy ne equm` tb (5)= 2

].: _rjte emlg bd tge db``bwjio ji expbieitjm` dbrh6

(j)

====

>

<

(jj)

>>>

5

>

5

>

5

>

].=3 Evmùmte6

(j) 5> (jj) (

(jjj)

1

=

].== Express emlg bd tge db``bwjio ms m rmtjbim`

iuhner ji tge dbrhq

p6

(j) 4=

(jj) (?)=

(jjj)

=

1

=

(jv)

==

>

\jhp`jdy

(j) >==


20/26


].=1 Express emlg bd tge db``bwjio rmtjbim` iuhners

wjtg m pbsjtjve expbieit6

(j)

>

1

<

(jj)

=

(jv)

=>

>

1

=

(v) ==

1

4

].=4 Ny wgmt iuhner sgbuà 5= ne hu`tjp`jea sb tgmt

tge prbault hmy ne equm` tb (?)=2

].=? Ny wgmt iuhner sgbuà

=

>

=

ne hu`tjp`jea sb

tgmt tge prbault hmy ne equm` tb

=

?

12

].=7 Ny wgmt iuhner sgbuà (=5)=

ne ajvjaea sbtgmt tge qubtjeit hmy ne equm` tb (5)= 2

].=: Ny wgmt iuhner sgbuà

>

7=:

>

=

>

=

>

=

(jjj)

=x>5

<

(jv)

x=5

5

>

5

>

(v)

51x

1

5

1

5

1

5

(vj)

>x5=x>

.

(jj) Jd

>>

1

=

5

1x

, djia tge vmùe bd x=

].>> Djia tge vmùe bd x dbr wgjlg 5>x 5< 8 55.

].>< Express tge db``bwjio iuhners ji stmiamra dbrh6

(j) 43>3333333333333 (jj)3.33333333333:1>

(jjj) 3.33333333375 (jv) 714 =3?

(v) :71

(vjj) 3.3331


21/26


].>? Qge suh bd twb iuhners js7 Qge suh bd twb iuhners js : Qge suh bd twb rmtjbim` iuhners js7. Jd bie bd

tge iuhners js?

=5, djia tge btger..

].


22/26


23/26


].= Jd x 8 3.? , tgei >x js -

(M) 1.= (N) 5.=

(L) 51.= (A) 15.=

].> Evmùmte

41

=

(M) 1 (N) =4 (L) (A) 41

].< Qge vmùe bd (>54)3.=4 (>54)3.3: js -

(M) 41 (N) >54.>5

(L) =4 (A) 1

].1 Jd m 8 > + < (N) > + < mia n 8 > > n

=

m

= js

equm` tb

(M) =1 (N) =1

(L) 7 < (A) 7 y

=

x

= js

equm` tb -

(M) 7

].7 )>(>

>>>>i

i (>i) (N) >i+< 1

=

(L) = 1

=(A) =

>

=

].:

3

41

=

+ 41=/> ()1/5 js equm` tb

(M) =57

?(N) =4

7

=

(L) =17

?(A) =?

7

=

].=3 M rmtjbim` iuhner netweei > mia

(N)

>

(L) =.1 (A) =.5

].== Jd x 8 4=.3 , tgei Djia tge vmùe bd x wgei

5

<

8

(N) x 8 >

(L) x 8 = (A) x 8 =

].=< Jd >x >x= 8 1, tgei xx js equm` tb

(M) = (N) >?(L) < (A) Ibie bd tgese

].=1 Qge vmùe bd

=

///< (A) =/>

].=5 _gmt hust ne maaea tb tge suh bd

1x> + + :x < (N) < :x ?x>

(L) ?x> + :x > (A) Ibie bd tgese

].=4 /5

].=? 575.3 js equm` tb

(M)::

575(N)

:::

575

(L)575

:::(A) ibie bd tgese

EWE^LJ\E # >


24/26


].=7 >.5 js equm` tb

(M) 15/: (N) 14/:

(L) 1?/: (A) Ibie bd tgese

].=: _gjlg bd tge db``bwjio iuhners js ajddereit

drbh btgers 2

(M) > (N) < (L) 1 (A) 5

].>3 _gjlg bd tge db``bwjio iuhners js ajddereit

drbh btgers 2

(M) ? (N) 7 (L) =< (A) =4

].>= Qge prbault bd 41 mia >1< js

(M) =>1 (N) => Jd m 8>>> + n> js

(M)


25/26


Gjits & \b`utjbi # =

\b`.= (j)7

=(jj)

=4

=(jjj) : (jv) (v)

1

:

\b`.> (j)=>

?(jj) 5 (jjj) = (jv) =

\b`.< (j) : (jj) >: (jjj)>

=(jv)

5

5

>=4(jjj)

5

4(jv)

5

=>

\b`.5 (j)7

=

\b`.=3 (j)>5

=(jj)

:

=(jjj) 7= (jv) >

\b`.== (j)4

=(jj)

?

=(jjj) 1 (jv)

4

=(v)

\b`.=> (j)=11

=(jj)

=>5

>=4(jjj)

5

4(jv)

5

=>(v)

>3

=3=5 (jj) :.1> =3=>

(jjj) 7.5 =3=3 (jv) 7.14 =3:

(v) :71 =31

(vjj) 1.

=3


26/26


].Ib = > < 1 5 4 ? 7 : =3 == => =< =1 =5 =4 =? =7

Mis. N N A L A M M L L A A M N N N L N L

].Ib =: >3 >= >> >< >1 >5 >4 >? >7 >: m

=8 >)(

=

8

>>i

ii=

i=.>

>.>>

8

>

8

>.>>.> 8

)>(>

)>>(>> 8 =

1

=

==.[AX x 8 4=.3 ....(j)

=3 x 8 4.= ....(jj)

=33 x 8 4.=4

\untrmltjio (j) drbh (jj) :3x 8 =5

x 84

=

=8 =.5

=1.[NX =

Documents

01. Rational Numbers