3. ge i<saxs&
okLrfod la[;k,a (Real Numbers)
;wfDyM foHkktu izesf;dk (Euclid’s Division Lemma)
iwoZ Kku%& ge tkurs gSa----
izkd`r la[;k,a¼N½----------------1]2]3]4----------
iw.kZ la[;k,a ¼W½ -----------------0]1]2]3]4---------
ifjes; la[;k,a ¼Q½--------------izkd`r la[;k,a]iw.kZ la[;k,a]iw.kkZad
ds vfrfjDr---------1@2]3@4-----------
Rational no. means Ratio …. r = p/q q ≠ 0
4. vifjes; la[;k,a ¼S½-------------tks ifjes; la[;k ugha gS vFkkZr
√2] √3] π ] 0-10110111011110 vkfnA
vifjfer :Ik ls tSls ifjes; la[;k vusd gksrh gS oSls gh
vifjes; la[;k Hkh vusd gksrh gSA
,d lkFk yh xbZ ifjes; la[;k o vifjes; la[;kvksa ds
laxzg dks gh okLrfod la[;k,a dgrs gSaA
5. ,d lkFk yh xbZ ifjes; la[;k o vifjes; la[;kvksa ds laxzg
dks gh okLrfod la[;k,a dgrs gSaA
6. ;wfDyM foHkktu izesf;dk (Euclid’s Division Lemma)
;fn a o b nks /kukRed iw.kkZad gks rks q o r ,slh nks vf}rh;
iw.kZ la[;kvksa dk vfLrRo gksxk ftlds fy,
a = bq + r tcfd 0 ≤ r < b
tgkWa& a = HkkT;
b = Hkktd
q = HkkxQy
r = ’ks"kQy
”
”””””
8. mnkgj.k%&n’kkZb;s fd izR;sd /kukRed le iw.kkZad 2q ds :Ik dk gksrk
gS rFkk izR;sd /kukRed fo"ke iw.kkZad 2q+1 ds :i dk gksrk gSA
gy%& ekuk fd
a = ,d /kukRed iw.kkZad gS
b = 2
;wfDyM foHkktu izesf;dk a = bq + r ls
a = 2q + r tcfd 0 ≤ r < 2
vFkkZr r = 0 or r = 1
so … a = 2q or a = 2q + 1
;fn a = 2q ; rks a = ,d /kukRed le iw.kkZad gS D;ksafd ;s 2 ls foHkkT;
gSA ¼ 2 ls Hkkx ns ldrs ½
” ;fn a = 2q + 1 ; rks a = ,d /kukRed fo"ke iw.kkZad gS D;ksafd ;s 2 ls
foHkkT; ugha gSA ¼ 2 ls Hkkx ugha ns ldrs ½
vr% izR;sd /kukRed le iw.kkZad 2q ds :Ik dk gksrk gS rFkk izR;sd /kukRed
fo"ke iw.kkZad 2q+1 ds :i dk gksrk gSA
9. mnkgj.k%&n’kkZb;s fd ,d /kukRed fo"ke iw.kkZad 4q+1 ;k 4q+3
ds :i dk gksrk gSA
gy%& ekuk fd
a = ,d /kukRed fo"ke iw.kkZad gS
b = 4
;wfDyM foHkktu izesf;dk a = bq + r ls
a = 4q + r tcfd 0 ≤ r < 4
vFkkZr r = 0 ,1, 2, or 3
so … a = 4q, a = 4q + 1, a = 4q + 2, a = 4q + 3 gks ldrk gSA
a = 4q o a = 4q + 2 le iw.kkZad gS D;ksafd ;s 2 ls foHkkT; gSA ¼ 2 ls Hkkx ns ldrs
½
” a = 4q + 1 o a = 4q + 3 fo"ke iw.kkZad gS D;ksafd ;s 2 ls foHkkT; ugha gSA ¼ 2 ls
Hkkx ugha ns ldrs ½
vr% ,d /kukRed fo"ke iw.kkZad 4q+1 ;k 4q+3 ds :i dk gksrk gSA
10. ;wfDyM foHkktu izesf;dk
}kjk HCF Kkr djuk
HCF ¼egRre lekiorZd½ ---- nks ;k vf/kd la[;kvksa dk
egRre lekiorZd og lcls cMh la[;k gS tks nh xbZ lHkh
la[;kvksa dks iw.kZr% foHkkftr djrh gSA
;wfDyM foHkktu izesf;dk alogorithm nks /kukRed iw.kkZadksa ds
HCF Kkr djus dh pj.kc) izfdz;k gSA
11. mnkgj.k %&
;wfDyM foHkktu ,YxksfjFke dk iz;ksx djds 420 rFkk 130 dk
HCF Kkr dhft,A
gy%& 420 o 130
130)420(3
390
30)130(4 420 = 130X3+30
120 130 = 30X4+10
10)30(3 30 = 10X3+0
30
00 mRrj%& 420 o 130 dk HCF 10 gSA
12. mnkgj.k %&
;wfDyM foHkktu ,YxksfjFke dk iz;ksx djds 135 rFkk 225 dk
HCF Kkr dhft,A
gy%& 135 o 225
135)225(1
135
90)135(1 225 = 135X1+90
90 135 = 90X1+45
45)90(2 90 = 45X2+0
90
00 mRrj%& 135 o 225 dk HCF 45 gSA
13. vH;kl iz’u
1- fuEufyf[kr esa ifjes; la[;k gS&
aaaa) √2 b) √3 c) π d) 1.5
2- fuEufyf[kr esa vifjes; la[;k gS&
aaaa) 1.5 b) 1.05 c) 1.005 d) 1.05005
3- nks vifjes; la[;kvksa dk ;ksx lnSo ----------------------la[;k gksrh gSaA
4- nks /kukRed iw.kkZadksa dk HCF ifjdfyr djus dh rduhd dks -------dgrs
gSaA
5- n’kkZb;s fd ,d /kukRed fo"ke iw.kkZad 6q+1 ;k 4q+3 ;k 4q+5 ds
:i dk gksrk gSA tgkWa q dksbZ iw.kkZad gSA
6- ;wfDyM foHkktu ,YxksfjFke dk iz;ksx djds 867 rFkk 25 dk HCF Kkr
dhft,A
