Bangla Math By Rakes Prasad
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Rakes prasad
°h¢cL Aˆ
Digitally signed by Rakes prasad DN: cn=Rakes prasad, c=IN, o=prasadonlineproject, email=prasadrakes@gmail. com Reason: I am the editor of this document Location: mollarpur,birbhum, w.b.,india Date: 2009.01.11 22:36:12 +05'30'
Bjl¡ Hh¡l Bj¡−cl j¡a«i¡o¡u Aˆ ¢nMh , −cM−h¡ ¢L L−l A¢a pq−S …Z Ll¡ k¡u z a¡l SeÉ fËb−j clL¡l ¢LR¥ e¡ja¡ j¤MÙ¹ Ll¡l z e£−Ql RL …−m¡ −cMz RL 1.
1 2
1
2
3
4
5
6
7
8
9
10
1
2
3
4
5
6
7
8
9
10
4
6
8
10
12
14
16
18
20
9
12
15
18
21
24
27
30
16
20
24
28
32
36
40
25
30
35
40
45
50
36
42
48
54
60
49
56
63
70
64
72
80
81
90
3 4 5 6 7 8 9 10
8*4 = 4 *8,..
100
RL 2. 1
2
3
4
5
6
7
8
9
10
11
11
22
33
44
55
66
77
88
99
110
12
12
24
36
48
60
72
84
96
108
120
13
13
26
39
52
65
78
91
104
117
130
14
14
28
42
56
70
84
98
112
126
140
15
15
30
45
60
75
90
105
120
135
150
16
16
32
48
64
80
96
112
128
144
160
17
17
34
51
68
85
102
119
136
153
170
18
18
36
54
72
90
108
126
144
162
180
19
19
38
57
76
95
114
133
152
171
190
20
20
40
60
80
100
120
140
160
180
200
21
21
42
63
84
105
126
147
168
189
210
22
22
44
66
88
110
132
154
176
198
220
23
23
46
69
92
115
138
161
184
207
230
24
24
48
72
96
120
144
168
192
216
240
25
25
50
75
100
125
150
175
200
225
250
RL 3.
11 12 13 14 15 16 17
11
12
13
14
15
16
17
18
19
20
121
132
143
154
165
176
187
198
209
220
144
156
168
180
192
204
216
228
240
169
182
195
208
221
234
247
260
196
210
224
238
252
266
280
225
240
255
270
285
300
256
272
288
304
320
289
306
323
340
324
342
360
361
380
18 19
400
20
pÇf¡ce¡u l¡−Ln fËp¡c, 2008
Hh¡l Bjl¡ hNÑ , hNÑj§m, Oe, Oej§m CaÉ¡¢c ¢nMh z HM¡−e ¢LR¥ ¢Q−œl j¡dÉ−j fÜ¢a…−m¡ −h¡T¡h¡l −Qø¡ Ll¡ q−u−Rz ¢euj 01. fËb−j 1 −b−L 32 fkÑ¿¹ pwMÉ¡l hNÑ j¤MÙ¹ Ll¡ clL¡l k¡ e£−Ql R−L −cJu¡ q−m¡z No
Square
No
Sq
No
Sq
No
Sq
no
sq
1
1
9
81
17
289
25
625
33
1089
2
4
10
100
18
324
26
676
34
1156
3
9
11
121
19
361
27
729
35
1225
4
16
12
144
20
400
28
784
36
1296
5
25
13
169
21
441
29
841
38
1444
6
36
14
196
22
484
30
900
41
16 81
7
49
15
225
23
529
31
961
8
64
16
256
24
576
32
1024
¢euj 02. −k pjÙ¹ pwMÉ¡l −no pwMÉ¡ 5 a¡−cl hNÑ ¢eÑZu zHLV¡ Ec¡lqZ ¢e−u hÉ¡f¡lV¡ −h¡T¡l −Qø¡ Ll¡ k¡Lz −kje d¢l 25 ,Hl −no pwMÉ¡ 5 Hhw B−Nl pwMÉ¡ 2 za¡q−m Bjl¡ B−Nl pwMÉ¡¢V−L a¡l f−ll pwMÉ¡ ¢c−u …Z Llh AbÑ¡v 2 −L 3 ¢c−u AbÑ¡v 2x3 / 5x5=6 / 25 = 625 AbÑ¡v (25) =625 HLCi¡−h 105P2P= 10 X 11/25 = 11025; 135P2P= 13 X 14/25 = 18225
e£−Ql ¢Qœ ¢V −cM z
…
Bjl¡ Hh¡l ¢euj¢V−L fËp¡¢la Ll¢R. ¢euj 03 : k¢c −no pwMÉ¡ c¤¢Vl −k¡Ngm 10 qu a¡q−m Bjl¡ HC ¢euj¢V fË−u¡N Ll−a f¡lh ¢L¿¹¥ h¡j q¡−al pwMÉ¡ c¤¢V−L (L.H.S) HLC q−a q−hz Ex 1 : 47 X 43 . HM¡−e −no pwMÉ¡ c¤¢Vl −k¡Ngm 10 AbÑ¡v7 + 3 = 10 ;a¡q−m 47 x 43 = ( 4 + 1 ) x 4 / 7 x 3 = 20 / 21 = 2021. Ex2: 62 x 68 2 + 8 = 10, h¡j q¡−al c¤¢V−L (L.H.S)HLC AbÑ¡v 6. 6 Hl f−ll pwMÉ¡ qm 7 Hhw 62 x 68 = ( 6 x 7 ) / ( 2 x 8 ) = 42 / 16 = 4216.
Ex 3: 127 x 123 127 x 123 = 12 x 13 / 7 x 3 = 156 / 21 = 15621. U(e£−Ql ¢Qœ ¢V −cM )U
Ex 4. 395P2P 395P2P = 395 x 395 = 39 x 40 / 5 x 5 = 1560 / 25 = 156025. .Hi¡−hJ ¢euj¢V−L j−e l¡M¡ −k−a f¡−l 125P2P =(12P2P +12)25=(144+12)25=15625 . .Hl g−m h−s¡ pwMÉ¡l −r−œ p¤¤¢hd¡ q−hz ¢euj 04 : k¢c −no pwMÉ¡ c¤¢Vl −k¡Ngm 5 qu a¡q−m Bjl¡ HC ¢euj¢V fË−u¡N Ll−a f¡lh ¢L¿¹¥ h¡j q¡−al pwMÉ¡ c¤¢V−L (L.H.S)HLC q−a q−hz Ex 1. 82 x 83=(8P2P+8/2) /3 x 2=(64+4) / 06=6806
AbÑ¡v U( nU2 PU UP +n/2) ,n kMe −S¡l U pwMÉ¡ (even no).a−h mr
L−l¡ −no …Zgm¢V 06 AbÑ¡v c¤C Ar−ll ,öd¤ 6 euz
Ex 2.181 x 184=(18P2P +9) / 04=(324+9) / 04=33304. ¢L¿¹¥ k¢c n ¢h−S¡l qu a¡q−m ¢L q−h ? aMe Bjl¡ Hi¡−h pj¡d¡e Llh: Ex 3: 91 x 94= (9P2P +9/2) / 04=(81+4 ½ ) / 04= (81+4 ) / 54 Ah¡L m¡N−R ? Bjl¡ −Lhm ½ −L f−ll O−l 5 ¢m−M¢R L¡lZ ½ na−Ll O−l B−R a¡C 1/2= 1/2 x 100=50. H−r−œ 5 −L Hi¡−h p¢l−u ¢e−a quz e£−Ql ¢Qœ ¢V −cM z
…
Ex 4 . 51 x 54= (25+2) / 54 =2754 54=29754
Ex 5. 171 x 174= (289+8) /
¢euj 05 : Hh¡l Bjl¡ −k −L¡e pwMÉ¡l hNÑ Ll¡l p¡d¡lZ fÜ¢a ¢e−u B−m¡Qe¡ LlhzB−N e£−Ql RL¢V i¡mi¡−h mr L−l¡z base 50 100
range 26-74 76-126
Trick
alternative −cM 5P2P =25
25(±)( read as 25 plus minus) 100(±)2(±)
150
126-174
225(±)3(±)
200
176-224
400(±)4(±)
250
226-274
625(±)5(±)
300
276-224
900(±)6(±)
350
326-374
1225(±)7(±)
400
376-424
1600(±)8(±)
450
426-474
2025(±)9(±)
500
476-524
2500(±)10(±)
………
…………..
……………………….
1000
976-1024
100,00(±)20(±)
1500
1476-1524
225,00,00(±)30(±)
2000
1976-2024
4,000,000(±)40(±)
Given no(±)
Hhw 10P2P = 100 Hi¡−h
.
50 base Hl SeÉ: 26 −b−L 74 fkÑ¿¹ pwMÉ¡…−m¡l SeÉ base 50 dl¡ qu,AbÑ¡v HC p£j¡l (range) j−dÉ pwMÉ¡…−m¡l SeÉ 25+- ¢euj¢V M¡−Vz fËb−j −cM−a qu pwMÉ¡¢V 25 Hl −R¡−V¡ e¡ h−s¡ ,−R¡−V¡ q−m 25Hhw h−s¡ q−m 25+ ¢euj¢V hÉhq¡l Ll−a quz Ec¡lq−Zl p¡q¡−kÉ J d¡f BL¡−l ¢hou¢V −h¡T¡−e¡ qmz .Ex 1. 43P2P d¡f 1:
base −b−L pwMÉ¡¢Vl ¢h−k¡Ngm −hl L−l¡z −kje 50-43=07 d¡f 2: fË¡ç de¡aÆL pwMÉ¡¢Vl hNÑ L−l¡ J H¢V−L X¡e ¢c−L −mM , −kje 07 2 P P=49. d¡f 3: −k−qa¥ 43 pwMÉ¡¢V 50 Hl −Q−u −R¡−V¡ a¡C HM¡−e . 25- ¢euj¢V M¡−V, AbÑ¡v 25 −b−L 07 ¢h−k¡N c¡J J ¢h−k¡Ngm−L h¡j¢c−L −mMz 25-07=18 d¡f 4: H¢VC qm Ešl AbÑ¡v 1849 Ex 2. 57P2P : d¡f 1.
base −b−L pwMÉ¡¢Vl ¢h−k¡Ngm −hl L−l¡z −kje 57-50=7P .
d¡f 2:
fË¡ç pwMÉ¡¢Vl de¡aÆL j¡e¢V hNÑ L−l¡ J H¢V−L X¡e ¢c−L −mM , −kje 7P2P=49
d¡f 3 :
−k−qa¥ 57 pwMÉ¡¢V 50 Hl −Q−u h−s¡ a¡C HM¡−e 25+ ¢euj¢V M¡−V, AbÑ¡v 25 −b−L 07 −k¡N c¡J J −k¡Ngm−L h¡j¢c−L −mMz 25+7=32
Ešl qm 32 49. Ex 3. 69 2P P d¡f4:
d¡f 1: 69-50=19 , 19P2P =361 d¡f 2: H¢M−e 3 −L q¡−a e¡J J X¡e ¢c−L 61 −L −mMz L¡lZ Bjl¡ c¤C Ar−ll −h¢n ¢e−a f¡lh e¡z d¡f 3: 25+19=44, ¢L¿¹¥ HC 44 Hl p−‰ 3 −L (q¡−a l¡M¡) −k¡N L−l¡ J h¡j¢c−L −mMzAbÑ¡v (25+19+3) d¡f 8:
Ešl qm 47 61
100 base Hl SeÉ:
76 −b−L 124 fkÑ¿¹ pwMÉ¡…−m¡l SeÉ base 100 dl¡ qu,AbÑ¡v HC p£j¡l (range) j−dÉ pwMÉ¡…−m¡l SeÉ 100(+-)2(+-) ¢euj¢V M¡−Vz Ex 5. 89P2P d¡f 1: 100-89=11 ,11P2P =121 ,q¡−a b¡L−m¡ 1 d¡f 2: . "
100-(2 x 11 )+1=79,AbÑ¡v Ešl qm 7921
HM¡−e ¢hLÒf ¢euj¢V qm ”given no (±)” AbÑ¡v fËcš pwMÉ (±) d¡f 1: 100-89=11 ,11P2P =121 ,q¡−a b¡L−m¡ 1 d¡f 2: 89-11+1=79 ,AbÑ¡v Ešl qm 7921
# HM¡−e fËcš pwMÉ¢V qm 89 ,−pM¡e −b−L ¢h−k¡Ngm 11 −L ¢h−k¡N −cJu¡ q−u−R ¢L¿¹¥ j−e l¡M−h q¡−a l¡M¡ 1 −L phpju −k¡N Ll−a quz # HC ¢hLÒf ¢euj¢V H−r−œ hÉhq¡l Ll¡ i¡m z Ex 6. 117² d¡f 1: 117-17=17 , 17P2P=289 , q¡−a b¡L−m¡ 2 d¡f 2: .
117+17=134, 134+2=136 AbÑ¡v Ešl qm 13689.
h¡L£ −r−œJ HLCi¡−h ¢euj…−m¡−L fË−u¡N Ll−a q−hz P P
Ex 6. 139² 1. 150-139=11 , hNÑ 121 , q¡−a b¡L−m¡ 1 2. 225-3×11=225-33=192, 192+1=193 , Ešl qm 19, 321 Ex 7. 164² 1. 164-150=14, 14²=196, q¡−a b¡L−m¡ 1 2. 225+3(14)=225+42=267 ,267+1=268 , Ešl qm 26,896 Ex 8. 512² 1. 12²=144 , q¡−a b¡L−m¡ 1, 2500+10(12)+1=2621, Ešl qm 262144 Hh¡l −a¡jl¡ ¢e−Sl¡ ¢LR¥ Ec¡lqZ AiÉ¡p L−l¡ z ¢euj 06 : Hh¡l Bjl¡ c¤¢V pwMÉ¡l Oe ¢nMh . HM¡−e HL¢V EQ¡lq−el p¡q¡−kÉ hÉ¡f¡l¢V −h¡T¡h¡l −Qø¡ Ll¡ qmz −kje dl Ex 1. (18)P 3
d¡f 1. fËb−j pwMÉ¡c¤¢Vl Ae¤f¡a −hl Ll−a q−h −kje HM¡−e 1:8
d¡f 2. Hh¡l fËbj pwMÉ¡¢Vl Oe Ll J a¡lfl Oe pwMÉ¡¢V−L Ae¤f¡a ¢c−u flfl ¢aeh¡l …Z L−l HL¢V m¡C−e −mMz −kje : 1P3P=1 / 1 x 8=8 / 8 x 8=64 / 64 x 8 =512 d¡f 3. ¢àa£u J a«a£u pwMÉ¡¢Vl ¢à…Z L−l a¡−cl AhÙÛ¡−el ¢WL e£−Q a¡¢c−L −mM z d¡f 4. Aa:fl Efl J e£−Ql pwMÉ¡…−m¡−L −k¡N Ll ( HL Ar−ll −hn£ q−m q¡−a ¢e−a q−h)z a¡q−mC Ešl¢V f¡Ju¡ k¡−hz e£−Ql ¢Qœ¢V −cMz 1. 2. 4.
18
Abѡv 1 :8
4
3
1 =1 / P
P
24
51
1x8=8
/ 8 x 8=64
/ 64 x 8 =512
8 x 2=16 64 x 2=128 -----------------------------------------------------------(4+1) 5
(24+8+16) =48
(51+64+128) =243
51
2
Ex 2. (33) P3
Hh¡l −a¡jl¡ ¢e−Sl¡ ¢LR¥ Ec¡lqZ AiÉ¡p L−l¡ z ¢euj 07 : Hh¡l Bjl¡ −k −L¡e pwMÉ¡l hNÑjm § Ll¡l p¡d¡lZ fÜ¢a ¢e−u B−m¡Qe¡ LlhzB−N e£−Ql RL¢V i¡mi¡−h mr L−l¡z
f¤ZÑhNÑ l¡¢nl HLL O−ll Aˆ hNÑjm § l¡¢nl HLL O−ll Aˆ
P1P
P4P
5 P P
6 P P
P9P
P1,9
P2,8
5 P
P4,6
P3,7
P1+9=10P
P2+8=10P
P5+5=10P
P4+6=10P
P3+7=10P
#1. −L¡−e¡ pwMÉ¡l −no Arl¢V k¢c pwMÉ¡¢Vl hNÑj§m f¡Ju¡ pñh euz
2,3,7,8 qu
,a¡q−m −pC
#2. Arl¢V−a k¢c ¢h−S¡l pwMÉ¡L n§eÉ b¡−L, a¡q−m −pC pwMÉ¡¢Vl hNÑj§m f¡Ju¡ pñh euz e£−Ql RL¢V i¡mi¡−h mr L−l¡z p£j¡
hNÑ
hNÑ
P
1²=1
1-3
P
P
P
P
p£j¡
9²=81 P
hNÑ
P
P
P
81-99 P
p£j¡ P
P
P
17²=289
289-323
P
P
P
hNÑ P
P
p£j¡ P
P
25²=625 P
P
625-675 P
P
2²=4
4-8
P
P
P
10²=100
P
P
P
3²=9
9-15
11²=121
P
P
P
P
4²=16 P
16-24
P
P
P
P
P
P
7²=49 P
P
P
P
P
64-80 P
P
225-255
P
P
P
361-399 P
P
400-440 P
P
441-483
P
P
484-528 P
529-575 P
256-288 P
P
P
576-624 P
P
P
P
900-960
P
P
P
31²=961 P
P
841-899 P
30²=900
P
24²=576
P
P
P
P
784-840 P
29²=841
P
23²=529
P
P
P
P
P
P
28²=784
P
22²=484
729-783
P
P
21²=441
P
27²=729
P
P
P
P
P
676-728
P
P
20²=400
P
16²=256 P
P
P
P
15²=225
P
8²=64
196-224 P
26²=676
P
19²=361
P
P
49-63
P
169-195 P
14²=196
P
P
P
P
P
36-48
P
144-168 P
P
324-360
P
P
13²=169
P
6²=36
P
P
25-35
P
121-143
P
P
18²=324
P
12²=144
P
5²=25
P
P
100-120
P
961-1023
P
P
32²=1024
P
P
P
1024P
1088 P
¢euj 1: H−Lh¡−l X¡e¢c−Ll −no c¤¢V pwMÉ¡−L Bm¡c¡ L−l ¢m−M . e¡Jz 2: f−s b¡L¡ pwMÉ¡¢V −L¡e p£j¡l j−dÉ f−l −p¢V h¡l L−l¡ J . a¡−L h¡j¢c−L −mMz (H¢V qm Eš−ll h¡jfr) 3: p£j¡¢V−L a¡l f−ll pwMÉ¡ ¢c−u …Z L−l¡ J −cM Bm¡c¡ . . Ll¡ pwMÉ¡¢V JC …Zgm¢Vl −Q−u h−s¡ e¡ −R¡−V¡. 4: k¢c h−s¡ qu ,a¡q−m hNÑj§m l¡¢nl HLL O−ll A−ˆl h−s¡ . pwMÉ¡¢V qm X¡efr (R.H.S) . Ex1.√ (6241)
hÉMÉ¡: 1. X¡e¢c−Ll −no c¤¢V pwMÉ¡ qm 41 k¡−L Bm¡c¡ L−l ¢m−M ¢em¡jz 2. f−s b¡L¡ pwMÉ¡¢V AbÑ¡v 62-Hl p£j¡ qm 7,a¡C H¢V qm Eš−ll h¡jfr. 3. 7 −L a¡l f−ll pwMÉ¡ ¢c−u …Z L−l −cMm¡j 56 Hl −Q−u 62 h−s¡z 4. a¡C 1 Hl SeÉ fcš h−s¡ pwMÉ¡¢V AbÑ¡v 9 −L Bjl¡ X¡efr ¢qp¡−h ¢em¡jz p¤¤al¡w Ešl¢V qm 79z Ex2 .√ (2601) 1. 26 / 01 2. 26 Hl
p£j¡ qm
5 ,h¡j¢c−L 5
−mMz 3. HMe 5×6=30 ,26 qm −R¡V 30 Hl −Q−uz 4. 1 −L Bjl¡ a¡C X¡efr ¢qp¡−h ¢em¡j J Ešl qm 51. Ex1.√ (2704) 1. 27 / 04 2. 27 Hl
p£j¡ qm h¡j¢c−L
5
−mMz
3. HMe 5×6=30 , 27 qm
−R¡V 30 Hl −Q−uz a¡C 2 −L Bjl¡ a¡C X¡efr ¢qp¡−h ¢em¡j J Ešl qm 52
a−h
5
¢c−u −no pwMÉ¡ …¢ml −r−š pq−SC Ešl f¡Ju¡ u¡uz
Ex 4. √(99225) 1. 992 /25 2. 992 Hl
p£j¡ qm 31 . 3. HM¡−e X¡efr phpju q−h 5 L¡lZ R−L Bj¡−cl L¡−R Bl ¢LR¥ e¡Cz 4. Ešl qm 315
.
Exercise 1.√34225
.
2.√105625
3.√(0.00126025)
4. √2209
5. √2916
6. √2116
7. √15129
8.√ 16129
9. √55696
10. √66564
11. √8.8804
12. √0.00101124
¢euj 08 : Hh¡l Bjl¡ −k −L¡e pwMÉ¡l Oej§m Ll¡l p¡d¡lZ fÜ¢a ¢e−u B−m¡Qe¡ LlhzB−N e£−Ql RL¢V i¡mi¡−h mr L−l¡z f¤ZÑOe l¡¢nl HLL O−ll Aˆ
1
2
3
4
5
6
4
5
6
Oe l¡¢nl HLL 1 O−ll Aˆ
8
7
2+8=10
3+7=10
Oe
Oe
P
p£j¡
P
1 =1 P
1-7 P
P
p£j¡ P
216-342 P
8
9
3
2
9
7+3=10
8+2=10
Oe
P
6 =216 P
7
P
P
p£j¡
11 =1331 P
P
1331-1727 P
P
2 =8
8-26
7 =343
P
P
P
P
3 =27 P
27-63
P
P
P
P
64-124
P
343-511
P
P
P
P
125-215
10 =1000
P
P
P
P
P
21=9261
25=15625
P
P
Ex1. (46656) 1. 2. 3. 4.
P
P
2197-2743
P
P
14 =2744
P
P
1000-1330 P
P
13 =2197
P
P
1728-2196
P
729-999
P
5 =125 P
P
512-728
P
9 =729
P
12 =1728
P
8 =512
P
4 =64 P
P
P
P
2744-3374
P
P
P
15 =3375
3375-4095
P
P
P
P
1/3 P
P
46 / 656 6 Hl SeÉ U
U
X¡e¢c−L 46 Hl p£j¡ qm 3 Ešl qm 36 P
Ex2 . (0.0020048383)
n.b. (1 cn¢jL
6 −mMz
1/3 P
P
1.
0.0020048 / 383
2.
3 Hl
3.
2048 Hl
3.
Ešl qm
U
SeÉ X¡e¢c−L 7 −mMz
O−ll SeÉ Eš−ll
p£j¡ qm
12
0 .123 3
Ol B−N cn¢jL hp¡J )
Hh¡l −a¡jl¡ ¢e−Sl¡ ¢LR¥ Ec¡lqZ AiÉ¡p L−l¡ z
→ →
! #
$#
" %
$#
→
! #
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Digitally signed by Rakes prasad DN: cn=Rakes prasad, c=IN, o=prasadonlineproject, email=prasadrakes@gmail. com Reason: I am the editor of this document Location: mollarpur,birbhum, w.b.,india Date: 2009.01.11 22:36:12 +05'30'
Bjl¡ Hh¡l Bj¡−cl j¡a«i¡o¡u Aˆ ¢nMh , −cM−h¡ ¢L L−l A¢a pq−S …Z Ll¡ k¡u z a¡l SeÉ fËb−j clL¡l ¢LR¥ e¡ja¡ j¤MÙ¹ Ll¡l z e£−Ql RL …−m¡ −cMz RL 1.
1 2
1
2
3
4
5
6
7
8
9
10
1
2
3
4
5
6
7
8
9
10
4
6
8
10
12
14
16
18
20
9
12
15
18
21
24
27
30
16
20
24
28
32
36
40
25
30
35
40
45
50
36
42
48
54
60
49
56
63
70
64
72
80
81
90
3 4 5 6 7 8 9 10
8*4 = 4 *8,..
100
RL 2. 1
2
3
4
5
6
7
8
9
10
11
11
22
33
44
55
66
77
88
99
110
12
12
24
36
48
60
72
84
96
108
120
13
13
26
39
52
65
78
91
104
117
130
14
14
28
42
56
70
84
98
112
126
140
15
15
30
45
60
75
90
105
120
135
150
16
16
32
48
64
80
96
112
128
144
160
17
17
34
51
68
85
102
119
136
153
170
18
18
36
54
72
90
108
126
144
162
180
19
19
38
57
76
95
114
133
152
171
190
20
20
40
60
80
100
120
140
160
180
200
21
21
42
63
84
105
126
147
168
189
210
22
22
44
66
88
110
132
154
176
198
220
23
23
46
69
92
115
138
161
184
207
230
24
24
48
72
96
120
144
168
192
216
240
25
25
50
75
100
125
150
175
200
225
250
RL 3.
11 12 13 14 15 16 17
11
12
13
14
15
16
17
18
19
20
121
132
143
154
165
176
187
198
209
220
144
156
168
180
192
204
216
228
240
169
182
195
208
221
234
247
260
196
210
224
238
252
266
280
225
240
255
270
285
300
256
272
288
304
320
289
306
323
340
324
342
360
361
380
18 19
400
20
pÇf¡ce¡u l¡−Ln fËp¡c, 2008
Hh¡l Bjl¡ hNÑ , hNÑj§m, Oe, Oej§m CaÉ¡¢c ¢nMh z HM¡−e ¢LR¥ ¢Q−œl j¡dÉ−j fÜ¢a…−m¡ −h¡T¡h¡l −Qø¡ Ll¡ q−u−Rz ¢euj 01. fËb−j 1 −b−L 32 fkÑ¿¹ pwMÉ¡l hNÑ j¤MÙ¹ Ll¡ clL¡l k¡ e£−Ql R−L −cJu¡ q−m¡z No
Square
No
Sq
No
Sq
No
Sq
no
sq
1
1
9
81
17
289
25
625
33
1089
2
4
10
100
18
324
26
676
34
1156
3
9
11
121
19
361
27
729
35
1225
4
16
12
144
20
400
28
784
36
1296
5
25
13
169
21
441
29
841
38
1444
6
36
14
196
22
484
30
900
41
16 81
7
49
15
225
23
529
31
961
8
64
16
256
24
576
32
1024
¢euj 02. −k pjÙ¹ pwMÉ¡l −no pwMÉ¡ 5 a¡−cl hNÑ ¢eÑZu zHLV¡ Ec¡lqZ ¢e−u hÉ¡f¡lV¡ −h¡T¡l −Qø¡ Ll¡ k¡Lz −kje d¢l 25 ,Hl −no pwMÉ¡ 5 Hhw B−Nl pwMÉ¡ 2 za¡q−m Bjl¡ B−Nl pwMÉ¡¢V−L a¡l f−ll pwMÉ¡ ¢c−u …Z Llh AbÑ¡v 2 −L 3 ¢c−u AbÑ¡v 2x3 / 5x5=6 / 25 = 625 AbÑ¡v (25) =625 HLCi¡−h 105P2P= 10 X 11/25 = 11025; 135P2P= 13 X 14/25 = 18225
e£−Ql ¢Qœ ¢V −cM z
…
Bjl¡ Hh¡l ¢euj¢V−L fËp¡¢la Ll¢R. ¢euj 03 : k¢c −no pwMÉ¡ c¤¢Vl −k¡Ngm 10 qu a¡q−m Bjl¡ HC ¢euj¢V fË−u¡N Ll−a f¡lh ¢L¿¹¥ h¡j q¡−al pwMÉ¡ c¤¢V−L (L.H.S) HLC q−a q−hz Ex 1 : 47 X 43 . HM¡−e −no pwMÉ¡ c¤¢Vl −k¡Ngm 10 AbÑ¡v7 + 3 = 10 ;a¡q−m 47 x 43 = ( 4 + 1 ) x 4 / 7 x 3 = 20 / 21 = 2021. Ex2: 62 x 68 2 + 8 = 10, h¡j q¡−al c¤¢V−L (L.H.S)HLC AbÑ¡v 6. 6 Hl f−ll pwMÉ¡ qm 7 Hhw 62 x 68 = ( 6 x 7 ) / ( 2 x 8 ) = 42 / 16 = 4216.
Ex 3: 127 x 123 127 x 123 = 12 x 13 / 7 x 3 = 156 / 21 = 15621. U(e£−Ql ¢Qœ ¢V −cM )U
Ex 4. 395P2P 395P2P = 395 x 395 = 39 x 40 / 5 x 5 = 1560 / 25 = 156025. .Hi¡−hJ ¢euj¢V−L j−e l¡M¡ −k−a f¡−l 125P2P =(12P2P +12)25=(144+12)25=15625 . .Hl g−m h−s¡ pwMÉ¡l −r−œ p¤¤¢hd¡ q−hz ¢euj 04 : k¢c −no pwMÉ¡ c¤¢Vl −k¡Ngm 5 qu a¡q−m Bjl¡ HC ¢euj¢V fË−u¡N Ll−a f¡lh ¢L¿¹¥ h¡j q¡−al pwMÉ¡ c¤¢V−L (L.H.S)HLC q−a q−hz Ex 1. 82 x 83=(8P2P+8/2) /3 x 2=(64+4) / 06=6806
AbÑ¡v U( nU2 PU UP +n/2) ,n kMe −S¡l U pwMÉ¡ (even no).a−h mr
L−l¡ −no …Zgm¢V 06 AbÑ¡v c¤C Ar−ll ,öd¤ 6 euz
Ex 2.181 x 184=(18P2P +9) / 04=(324+9) / 04=33304. ¢L¿¹¥ k¢c n ¢h−S¡l qu a¡q−m ¢L q−h ? aMe Bjl¡ Hi¡−h pj¡d¡e Llh: Ex 3: 91 x 94= (9P2P +9/2) / 04=(81+4 ½ ) / 04= (81+4 ) / 54 Ah¡L m¡N−R ? Bjl¡ −Lhm ½ −L f−ll O−l 5 ¢m−M¢R L¡lZ ½ na−Ll O−l B−R a¡C 1/2= 1/2 x 100=50. H−r−œ 5 −L Hi¡−h p¢l−u ¢e−a quz e£−Ql ¢Qœ ¢V −cM z
…
Ex 4 . 51 x 54= (25+2) / 54 =2754 54=29754
Ex 5. 171 x 174= (289+8) /
¢euj 05 : Hh¡l Bjl¡ −k −L¡e pwMÉ¡l hNÑ Ll¡l p¡d¡lZ fÜ¢a ¢e−u B−m¡Qe¡ LlhzB−N e£−Ql RL¢V i¡mi¡−h mr L−l¡z base 50 100
range 26-74 76-126
Trick
alternative −cM 5P2P =25
25(±)( read as 25 plus minus) 100(±)2(±)
150
126-174
225(±)3(±)
200
176-224
400(±)4(±)
250
226-274
625(±)5(±)
300
276-224
900(±)6(±)
350
326-374
1225(±)7(±)
400
376-424
1600(±)8(±)
450
426-474
2025(±)9(±)
500
476-524
2500(±)10(±)
………
…………..
……………………….
1000
976-1024
100,00(±)20(±)
1500
1476-1524
225,00,00(±)30(±)
2000
1976-2024
4,000,000(±)40(±)
Given no(±)
Hhw 10P2P = 100 Hi¡−h
.
50 base Hl SeÉ: 26 −b−L 74 fkÑ¿¹ pwMÉ¡…−m¡l SeÉ base 50 dl¡ qu,AbÑ¡v HC p£j¡l (range) j−dÉ pwMÉ¡…−m¡l SeÉ 25+- ¢euj¢V M¡−Vz fËb−j −cM−a qu pwMÉ¡¢V 25 Hl −R¡−V¡ e¡ h−s¡ ,−R¡−V¡ q−m 25Hhw h−s¡ q−m 25+ ¢euj¢V hÉhq¡l Ll−a quz Ec¡lq−Zl p¡q¡−kÉ J d¡f BL¡−l ¢hou¢V −h¡T¡−e¡ qmz .Ex 1. 43P2P d¡f 1:
base −b−L pwMÉ¡¢Vl ¢h−k¡Ngm −hl L−l¡z −kje 50-43=07 d¡f 2: fË¡ç de¡aÆL pwMÉ¡¢Vl hNÑ L−l¡ J H¢V−L X¡e ¢c−L −mM , −kje 07 2 P P=49. d¡f 3: −k−qa¥ 43 pwMÉ¡¢V 50 Hl −Q−u −R¡−V¡ a¡C HM¡−e . 25- ¢euj¢V M¡−V, AbÑ¡v 25 −b−L 07 ¢h−k¡N c¡J J ¢h−k¡Ngm−L h¡j¢c−L −mMz 25-07=18 d¡f 4: H¢VC qm Ešl AbÑ¡v 1849 Ex 2. 57P2P : d¡f 1.
base −b−L pwMÉ¡¢Vl ¢h−k¡Ngm −hl L−l¡z −kje 57-50=7P .
d¡f 2:
fË¡ç pwMÉ¡¢Vl de¡aÆL j¡e¢V hNÑ L−l¡ J H¢V−L X¡e ¢c−L −mM , −kje 7P2P=49
d¡f 3 :
−k−qa¥ 57 pwMÉ¡¢V 50 Hl −Q−u h−s¡ a¡C HM¡−e 25+ ¢euj¢V M¡−V, AbÑ¡v 25 −b−L 07 −k¡N c¡J J −k¡Ngm−L h¡j¢c−L −mMz 25+7=32
Ešl qm 32 49. Ex 3. 69 2P P d¡f4:
d¡f 1: 69-50=19 , 19P2P =361 d¡f 2: H¢M−e 3 −L q¡−a e¡J J X¡e ¢c−L 61 −L −mMz L¡lZ Bjl¡ c¤C Ar−ll −h¢n ¢e−a f¡lh e¡z d¡f 3: 25+19=44, ¢L¿¹¥ HC 44 Hl p−‰ 3 −L (q¡−a l¡M¡) −k¡N L−l¡ J h¡j¢c−L −mMzAbÑ¡v (25+19+3) d¡f 8:
Ešl qm 47 61
100 base Hl SeÉ:
76 −b−L 124 fkÑ¿¹ pwMÉ¡…−m¡l SeÉ base 100 dl¡ qu,AbÑ¡v HC p£j¡l (range) j−dÉ pwMÉ¡…−m¡l SeÉ 100(+-)2(+-) ¢euj¢V M¡−Vz Ex 5. 89P2P d¡f 1: 100-89=11 ,11P2P =121 ,q¡−a b¡L−m¡ 1 d¡f 2: . "
100-(2 x 11 )+1=79,AbÑ¡v Ešl qm 7921
HM¡−e ¢hLÒf ¢euj¢V qm ”given no (±)” AbÑ¡v fËcš pwMÉ (±) d¡f 1: 100-89=11 ,11P2P =121 ,q¡−a b¡L−m¡ 1 d¡f 2: 89-11+1=79 ,AbÑ¡v Ešl qm 7921
# HM¡−e fËcš pwMÉ¢V qm 89 ,−pM¡e −b−L ¢h−k¡Ngm 11 −L ¢h−k¡N −cJu¡ q−u−R ¢L¿¹¥ j−e l¡M−h q¡−a l¡M¡ 1 −L phpju −k¡N Ll−a quz # HC ¢hLÒf ¢euj¢V H−r−œ hÉhq¡l Ll¡ i¡m z Ex 6. 117² d¡f 1: 117-17=17 , 17P2P=289 , q¡−a b¡L−m¡ 2 d¡f 2: .
117+17=134, 134+2=136 AbÑ¡v Ešl qm 13689.
h¡L£ −r−œJ HLCi¡−h ¢euj…−m¡−L fË−u¡N Ll−a q−hz P P
Ex 6. 139² 1. 150-139=11 , hNÑ 121 , q¡−a b¡L−m¡ 1 2. 225-3×11=225-33=192, 192+1=193 , Ešl qm 19, 321 Ex 7. 164² 1. 164-150=14, 14²=196, q¡−a b¡L−m¡ 1 2. 225+3(14)=225+42=267 ,267+1=268 , Ešl qm 26,896 Ex 8. 512² 1. 12²=144 , q¡−a b¡L−m¡ 1, 2500+10(12)+1=2621, Ešl qm 262144 Hh¡l −a¡jl¡ ¢e−Sl¡ ¢LR¥ Ec¡lqZ AiÉ¡p L−l¡ z ¢euj 06 : Hh¡l Bjl¡ c¤¢V pwMÉ¡l Oe ¢nMh . HM¡−e HL¢V EQ¡lq−el p¡q¡−kÉ hÉ¡f¡l¢V −h¡T¡h¡l −Qø¡ Ll¡ qmz −kje dl Ex 1. (18)P 3
d¡f 1. fËb−j pwMÉ¡c¤¢Vl Ae¤f¡a −hl Ll−a q−h −kje HM¡−e 1:8
d¡f 2. Hh¡l fËbj pwMÉ¡¢Vl Oe Ll J a¡lfl Oe pwMÉ¡¢V−L Ae¤f¡a ¢c−u flfl ¢aeh¡l …Z L−l HL¢V m¡C−e −mMz −kje : 1P3P=1 / 1 x 8=8 / 8 x 8=64 / 64 x 8 =512 d¡f 3. ¢àa£u J a«a£u pwMÉ¡¢Vl ¢à…Z L−l a¡−cl AhÙÛ¡−el ¢WL e£−Q a¡¢c−L −mM z d¡f 4. Aa:fl Efl J e£−Ql pwMÉ¡…−m¡−L −k¡N Ll ( HL Ar−ll −hn£ q−m q¡−a ¢e−a q−h)z a¡q−mC Ešl¢V f¡Ju¡ k¡−hz e£−Ql ¢Qœ¢V −cMz 1. 2. 4.
18
Abѡv 1 :8
4
3
1 =1 / P
P
24
51
1x8=8
/ 8 x 8=64
/ 64 x 8 =512
8 x 2=16 64 x 2=128 -----------------------------------------------------------(4+1) 5
(24+8+16) =48
(51+64+128) =243
51
2
Ex 2. (33) P3
Hh¡l −a¡jl¡ ¢e−Sl¡ ¢LR¥ Ec¡lqZ AiÉ¡p L−l¡ z ¢euj 07 : Hh¡l Bjl¡ −k −L¡e pwMÉ¡l hNÑjm § Ll¡l p¡d¡lZ fÜ¢a ¢e−u B−m¡Qe¡ LlhzB−N e£−Ql RL¢V i¡mi¡−h mr L−l¡z
f¤ZÑhNÑ l¡¢nl HLL O−ll Aˆ hNÑjm § l¡¢nl HLL O−ll Aˆ
P1P
P4P
5 P P
6 P P
P9P
P1,9
P2,8
5 P
P4,6
P3,7
P1+9=10P
P2+8=10P
P5+5=10P
P4+6=10P
P3+7=10P
#1. −L¡−e¡ pwMÉ¡l −no Arl¢V k¢c pwMÉ¡¢Vl hNÑj§m f¡Ju¡ pñh euz
2,3,7,8 qu
,a¡q−m −pC
#2. Arl¢V−a k¢c ¢h−S¡l pwMÉ¡L n§eÉ b¡−L, a¡q−m −pC pwMÉ¡¢Vl hNÑj§m f¡Ju¡ pñh euz e£−Ql RL¢V i¡mi¡−h mr L−l¡z p£j¡
hNÑ
hNÑ
P
1²=1
1-3
P
P
P
P
p£j¡
9²=81 P
hNÑ
P
P
P
81-99 P
p£j¡ P
P
P
17²=289
289-323
P
P
P
hNÑ P
P
p£j¡ P
P
25²=625 P
P
625-675 P
P
2²=4
4-8
P
P
P
10²=100
P
P
P
3²=9
9-15
11²=121
P
P
P
P
4²=16 P
16-24
P
P
P
P
P
P
7²=49 P
P
P
P
P
64-80 P
P
225-255
P
P
P
361-399 P
P
400-440 P
P
441-483
P
P
484-528 P
529-575 P
256-288 P
P
P
576-624 P
P
P
P
900-960
P
P
P
31²=961 P
P
841-899 P
30²=900
P
24²=576
P
P
P
P
784-840 P
29²=841
P
23²=529
P
P
P
P
P
P
28²=784
P
22²=484
729-783
P
P
21²=441
P
27²=729
P
P
P
P
P
676-728
P
P
20²=400
P
16²=256 P
P
P
P
15²=225
P
8²=64
196-224 P
26²=676
P
19²=361
P
P
49-63
P
169-195 P
14²=196
P
P
P
P
P
36-48
P
144-168 P
P
324-360
P
P
13²=169
P
6²=36
P
P
25-35
P
121-143
P
P
18²=324
P
12²=144
P
5²=25
P
P
100-120
P
961-1023
P
P
32²=1024
P
P
P
1024P
1088 P
¢euj 1: H−Lh¡−l X¡e¢c−Ll −no c¤¢V pwMÉ¡−L Bm¡c¡ L−l ¢m−M . e¡Jz 2: f−s b¡L¡ pwMÉ¡¢V −L¡e p£j¡l j−dÉ f−l −p¢V h¡l L−l¡ J . a¡−L h¡j¢c−L −mMz (H¢V qm Eš−ll h¡jfr) 3: p£j¡¢V−L a¡l f−ll pwMÉ¡ ¢c−u …Z L−l¡ J −cM Bm¡c¡ . . Ll¡ pwMÉ¡¢V JC …Zgm¢Vl −Q−u h−s¡ e¡ −R¡−V¡. 4: k¢c h−s¡ qu ,a¡q−m hNÑj§m l¡¢nl HLL O−ll A−ˆl h−s¡ . pwMÉ¡¢V qm X¡efr (R.H.S) . Ex1.√ (6241)
hÉMÉ¡: 1. X¡e¢c−Ll −no c¤¢V pwMÉ¡ qm 41 k¡−L Bm¡c¡ L−l ¢m−M ¢em¡jz 2. f−s b¡L¡ pwMÉ¡¢V AbÑ¡v 62-Hl p£j¡ qm 7,a¡C H¢V qm Eš−ll h¡jfr. 3. 7 −L a¡l f−ll pwMÉ¡ ¢c−u …Z L−l −cMm¡j 56 Hl −Q−u 62 h−s¡z 4. a¡C 1 Hl SeÉ fcš h−s¡ pwMÉ¡¢V AbÑ¡v 9 −L Bjl¡ X¡efr ¢qp¡−h ¢em¡jz p¤¤al¡w Ešl¢V qm 79z Ex2 .√ (2601) 1. 26 / 01 2. 26 Hl
p£j¡ qm
5 ,h¡j¢c−L 5
−mMz 3. HMe 5×6=30 ,26 qm −R¡V 30 Hl −Q−uz 4. 1 −L Bjl¡ a¡C X¡efr ¢qp¡−h ¢em¡j J Ešl qm 51. Ex1.√ (2704) 1. 27 / 04 2. 27 Hl
p£j¡ qm h¡j¢c−L
5
−mMz
3. HMe 5×6=30 , 27 qm
−R¡V 30 Hl −Q−uz a¡C 2 −L Bjl¡ a¡C X¡efr ¢qp¡−h ¢em¡j J Ešl qm 52
a−h
5
¢c−u −no pwMÉ¡ …¢ml −r−š pq−SC Ešl f¡Ju¡ u¡uz
Ex 4. √(99225) 1. 992 /25 2. 992 Hl
p£j¡ qm 31 . 3. HM¡−e X¡efr phpju q−h 5 L¡lZ R−L Bj¡−cl L¡−R Bl ¢LR¥ e¡Cz 4. Ešl qm 315
.
Exercise 1.√34225
.
2.√105625
3.√(0.00126025)
4. √2209
5. √2916
6. √2116
7. √15129
8.√ 16129
9. √55696
10. √66564
11. √8.8804
12. √0.00101124
¢euj 08 : Hh¡l Bjl¡ −k −L¡e pwMÉ¡l Oej§m Ll¡l p¡d¡lZ fÜ¢a ¢e−u B−m¡Qe¡ LlhzB−N e£−Ql RL¢V i¡mi¡−h mr L−l¡z f¤ZÑOe l¡¢nl HLL O−ll Aˆ
1
2
3
4
5
6
4
5
6
Oe l¡¢nl HLL 1 O−ll Aˆ
8
7
2+8=10
3+7=10
Oe
Oe
P
p£j¡
P
1 =1 P
1-7 P
P
p£j¡ P
216-342 P
8
9
3
2
9
7+3=10
8+2=10
Oe
P
6 =216 P
7
P
P
p£j¡
11 =1331 P
P
1331-1727 P
P
2 =8
8-26
7 =343
P
P
P
P
3 =27 P
27-63
P
P
P
P
64-124
P
343-511
P
P
P
P
125-215
10 =1000
P
P
P
P
P
21=9261
25=15625
P
P
Ex1. (46656) 1. 2. 3. 4.
P
P
2197-2743
P
P
14 =2744
P
P
1000-1330 P
P
13 =2197
P
P
1728-2196
P
729-999
P
5 =125 P
P
512-728
P
9 =729
P
12 =1728
P
8 =512
P
4 =64 P
P
P
P
2744-3374
P
P
P
15 =3375
3375-4095
P
P
P
P
1/3 P
P
46 / 656 6 Hl SeÉ U
U
X¡e¢c−L 46 Hl p£j¡ qm 3 Ešl qm 36 P
Ex2 . (0.0020048383)
n.b. (1 cn¢jL
6 −mMz
1/3 P
P
1.
0.0020048 / 383
2.
3 Hl
3.
2048 Hl
3.
Ešl qm
U
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