विनजीत वैदिक अंकगणित पुस्तक || 1 || अध्याय 12.01 || गुणा (01) अन्त्योय:दशकेऽपि विधि
Vinjeet Vedic Arithmetic Book || 1 || Chapter 12.01 || Multiplication (01) Antoy:Dashkeipi Method
Author
लेखक
ॐ जितेन्द्र सिंह तोमर
(M.A., B. Ed., MASSCOM, DNYS )
(Specialist in Basic and Vedic Maths)
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अध्याय 12.03
गुणन
– 5 अंत संख्याओं का
– अन्त्योदश केऽपि संख्याओं द्वारा
A B
A D जहां B + D =10
AוA/B×D
Chapter 12.01
Multiplication
– 5 ending numbers
– By Antyodash Keapi numbers
A B
A D where B + D =10
AוA/B×D
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– पूर्व्योदश केऽपि संख्याओं द्वारा
51 से 59 तक के गुणा करना
– by poorvyodash keapi sankhyaon
Multiplying 51 to 59
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A B
C D जहां A + C =10
A×C+B/B×B
– अन्त्योशत केऽपि संख्याओं द्वारा
– अन्त्योसहस्र केऽपि संख्याओं द्वारा
– By Antyayo sat keapi numbers
– By Antyoshasra Keapi numbers
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Ekadhikane Purvena Subsutra
(Antyayohdasakeapi अन्त्योय:दशकेऽपि)
Multiplycation of numbers ending whose sum of unit is 10.
गुणा
निखिलांक उपप्रमेय नंबर 2
→अन्तोदर्शकेऽपि
अब किसी भी एक 5 से अंत होने वाली संख्या के वर्ग पर विचार कीजिये :
25² = 25 x 25
A B × A B जहां पर (B + B = 10)
हम देखते हैं कि इनमें दोनों के 'पूर्व' अंक एक समान ही हैं अर्थात 2 है और अंतिम' अंकों का योग 10 है। अर्थात 'अंतिम' अंक 5 और 5 हैं जिनका योग 10 है।
जिन अन्तोदर्श अंकों का योग और 10 या 10 का गुणज होता है उन्हें अन्तोदर्शांक कहते हैं।
A B × A C जहां पर (B + C = 10)
A × •A / B × C
Multiply
Nikhilank Corollary No. 2
→Antyodash Keapi
Now consider the square of any number ending with 5:
25² = 25 x 25
A B × A B where (B + B = 10)
We see that both of them have the same 'first' digit i.e. 2 and the sum of the 'last' digits is 10. That means the 'last' numbers are 5 and 5 whose sum is 10.
The antodarshan numbers whose sum and number are 10 or multiples of 10 are called antodarshan numbers.
A B × A C where (B + C = 10)
A × •A / B × C
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अन्तोदर्शकेऽपि विधि से वर्ग
सूत्र
=(निखिलांक) × (निखिलांक का एकाधिकेन) / (अन्तोदर्शांक)1 × (अन्तोदर्शांक)2
इस प्रकार यह नियम उन सभी संख्याओं पर भी लागू होता है जिनमें अंतिम अंको का योग 10 हो।
Squaring by Antyodash Keapi Method
Formula
=(Nikhil Ank) × (Nikhil Ank Ka Ekaadhiken) / (Antodarshaank)1 × (Antodarshaank)2
Thus, this rule also applies to all those numbers in which the sum of the last digits is 10.
सूत्र - एकाधकेन पूर्वेण -
पहले से एक अधिक के द्वारा
(One more than the existing one)
प्रथम स्थिति (First Condition)
उपसूत्र - अन्तोदर्शकेऽपि
अंतिम अंकों का योग दस।
(Sum of last digits is ten.)
Sutra – Ekadhaken Purvena –
by one more than before
(One more than the existing one)
First Condition
Upsutra - Antyodash Keapi
The sum of the last digits is ten.
(Sum of last digits is ten.)
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जब किन्हीं दो संख्याओं के इकाई-अंकों (onces-place) का योग (Sum) दस (ten) हो तथा दहाई (tens) या शेष (rest) अंक (digits) समान ( equal) हो तो हम इस सूत्र का उपयोग करते हैं।
When the sum of the onces-place digits of any two numbers is ten and the tens or rest digits are equal, then we use this formula. .
As-
जैसे-
11x19, (1+9=10)
22 x 28, (2+8=10)
57 x 43 (7+3=10)
51x 49 (1+9=10)
98 x 92 (8+2=10)
Q. Find the product of (ab × ac), (Where b + c = 10)
a b
× a c
a(•a) / b×b
(Where b + c = 10)
Q.1 Find the product of 16 × 14
1 6
× 1 4
1 (•1) / 4 × 6
1×2 / 24
2/24
Q.2 Find the product of 26 × 24
2 6
× 2 4
2 (•2) / 4 × 6
2×3 / 24
6/24
Q.3 Find the product of 37 × 33
3 7
× 3 3
3 (•3) / 3 × 7
3×4 / 22
12/24
Q.4 Find the product of 97 × 93
9 7
× 9 3
9 (•9) / 3 × 7
9×10 / 21
90/21
यदि संख्या में इकाई के अंकों का योग 10 है।
When the sum of the onces-place digits is 10.
25 × 25, 48 × 42, 56 × 54, 53 × 57, 69 ×61, 105 × 105, 125 × 125, 138 × 132, 416 × 414, 513 × 517.
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इनके लिए एक उपसुत्र अन्तोदर्शकेऽपि का प्रयोग करते है।
(अन्तो+दर्श+केऽपि = दस से अंत होने वाले के लिए भी)
वैसे 'अन्तोदर्शकेऽपि' का अर्थ है: अंतिम वाले का योग 10 हो या 10ⁿ हो। जहां n एक प्राकृत संख्या है।
अर्थात् यह 10, या 100, या 1000, या 10000 या .....। 10 का कोई भी गुणज हो सकता है।
For these a sub-thread endoscope is used.
(Anto+darsh+keipi = also for the one ending with ten)
By the way, 'Antodarshkepi' means: The sum of the last one should be 10 or 10ⁿ. Where n is a natural number.
That means it is 10, or 100, or 1000, or 10000 or ..... Can be any multiple of 10.
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अन्तोदर्शकेऽपि (आधार 10 के लिए)
Antodarshkepi (for base 10)
102x108, (2+8=10)
1998x1992 (8+2=10)
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5 से अंत होने वाली संख्याओं के लिए।
For 5 ending numbers or numbers ended with 5
(a5)² = ( 10 a + 5)²
= ( 10a + 5) ( 10a + 5)
= 10a (10a + 5) + 5 ( 10a + 5)
= 100a² + 50a + 50a + 25
= 100a² + 100a + 25
= 100s(a + 1) + 25
= a(a + 1) सैकड़ा (hundreds) / 25
इस प्रकार हमें [ a(a+1) सैकड़ा (hundreds) / 25 ] निम्न संख्या प्राप्त हुई ।
इसे हम इस प्रकार भी लिख सकते हैं।
In this way we got the following number [ a(a+1) hundreds / 25].
We can also write it this way.
[ a(•a) / 5² ]
उदाहरण :- निम्न संख्या प्राप्त हुई
Example :- Following numbers were obtained
(1) 25 × 25
= 2 × (•2) / 5²
= 2 × 3 / 25
= 6 25
(2) 35 × 35
= 3 × (•3) / 5²
= 3 × 4 / 25
= 12 25
(3) 45 × 45
= 4 × (•4) / 5²
= 4 × 5 / 25
= 20 25
(4) 55 × 55
= 5 × (•5) / 5²
= 5 × 6 / 25
= 30 25
जब किन्हीं दो संख्याओं के इकाई-अंकों (ones-place) का योग (Sum) दस (ten) हो तथा दहाई (tens) अंक (digits) समान ( equal) हो तो
When the sum of the ones-place digits of any two numbers is ten and the tens digits are equal, then
ab × ac. (b+c=10)
= (a+b)(a+c)
इस प्रकार हमें [ a(a+1) सैकड़ा (hundreds) / 25 ] निम्न संख्या प्राप्त हुई ।
इसे हम इस प्रकार भी लिख सकते हैं।
In this way we got the following number [ a(a+1) hundreds / 25].
We can also write it this way.
[ a(•a) / b×c ]
उदाहरण :- निम्न संख्या प्राप्त हुई
Example :- Following numbers were obtained
इसी तरह से
in the same way
(1) 11 × 19
= 1 × •1/1x9
= 1 × 2/1x9
= 2/09
= 209
ध्यान दें कि
चरमांक 1 × 9=9 आता है । जबकि हमें दो अंक चाहिए। अतः हमें 09 लिखना चाहिए।
अतः 11 × 19 =209 होगा।
Note That
The extremum is 1 × 9=9. Whereas we need two digits. Hence we should write 09.
Hence 11 × 19 = 209.
(2) 13x17
= 1 × •1/3 × 7
= 1 × 2/21
= 2/21
= 221
अतः 13 × 17 =221.
(3) 43x47
=4 × •4\3 × 7
=4 × 5\3 × 7
=20\21
=2021
अतः 43 × 47 =2021.
(4) 53 × 57
= 5 × (•5) / 3 × 7
= 5 × 6 / 21
= 30 21
(5) 54x56
=5 × •5\4 × 6
=5 × 6\4 × 6
=30\24
=3024
अतः 54 × 56 =3024.
(6) 25 × 25
= 2 × (•2) / 5 ×5
= 2 × 3 / 25
= 6 25
(7) 38 × 32
= 3 × (•3) / 8 × 2
= 3 × 4 / 16
= 12 16
(8) 46 × 44
= 4 × (•4) / 6 × 4
= 4 × 5 / 24
= 20 24
(9) 119x111
= 11(9) × 11(1)
=11 × •11\9 × 1
=11 × 12\9 × 1
=132\09
=13209
अतः 119 × 111 = 13209
(10) 125 × 125
= 12(5) × 12(5)
=12 × •12\5 × 5
=12 × 13\5 × 5
=156\25
=15625.
अतः 125 × 125 =15625
(11) 1998 × 1992
यहां 1998 × 1992 में 'पूर्व' 199 है और 'अंतिम' में 8 और 2 है।
1998 × 1992
= 199(8) × 199(2)
=199 × •199\8 × 2
=199 × 200\16
=39800\16
=3980016
अतः 1998 × 1992 =3980016
Practice Time (1)
Q.1 Find the product of 11 × 19
Q.2 Find the product of 12 × 18
Q.3 Find the product of 13 × 17
Q.4 Find the product of 14 × 16
Q.5 Find the product of 15 × 15
Q.6 Find the product of 16 × 14
Q.7 Find the product of 17 × 13
Q.8 Find the product of 18 × 12
Q.9 Find the product of 19 × 11
Q.10 Find the product of 29 × 21
Q.11 Find the product of 21 × 29
Q.12 Find the product of 22 × 28
Q.13 Find the product of 23 × 27
Q.14 Find the product of 24 × 26
Q.15 Find the product of 25 × 25
Q.16 Find the product of 26 × 24
Q.17 Find the product of 27 × 23
Q.18 Find the product of 28 × 22
Q.19 Find the product of 31 × 39
Q.20 Find the product of 32 × 38
Q.21 Find the product of 43 × 47
Q.22 Find the product of 54 × 56
Q.23 Find the product of 65 × 65
Q.24 Find the product of 76 × 74
Q.25 Find the product of 87 × 83
Q.26 Find the product of 98 × 92
Q.27 Find the product of 49 × 41
Q.28 Find the product of 59 × 51
Q.29 Find the product of 61 × 69
Q.30 Find the product of 72 × 78
Q.31 Find the product of 83 × 87
Q.32 Find the product of 94 × 96
Q.33 Find the product of 45 × 45
Q.34 Find the product of 56 × 54
Q.35 Find the product of 67 × 63
Q.36 Find the product of 78 × 72
Q.37 Find the product of 85 × 85
Q.38 Find the product of 96 × 94
Q.39 Find the product of 97 × 93
Q.40 Find the product of 88 × 82
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Four three Digits
ab c
× ab d
ab(•ab) / c×d
Q.4 Find the product of 107 × 103
10 7
× 10 3
10(•10) / 3×7
10(11) / 22
110 / 22
11021
Q.5 Find the product of 216 × 214
21 6
× 21 4
21(•21) / 4×6
21(22) / 24
462 / 25
46224
Practice Time 2
Q.1 Find the product of 106 × 104
Q.2 Find the product of 116 × 114
Q.3 Find the product of 137 × 133
Q.4 Find the product of 107 × 103
Q.5 Find the product of 116 × 114
Q.6 Find the product of 216 × 214
Q.7 Find the product of 218 × 212
Q.8 Find the product of 176 × 174
Q.9 Find the product of 206 × 204
Q.10 Find the product of 307 × 313
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