Squares of five digit numbers using Duplex by Bhārati Kṛṣṇa Tīrtha
To find the square of any number we use the " Dwandwa yoga " with " Ūrdhvatiryagbhyām "
द्वन्द्व योग
" Dwandwa yoga " " Duplex combination process " |
+ | ऊर्ध्वतिर्यग्भ्याम्
" Ūrdhvatiryagbhyām " " Vertical and crosswise " |
Here we will be learning the Squares of Fiver digit Numbers[1]. The detailed steps will be explained through the below examples.
Squares of Five digit Numbers
Example : 310522
Fifth Column | Fourth Column | Third Column | Second Column | First Column |
---|---|---|---|---|
3 | 1 | 0 | 5 | 2 |
Step 1: 3 1 0 5 2Starting from right take the duplex of the first column digit .D(2) = 22 = 4
Step 2 :3 1 0 5 2 Take the duplex of the second column and first column digit D(52) = 2(5 X 2) = 20
Step 3 : 3 1 0 5 2 Take the duplex of the third column , second column and first column digits D(052) = 2(0 X 2) + 52 = 0 + 25 = 25
Step 4 : 3 1 0 5 2 Take the duplex of the fourth column, third column , second column and first column D(1052) = 2(1 X 2) + 2(0 X 5) = 4 + 0 = 4
Step 5 : 3 1 0 5 2 Take the duplex of the fifth column, fourth column, third column , second column and first column D(31052) = 2(3 X 2) + 2(1 X 5) +02 = 12 + 10 + 0 =22
Step 6 : 3 1 0 5 2 Take the duplex of the fifth column, fourth column, third column , second column D(3105) = 2(3 X 5) + 2(1 X 0) = 30 + 0 = 30
Step 7 : 3 1 0 5 2 Take the duplex of the fifth column, fourth column, third column D(310) = 2(3 X 0) + 12 = 0 + 1 = 1
Step 8 : 3 1 0 5 2 Take the duplex of the fifth column, fourth column D(31) = 2(3 X 1) = 6
Step 9 : 3 1 0 5 2 Take the duplex of the fifth column D(3) = 32 = 9
Step 10 : Put the above values got from each step in the below table.
Step 9 | Step 8 | Step 7 | Step 6 | Step 5 | Step 4 | Step 3 | Step 2 | Step 1 |
---|---|---|---|---|---|---|---|---|
9 | 6 | 1 | 30 | 22 | 4 | 25 | 20 | 4 |
9 | 6 | 1 | 30 | 22 | 4 | 25 | Put 0 and carry over 2 | 4 |
9 | 6 | 1 | 30 | 22 | 4 | 25 + Carry over (2) | 0 | 4 |
9 | 6 | 1 | 30 | 22 | 4 | 27 | 0 | 4 |
9 | 6 | 1 | 30 | 22 | 4 | Put 7 and carry over 2 | 0 | 4 |
9 | 6 | 1 | 30 | 22 | 4 + Carry over (2) | 7 | 0 | 4 |
9 | 6 | 1 | 30 | 22 | 6 | 7 | 0 | 4 |
9 | 6 | 1 | 30 | Put 2 and carry over 2 | 6 | 7 | 0 | 4 |
9 | 6 | 1 | 30 + Carry over (2) | 2 | 6 | 7 | 0 | 4 |
9 | 6 | 1 | 32 | 2 | 6 | 7 | 0 | 4 |
9 | 6 | 1 | Put 2 and carry over 3 | 2 | 6 | 7 | 0 | 4 |
9 | 6 | 1 + Carry over (3) | 2 | 2 | 6 | 7 | 0 | 4 |
9 | 6 | 4 | 2 | 2 | 6 | 7 | 0 | 4 |
Answer : 310522 = 964226704
Example : 123452
Fifth Column | Fourth Column | Third Column | Second Column | First Column |
---|---|---|---|---|
1 | 2 | 3 | 4 | 5 |
Step 1: 1 2 3 4 5 Starting from right take the duplex of the first column digit .D(5) = 52 = 25
Step 2 :1 2 3 4 5 Take the duplex of the second column and first column digit D(45) = 2(4 X 5) = 40
Step 3 : 1 2 3 4 5 Take the duplex of the third column , second column and first column digits D(345) = 2(3 X 5) + 42 = 30 + 16 = 46
Step 4 : 1 2 3 4 5 Take the duplex of the fourth column, third column , second column and first column D(2345) = 2(2 X 5) + 2(3 X 4) = 20 + 24 = 44
Step 5 : 1 2 3 4 5 Take the duplex of the fifth column, fourth column, third column , second column and first column D(12345) = 2(1 X 5) + 2(2 X 4) + 32 = 10 + 16 + 9 = 35
Step 6 : 1 2 3 4 5 Take the duplex of the fifth column, fourth column, third column , second column D(1234) = 2(1 X 4) + 2(2 X 3) = 8 + 12 = 20
Step 7 : 1 2 3 4 5 Take the duplex of the fifth column, fourth column, third column D(123) = 2(1 X 3) + 22 = 6 + 4 = 10
Step 8 : 1 2 3 4 5 Take the duplex of the fifth column, fourth column D(12) = 2(1 X 2) = 4
Step 9 : 1 2 3 4 5 Take the duplex of the fifth column D(1) = 12 = 1
Step 10 : Put the above values got from each step in the below table.
Step 9 | Step 8 | Step 7 | Step 6 | Step 5 | Step 4 | Step 3 | Step 2 | Step 1 |
---|---|---|---|---|---|---|---|---|
1 | 4 | 10 | 20 | 35 | 44 | 46 | 40 | 25 |
1 | 4 | 10 | 20 | 35 | 44 | 46 | 40 | Put 5 and carry over 2 |
1 | 4 | 10 | 20 | 35 | 44 | 46 | 40 + Carry over (2) | 5 |
1 | 4 | 10 | 20 | 35 | 44 | 46 | 42 | 5 |
1 | 4 | 10 | 20 | 35 | 44 | 46 | Put 2 and carry over 4 | 5 |
1 | 4 | 10 | 20 | 35 | 44 | 46 + Carry over(4) | 2 | 5 |
1 | 4 | 10 | 20 | 35 | 44 | 50 | 2 | 5 |
1 | 4 | 10 | 20 | 35 | 44 | Put 0 and carry over 5 | 2 | 5 |
1 | 4 | 10 | 20 | 35 | 44 + Carry over 5 | 0 | 2 | 5 |
1 | 4 | 10 | 20 | 35 | 49 | 0 | 2 | 5 |
1 | 4 | 10 | 20 | 35 | Put 9 and carry over 4 | 0 | 2 | 5 |
1 | 4 | 10 | 20 | 35 + Carry over (4) | 9 | 0 | 2 | 5 |
1 | 4 | 10 | 20 | 39 | 9 | 0 | 2 | 5 |
1 | 4 | 10 | 20 + Carry over (3) | Put 9 and carry over 3 | 9 | 0 | 2 | 5 |
1 | 4 | 10 | 23 | 9 | 9 | 0 | 2 | 5 |
1 | 4 | 10 | Put 3 and carry over 2 | 9 | 9 | 0 | 2 | 5 |
1 | 4 | 10 + Carry over (2) | 3 | 9 | 9 | 0 | 2 | 5 |
1 | 4 | 12 | 3 | 9 | 9 | 0 | 2 | 5 |
1 | 4 | Put 2 and carry over 1 | 3 | 9 | 9 | 0 | 2 | 5 |
1 | 4 + Carry over (1) | 2 | 3 | 9 | 9 | 0 | 2 | 5 |
1 | 5 | 2 | 3 | 9 | 9 | 0 | 2 | 5 |
Answer : 123452 = 152399025
See Also
द्वैध उपयोग से पांच अंकों की संख्याओं का वर्ग - भारती कृष्ण तीर्थ
References
- ↑ Singhal, Vandana (2007). Vedic Mathematics For All Ages - A Beginners' Guide. Delhi: Motilal Banarsidass. p. 232. ISBN 978-81-208-3230-5.