Multiplication of four digit numbers by Bhārati Kṛṣṇa Tīrtha
The general formula of multiplication which will be applicable to multiplication of 4 digit number with 4 digit or 3 digit or 2 digit number.
Here the Sūtra
ऊर्ध्वतिर्यग्भ्याम्
Ūrdhvatiryagbhyām
(vertical and crosswise)
will be used.
Multiplication of a 4 digit number with 4 digit or 3 digit or 2 digit number[1]
Columns are counted from the right side
Step 1 :Vertically multiply the two digits in the first column.
Step 2 : Cross multiply the first digit of the first column with the second digit of the second column. Second digit of the first column with the first digit of the second column and add the two products.
Step 3 : Cross multiply the first digit of the first column with the second digit of the third column. Second digit of the first column with the first digit of the third column. Vertically multiply the two digits in the second column and add the three products.
Step 4 : Cross multiply the first digit of the first column with the second digit of the fourth column. Second digit of the first column with the first digit of the fourth column. Cross multiply the first digit of the second column with the second digit of the third column. Second digit of the second column with the first digit of the third column and add the four products
Step 5 : Cross multiply the first digit of the second column with the second digit of the fourth column. Second digit of the second column with the first digit of the fourth column. Vertically multiply the two digits in the third column and add the three products.
Step 6 : Cross multiply the first digit of the third column with the second digit of the fourth column. Second digit of the third column with the first digit of the fourth column and add the two products.
Step 7 :Vertically multiply the two digits in the fourth column.
Multiplication of a 4 digit number with 4 digit number
Example: 3124 X 2015
| Fourth Column | Third Column | Second Column | First Column | |
|---|---|---|---|---|
| First Digit | 3 | 1 | 2 | 4 |
| Second Digit | 2 | 0 | 1 | 5 |
Step 1 :Vertically multiply the two digits in the first column.- 4 X 5 = 20
Step 2 : Cross multiply the first digit of the first column with the second digit of the second column. Second digit of the first column with the first digit of the second column and add the two products.- (4 X 1) + (5 X 2) = 4 + 10 = 14
Step 3 : Cross multiply the first digit of the first column with the second digit of the third column. Second digit of the first column with the first digit of the third column. Vertically multiply the two digits in the second column and add the three products.- (4 X 0) + (5 X 1) + (2 X 1) = 0 + 5 + 2 = 7
Step 4 : Cross multiply the first digit of the first column with the second digit of the fourth column. Second digit of the first column with the first digit of the fourth column. Cross multiply the first digit of the second column with the second digit of the third column. Second digit of the second column with the first digit of the third column and add the four products - (4 X 2) + (5 X 3) + (2 X 0) + (1 X 1) = 8 + 15 + 0 + 1 = 24
Step 5 : Cross multiply the first digit of the second column with the second digit of the fourth column. Second digit of the second column with the first digit of the fourth column. Vertically multiply the two digits in the third column and add the three products. - (2 x 2) + (1 X 3) + (1 X 0) = 4 + 3 + 0 = 7
Step 6 : Cross multiply the first digit of the third column with the second digit of the fourth column. Second digit of the third column with the first digit of the fourth column and add the two products.- (1 X 2) + (0 X 3) = 2 + 0 = 2
Step 7 :Vertically multiply the two digits in the fourth column.- 3 x 2 = 6
Step 8 : Put the above values got from each step in the below table.
| Step 7 | Step 6 | Step 5 | Step 4 | Step 3 | Step 2 | Step 1 |
| 6 | 2 | 7 | 24 | 7 | 14 | 20 |
| 6 | 2 | 7 | 24 | 7 | 14 | Put 0 and carry over 2 |
| 6 | 2 | 7 | 24 | 7 | 14 + Carry over (2) | 0 |
| 6 | 2 | 7 | 24 | 7 | 16 | 0 |
| 6 | 2 | 7 | 24 | 7 | Put 6 and carry over 1 | 0 |
| 6 | 2 | 7 | 24 | 7 + Carry over (1) | 6 | 0 |
| 6 | 2 | 7 | 24 | 8 | 6 | 0 |
| 6 | 2 | 7 | Put 4 and carry over 2 | 8 | 6 | 0 |
| 6 | 2 | 7+ Carry over (2) | 4 | 8 | 6 | 0 |
| 6 | 2 | 9 | 4 | 8 | 6 | 0 |
Answer : 3124 X 2015 = 6294860
Multiplication of a 4 digit number with 3 digit number
Example: 2436 X 372
2436 X 0372
| Fourth Column | Third Column | Second Column | First Column | |
|---|---|---|---|---|
| First Digit | 2 | 4 | 3 | 6 |
| Second Digit | 0 | 3 | 7 | 2 |
Step 1 :Vertically multiply the two digits in the first column.- 6 X 2 = 12
Step 2 : Cross multiply the first digit of the first column with the second digit of the second column. Second digit of the first column with the first digit of the second column and add the two products.- (6 X 7) + (2 X 3) = 42 + 6 = 48
Step 3 : Cross multiply the first digit of the first column with the second digit of the third column. Second digit of the first column with the first digit of the third column. Vertically multiply the two digits in the second column and add the three products.- (6 X 3) + (2 X 4) + (3 X 7) = 18 + 8 + 21 = 47
Step 4 : Cross multiply the first digit of the first column with the second digit of the fourth column. Second digit of the first column with the first digit of the fourth column. Cross multiply the first digit of the second column with the second digit of the third column. Second digit of the second column with the first digit of the third column and add the four products - (6 X 0) + (2 X 2) + (3 X 3) + (7 X 4) = 0 + 4 + 9 + 28 = 41
Step 5 : Cross multiply the first digit of the second column with the second digit of the fourth column. Second digit of the second column with the first digit of the fourth column. Vertically multiply the two digits in the third column and add the three products. - (3 x 0) + (7 X 2) + (4 X 3) = 0 + 14 + 12 = 26
Step 6 : Cross multiply the first digit of the third column with the second digit of the fourth column. Second digit of the third column with the first digit of the fourth column and add the two products.- (4 X 0) + (3 X 2) = 0 + 6 = 6
Step 7 :Vertically multiply the two digits in the fourth column.- 2 x 0 = 0
Step 8 : Put the above values got from each step in the below table.
| Step 7 | Step 6 | Step 5 | Step 4 | Step 3 | Step 2 | Step 1 |
| 0 | 6 | 26 | 41 | 47 | 48 | 12 |
| 0 | 6 | 26 | 41 | 47 | 48 | Put 2 and carry over 1 |
| 0 | 6 | 26 | 41 | 47 | 48 + Carry over (1) | 2 |
| 0 | 6 | 26 | 41 | 47 | 49 | 2 |
| 0 | 6 | 26 | 41 | 47 | Put 9 and carry over 4 | 2 |
| 0 | 6 | 26 | 41 | 47 + Carry over (4) | 9 | 2 |
| 0 | 6 | 26 | 41 | 51 | 9 | 2 |
| 0 | 6 | 26 | 41 | Put 1 and carry over 5 | 9 | 2 |
| 0 | 6 | 26 | 41 + Carry over 5 | 1 | 9 | 2 |
| 0 | 6 | 26 | 46 | 1 | 9 | 2 |
| 0 | 6 | 26 | Put 6 and carry over 4 | 1 | 9 | 2 |
| 0 | 6 | 26 + Carry over 4 | 6 | 1 | 9 | 2 |
| 0 | 6 | 30 | 6 | 1 | 9 | 2 |
| 0 | 6 | Put 0 and carry over3 | 6 | 1 | 9 | 2 |
| 0 | 6 + Carry over (3) | 0 | 6 | 1 | 9 | 2 |
| 0 | 9 | 0 | 6 | 1 | 9 | 2 |
Answer : 2436 X 372 = 906192
Multiplication of a 4 digit number with 2 digit number
Example: 4763 X 73
4763 X 0073
| Fourth Column | Third Column | Second Column | First Column | |
|---|---|---|---|---|
| First Digit | 4 | 7 | 6 | 3 |
| Second Digit | 0 | 0 | 7 | 3 |
Step 1 :Vertically multiply the two digits in the first column.- 3 X 3 = 9
Step 2 : Cross multiply the first digit of the first column with the second digit of the second column. Second digit of the first column with the first digit of the second column and add the two products.- (3 X 7) + (3 X 6) = 21 +18 = 39
Step 3 : Cross multiply the first digit of the first column with the second digit of the third column. Second digit of the first column with the first digit of the third column. Vertically multiply the two digits in the second column and add the three products.- (3 X 0 + (3 X 7) + (6 X 7) = 0 + 21 + 42 = 63
Step 4 : Cross multiply the first digit of the first column with the second digit of the fourth column. Second digit of the first column with the first digit of the fourth column. Cross multiply the first digit of the second column with the second digit of the third column. Second digit of the second column with the first digit of the third column and add the four products - (3 X 0) + (3 X 4) + (6 X 0) + (7 X 7) = 0 + 12 + 0+ 49 = 61
Step 5 : Cross multiply the first digit of the second column with the second digit of the fourth column. Second digit of the second column with the first digit of the fourth column. Vertically multiply the two digits in the third column and add the three products. - (6 x 0) + (7 X 4) + (7 X 0) = 0 + 28 + 0 = 28
Step 6 : Cross multiply the first digit of the third column with the second digit of the fourth column. Second digit of the third column with the first digit of the fourth column and add the two products.- (7 X 0) + (0 X 4) = 0 + 0 = 0
Step 7 :Vertically multiply the two digits in the fourth column.- 4 x 0 = 0
Step 8 : Put the above values got from each step in the below table.
| Step 7 | Step 6 | Step 5 | Step 4 | Step 3 | Step 2 | Step 1 |
| 0 | 0 | 28 | 61 | 63 | 39 | 9 |
| 0 | 0 | 28 | 61 | 63 | Put 9 and carry over 3 | 9 |
| 0 | 0 | 28 | 61 | 63 + Carry over (3) | 9 | 9 |
| 0 | 0 | 28 | 61 | 66 | 9 | 9 |
| 0 | 0 | 28 | 61 | Put 6 and carry over 6 | 9 | 9 |
| 0 | 0 | 28 | 61 + Carry over (6) | 6 | 9 | 9 |
| 0 | 0 | 28 | 67 | 6 | 9 | 9 |
| 0 | 0 | 28 | Put 7 and carry over 6 | 6 | 9 | 9 |
| 0 | 0 | 28 + Carry over (6) | 7 | 6 | 9 | 9 |
| 0 | 0 | 34 | 7 | 6 | 9 | 9 |
| 0 | 0 | Put 4 and carry over 3 | 7 | 6 | 9 | 9 |
| 0 | 0 + Carry over (3) | 4 | 7 | 6 | 9 | 9 |
| 0 | 3 | 4 | 7 | 6 | 9 | 9 |
Answer : 4763 X 73 = 347699
See Also
चार अंकों की संख्याओं का गुणन - भारती कृष्ण तीर्थ
References
- ↑ Singhal, Vandana (2007). Vedic Mathematics For All Ages - A Beginners' Guide. Delhi: Motilal Banarsidass. pp. 117–120. ISBN 978-81-208-3230-5.
