Multiplication of four digit numbers by Bhārati Kṛṣṇa Tīrtha

Introduction

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 is 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

Four digit multiplication steps

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