View source for 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

'''<big>ऊर्ध्वतिर्यग्भ्याम्</big>'''

'''<big>Ūrdhvatiryagbhyām</big>'''

'''<big>(vertical and crosswise)</big>'''

will be used.
==Multiplication of a 4 digit number with 4 digit or 3 digit or 2 digit number<ref>{{Cite book|last=Singhal|first=Vandana|title=Vedic Mathematics For All Ages - A Beginners' Guide|publisher=Motilal Banarsidass|year=2007|isbn=978-81-208-3230-5|location=Delhi|pages=117 - 120}}</ref>==
[[File:Four digit multiplication steps-latest.png|alt=Four digit multiplication steps|none|thumb|850x850px|Four digit multiplication steps|link=https://alpha.indicwiki.in/index.php?title=File:Four_digit_multiplication_steps-latest.png]]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===
{| class="wikitable"
|+
!
!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.
{| class="wikitable"
|'''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
{| class="wikitable"
|+
!
!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.
{| class="wikitable"
|'''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
{| class="wikitable"
|+
!
!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.
{| class="wikitable"
|'''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==
==References==
<references />
[[Category:Mathematics]]
[[Category:Bhārati Kṛṣṇa Tīrtha]]
[[Category:Multiplication]]