Squares of four 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 "

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ऊर्ध्वतिर्यग्भ्याम् " Ūrdhvatiryagbhyām " " Vertical and crosswise "

Here we will be learning the Squares of Four digit Numbers. The detailed steps will be explained through the below examples.

Squares of Four digit Numbers

Example : 2314²

Fourth Column	Third Column	Second Column	First Column
2	3	1	4

Step 1: 2 3 1 4 Starting from right take the duplex of the first column digit .D(4) = 4² = 16

Step 2 : 2 3 1 4 Take the duplex of the second column and first column digit D(14) = 2(1 X 4) = 8

Step 3 : 2 3 1 4 Take the duplex of the third column , second column and first column digits D(314) = 2(3 X 4) + 1² = 24 + 1 = 25

Step 4 : 2 3 1 4 Take the duplex of the fourth column, third column , second column and first column D(2314) = 2(2 X 4) + 2(3 X 1) = 16 + 6 = 22

Step 5 : 2 3 1 4 Take the duplex of the fourth column, third column , second column D(231) = 2(2 X 1) + 3² = 4 + 9 = 13

Step 6 : 2 3 1 4 Take the duplex of the fourth column, third column D(23) = 2(2 X 3) = 12

Step 7 : 2 3 1 4 Take the duplex of the fourth column D(2) = 2² = 4

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
4	12	13	22	25	8	16
4	12	13	22	25	8	Put 6 and carry over 1
4	12	13	22	25	8 + Carry over (1)	6
4	12	13	22	25	9	6
4	12	13	22	Put 5 and carry over 2	9	6
4	12	13	22 + Carry over (2)	5	9	6
4	12	13	24	5	9	6
4	12	13	Put 4 and carry over 2	5	9	6
4	12	13 + Carry over (2)	4	5	9	6
4	12	15	4	5	9	6
4	12	Put 5 and carry over 1	4	5	9	6
4	12 + Carry over (1)	5	4	5	9	6
4	13	5	4	5	9	6
4	Put 3 and carry over 1	5	4	5	9	6
4 + Carry over (1)	3	5	4	5	9	6
5	3	5	4	5	9	6

Answer : 2314² = 5354596

Example : 2736²

Fourth Column	Third Column	Second Column	First Column
2	7	3	6

Step 1: 2 7 3 6 Starting from right take the duplex of the first column digit .D(6) = 6² = 36

Step 2 : 2 7 3 6 Take the duplex of the second column and first column digit D(36) = 2(3 X 6) = 36

Step 3 : 2 7 3 6 Take the duplex of the third column , second column and first column digits D(736) = 2(7 X 6) + 3² = 84 + 9 = 93

Step 4 : 2 7 3 6 Take the duplex of the fourth column, third column , second column and first column D(2736) = 2(2 X 6) + 2(7 X 3) = 24 + 42 = 66

Step 5 : 2 7 3 6Take the duplex of the fourth column, third column , second column D(273) = 2(2 X 3) + 7² = 12 + 49 = 61

Step 6 : 2 7 3 6 Take the duplex of the fourth column, third column D(27) = 2(2 X 7) = 28

Step 7 : 2 7 3 6Take the duplex of the fourth column D(2) = 2² = 4

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
4	28	61	66	93	36	36
4	28	61	66	93	36	Put 6 and carry over 3
4	28	61	66	93	36 + Carry over (3)	6
4	28	61	66	93	39	6
4	28	61	66	93	Put 9 and carry over 3	6
4	28	61	66	93 + Carry over (3)	9	6
4	28	61	66	96	9	6
4	28	61	66	Put 6 and carry over 9	9	6
4	28	61	66 + Carry over (9)	6	9	6
4	28	61	75	6	9	6
4	28	61 + Carry over (7)	Put 5 and carry over 7	6	9	6
4	28	68	5	6	9	6
4	28	Put 8 and carry over 6	5	6	9	6
4	28 + Carry over (6)	8	5	6	9	6
4	34	8	5	6	9	6
4	Put 4 and carry over 3	8	5	6	9	6
4 + Carry over (3)	4	8	5	6	9	6
7	4	8	5	6	9	6

Answer : 2736² = 7485696

See Also

द्वैध उपयोग से चार अंकों की संख्याओं का वर्ग - भारती कृष्ण तीर्थ