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 "

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ऊर्ध्वतिर्यग्भ्याम् " Ūrdhvatiryagbhyām " " Vertical and crosswise "

Here we will be learning the Squares of Fiver digit Numbers. 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