Squares of three 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 Three digit Numbers. The detailed steps will be explained through the below examples.

Squares of Three digit Numbers

Example : 152²

This is a three digit number . Starting from the right we get the answer in five parts.

Here we will use only crosswise part of the sūra. We start from the right column and keep adding a column in each step. When we have taken all the columns we start leaving one column at a time from the right again till we are left with the left most column. Here we keep taking the duplex of the digits as we move from right to left. Let us understand it better with a few examples.

Third Column	Second Column	First Column
1	5	2

Step 1: 1 5 2 Starting from right take the duplex of the first column digit .D(2) = 2² = 4

Step 2 : 1 5 2 Take the duplex of the second column and first column digit D(52) = 2(5 X 2) = 20

Step 3 : 1 5 2 Take the duplex of the third column , second column and first column digits D(152) = 2(1 X 2) + 5² = 4 + 25 = 29

Step 4 : 1 5 2 Take the duplex of the third column and second column D(15) = 2(1 X 5) = 10

Step 5 : 1 5 2 Take the duplex of the third column D(1) = 1² = 1

Step 6 : Put the above values got from each step in the below table.

Step 5	Step 4	Step 3	Step 2	Step 1
1	10	29	20	4
1	10	29	Put 0 and carry over 2	4
1	10	29 + Carry over (2)	0	4
1	10	31	0	4
1	10	Put 1 and carry over 3	0	4
1	10 + Carry over (3)	1	0	4
1	13	1	0	4
1	Put 3 and carry over 1	1	0	4
1 + Carry over (1)	3	1	0	4
2	3	1	0	4

Answer : 152² = 23104

Example : 867²

Third Column	Second Column	First Column
8	6	7

Step 1: 8 6 7 Starting from right take the duplex of the first column digit .D(7) = 7² = 49

Step 2 : 8 6 7 Take the duplex of the second column and first column digit D(67) = 2(6 X 7) = 84

Step 3 : 8 6 7 Take the duplex of the third column , second column and first column digits D(867) = 2(8 X 7) + 6² = 112 + 36 = 148

Step 4 : 8 6 7 Take the duplex of the third column and second column D(86) = 2(8 X 6) = 96

Step 5 : 8 6 7 Take the duplex of the third column D(8) = 8² = 64

Step 6 : Put the above values got from each step in the below table.

Step 5	Step 4	Step 3	Step 2	Step 1
64	96	148	84	49
64	96	148	84	Put 9 and carry over 4
64	96	148	84 + Carry over (4)	9
64	96	148	88	9
64	96	148	Put 8 and carry over 8	9
64	96	148 + Carry over (8)	8	9
64	96	156	8	9
64	96	Put 6 and carry over 15	8	9
64	96 + Carry over 15	6	8	9
64	111	6	8	9
64	Put 1 and carry over 11	6	8	9
64 + Carry over 11	1	6	8	9
75	1	6	8	9

Answer : 867² = 751689

See Also

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