Square in Sadratnamālā

Introduction

Here we will be knowing a square of a number as mentioned in Sadratnamālā.

Verse 11

तुल्योभयहतिर्वर्ग एकतः क्रमशः पदैः ।

का वा धेनुस्तटे शुभ्रा तुङ्गो धावेद् वृषो यदि ॥ ११ ॥

The product of two equal numbers is the square of that number.

This verse gives the squares of single digit numbers from one to nine. The numbers are denoted using the Kaṭapayādi system of notation.

का	वा	धे नुः	त टे	शु रा	तु गः	धा वे	वृ षः	य दि
1	4	9 0	6 1	5 2	6 3	9 4	4 6	1 8
1	4	9	16	25	36	49	64	81

The squares of numbers from one to nine are one, four, nine, sixteen, twenty five, thirty six, forty nine, sixty four and eighty one in order.

Verse 12

स्थाप्योऽन्त्यवर्गः शेषोऽपि द्विघ्नान्त्यघ्नो निजोपरि ।

उपान्त्यादिम् अथोत्सार्य भूयोऽप्येवं क्रिया कृतिः ॥ १२ ॥

Having placed the square of the last digit (in the line of the square), the remaining part, multiplied by twice the last digit, is added (on the right of the square already placed). This procedure is repeated with the remaining digits (of the number).

Squaring a number of more than one digit is carried from left to right. The digit on the extreme left is called antya (the last) and that on its right is upāntya (the next to the last). Antya is squared and placed. The remaining part is multiplied from left to right by twice the last digit and placed as a part of the square already placed starting from the next place. The procedure is repeated until all the digits are finished.

The squares of the digits are to be added to alternate places from left to right. These places are called vargasthānas (the square places). The places in between vargasthāna are called avargasthānas (the non-square places). Hence the squares are to be placed in the vargasthāna and the product of twice the last digits and the remaining parts are to be placed in the avargasthāna.

Detailed procedure is explained with the following examples.

Example: Square of 12

Here

Antya	Upāntya
1	2

All the steps are done from bottom to top.

Step 1:

		Vargasthāna	Avargasthāna
A	2 X 1 X 2		4	Multiply twice of antya (1) with the rest of the digit (2) = 2 X 1 X 2 = 4 and write above 2
	12	1		Square of the antya (1) = 1 is placed above the antya (1)
	Given Number	1	2	Given number is written Here

Step 2: the above procedure to be repeated for the subsequent digits of the given number. Here the subsequent number is 2

		Vargasthāna	Avargasthāna	Vargasthāna
		1	4	4	Add the values in each column starting from A through B
B	22			4	square of the antya (2)= 4 is place above the antya (2)
A	2 X 1 X 2		4
	12	1
	Given Number	1	2
	1st Removal		1	2	Shift the number to one position to the right and strike off the antya (1). Here 2 will be antya

At this point we have reached the last digit of the given number. Hence the procedure will end. Add the values in each column starting from A through B. Hence Square of 12 = 144.

Example: Square of 123

Here

Antya	Upāntya
1	2	3

All the steps are done from bottom to top.

Step 1:

		Vargasthāna	Avargasthāna	Vargasthāna
A	2 X 1 X 3			6	Multiply twice of antya (1) with the rest of the digit (3) = 2 X 1 X 3 = 6 and write above 3
	2 X 1 X 2		4		Multiply twice of antya (1) with the rest of the digit (2) = 2 X 1 X 2 = 4 and write above 2
	12	1			Square of the antya (1) = 1 is placed above the antya (1)
	Given Number	1	2	3	Given number is written Here

Step 2: the above procedure to be repeated for the subsequent digits of the given number. Here the subsequent number is 2

		Varga	Avargasthāna	Vargasthāna	Avargasthāna
B	2 X 2 X 3			1	2	Multiply twice of antya (2) with the rest of the digit (3) = 2 X 2 X 3 = 12 and write above 3 such that 2 is above 3
	22			4		Square of the antya (2) = 4 placed above the antya (2)
A	2 X 1 X 3			6
	2 X 1 X 2		4
	12	1
	Given Number	1	2	3
	1st Removal		1	2	3	Shift the number to one position to the right and strike off the antya (1) . Here 2 will be antya and 3 will be upāntya

Step 3: the above procedure to be repeated for the subsequent digits of the given number. Here the subsequent number is 3

		Vargasthāna	Avargasthāna	Vargasthāna	Avargasthāna	Vargasthāna
		1	5	1	2	9	Add the values in each column starting from A through C
		1	4 + carry over (1)	1	2	9
		1	4	11 Put 1 and carry 1	2	9
C	32					9	Square of the antya (3) = 9 placed above the antya (3)
B	2 X 2 X 3			1	2
	22			4
A	2 X 1 X 3			6
	2 X 1 X 2		4
	12	1
	Given Number	1	2	3
	1st Removal		1	2	3
	2nd Removal			1	2	3	Shift the number to one position to the right and strike off the antya (2) . Here 3 will be antya.

At this point we have reached the last digit of the given number. Hence the procedure will end. Add the values in each column starting from A through C. Hence Square of 123 = 15129.

Example: Square of 1234

Here

Antya	Upāntya
1	2	3	4

All the steps are done from bottom to top.

Step 1:

		Vargasthāna	Avargasthāna	Vargasthāna	Avargasthāna
A	2 X 1 X 4				8	Multiply twice of antya (1) with the rest of the digit (4) = 2 X 1 X 4 = 8 and write above 4
	2 X 1 X 3			6		Multiply twice of antya (1) with the rest of the digit (3) = 2 X 1 X 3 = 6 and write above 3
	2 X 1 X 2		4			Multiply twice of antya (1) with the rest of the digit (2) = 2 X 1 X 2 = 4 and write above 2
	12	1				Square of the antya (1) = 1 is placed above the antya (1)
	Given Number	1	2	3	4	Given number is written Here

Step 2: the above procedure to be repeated for the subsequent digits of the given number. Here the subsequent number is 2

		Vargasthāna	Avargasthāna	Vargasthāna	Avargasthāna	Vargasthāna
B	2 X 2 X 4				1	6	Multiply twice of antya (2) with the rest of the digit (4) = 2 X 2 X 4 = 16 and write above 4 such that 6 is above 4
	2 X 2 X 3			1	2		Multiply twice of antya (2) with the rest of the digit (3) = 2 X 2 X 3 =12 and write above 3 such that 2 is above 3
	22			4			Square of the antya (2) = 4 placed above the antya (2)
A	2 X 1 X 4				8
	2 X 1 X 3			6
	2 X 1 X 2		4
	12	1
	Given Number	1	2	3	4
	1st Removal		1	2	3	4	Shift the number to one position to the right and strike off the antya (1) . Here 2 will be antya and 3 will be upāntya

Step 3: the above procedure to be repeated for the subsequent digits of the given number. Here the subsequent number is 3

		Vargasthāna	Avargasthāna	Vargasthāna	Avargasthāna	Vargasthāna	Avargasthāna
	2 X 3 X 4					2	4	Multiply twice of antya (3) with the rest of the digit (4) = 2 X 3 X 4 = 24 and write above 4 such that 4 is above 4
C	32					9		Square of the antya (3) = 9 placed above the antya (3)
B	2 X 2 X 4				1	6
	2 X 2 X 3			1	2
	22			4
A	2 X 1 X 4				8
	2 X 1 X 3			6
	2 X 1 X 2		4
	12	1
	Given Number	1	2	3	4
	1st Removal		1	2	3	4
	2nd Removal			1	2	3	4	Shift the number to one position to the right and strike off the antya (2). Here 3 will be antya and 4 will be upāntya .

Step 4: the above procedure to be repeated for the subsequent digits of the given number. Here the subsequent number is 4

		Vargasthāna	Avargasthāna	Vargasthāna	Avargasthāna	Vargasthāna	Avargasthāna	Vargasthāna
		1	5	2	2	7	5	6	Add the values in each column starting from A through D
		1	4 + carry over (1)	2	2	7	5	6
		1	4	12 put 2 and carry 1	2	7	5	6
		1	4	11 + carry over (1)	2	7	5	6
		1	4	11	12 put 2 and carry 1	7	5	6
		1	4	11	11 + carry over (1)	7	5	6
		1	4	11	11	17 put 7 and carry 1	5	6
D							1	6	Square of the antya (4) = 16 placed above the antya (4) such that 6 is above 4
C	2 X 3 X 4					2	4
	32					9
B	2 X 2 X 4				1	6
	2 X 2 X 3			1	2
	22			4
A	2 X 1 X 4				8
	2 X 1 X 3			6
	2 X 1 X 2		4
	12	1
	Given Number	1	2	3	4
	1st Removal		1	2	3	4
	2nd Removal			1	2	3	4
	3rd removal				1	2	3	4	Shift the number to one position to the right and strike off the antya (3). Here 4 will be antya.

At this point we have reached the last digit of the given number. Hence the procedure will end. Add the values in each column starting from A through D. Hence Square of 1234 = 1522756.

Verse 13

खण्डद्वयहतिर्द्विघ्नी खण्डद्विकृतियुत् कृतिः ।

यद्वाभीष्टोनाढ्यवधोऽभीष्टवर्गयुता कृतिः ॥ १३ ॥

The product of two parts (into which a number is split), multiplied by two and added to the sum of the squares of the parts is the square (of that number). Or the sum of the square of any arbitrary number and the product of the sum and difference of the given number and the arbitrary number is (also) the square (of that number).

Two more methods of finding the square are explained below.

First Method

In the first method, the given number is to be expressed as the sum of two parts. The sum of the squares of these parts to which twice the product of these parts is added, is the square. If a is the number, which is expressed as the sum of two numbers b and c, then

$a^{2}=b^{2}+c^{2}+2bc$

Example: Square of 75

a = 75 which is expressed as 70 + 5 here b = 70 , c = 5

$a^{2}=b^{2}+c^{2}+2bc$

$75^{2}=(70+5)^{2}=70^{2}+5^{2}+2\ X\ 70\ X\ 5$

$=70(70+10)+25$

$=70(80)+25$

$=5625$

Example: Square of 25

a = 25 which is expressed as 20 + 5 here b = 20 , c = 5

$a^{2}=b^{2}+c^{2}+2bc$

$25^{2}=(20+5)^{2}=20^{2}+5^{2}+2\ X\ 20\ X\ 5$

$=20(20+10)+25$

$=20(30)+25$

$=625$

Second Method

According to the second method, an arbitrary number is added to and subtracted from the given number and the product of these sum and difference is found. Add the square of arbitrary number to this product to get the square of the given number. If a is the given number and k is any arbitrary number, then

$a^{2}=(a+k)(a-k)+k^{2}$

Example: Square of 96

$a^{2}=(a+k)(a-k)+k^{2}$

a = 96 , k = 4

$96^{2}=(96+4)(96-4)+4^{2}$

$=(100)(92)+16$

$=9216$

Example: Square of 952

$a^{2}=(a+k)(a-k)+k^{2}$

a = 952 , k = 48

$952^{2}=(952+48)(952-48)+48^{2}$

$=(1000)(904)+2304$

$=906304$

See Also

सद्रत्नमाला में 'वर्ग'