Vedic Math Tricks for Square

 

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Vedic Math Tricks for the Square 

Introduction

Once a student develops an understanding of mental mathematics, he/she begin to think more systematically and creatively. Vedic Maths is a collection of techniques and sutras for solving mathematical problems quickly and easily. These tricks introduce wonderful applications of arithmetical computation, theory of numbers, mathematical and algebraic operations, higher-level mathematics, calculus, coordinate geometry, etc. 

Square: 

A square number is the product of a number multiplied by itself. It is commonly represented using exponent notation: a2 = a × a

Square 1 to 30 Table

Number	Square	Number	Square	Number	Square
(1)2	1	(11)2	121	(21)2	441
(2)2	4	(12)2	144	(22)2	484
(3)2	9	(13)2	169	(23)2	529
(4)2	16	(14)2	196	(24)2	576
(5)2	25	(15)2	225	(25)2	625
(6)2	36	(16)2	256	(26)2	676
(7)2	49	(17)2	289	(27)2	729
(8)2	64	(18)2	324	(28)2	784
(9)2	81	(19)2	361	(29)2	841
(10)2	100	(20)2	400	(30)2	900

Do you know how to square big numbers?

Squaring Tricks 

Type 1: Squaring of a number ending with 5.                           ("BY ONE MORE THAN THE PREVIOUS ONE")

Step 1: Add 1 to the first digit from the left and multiply by the number itself. 
Step 2: Add (25) at the end to the number obtained from step 1.

                        

Example: Find the square of 75 

Step 1: Add 1 to the left number (7) and multiply by the number itself, and 

7+1=8 and 7×8=56

Step 2: Add (25) at the end to the number obtained from step 1.

= 5,625  Ans

Example: Find the square of 205 

Step 1: Add 1 to the left number (20) and multiply by the number itself, and 

20+1=21 and 20×21=420

Step 2: Add (25) at the end to the number obtained from step 1.

= 42,025  Ans

Example: Find the square of 45 

Step 1: Add 1 to the left number (4) and multiply by the number itself, and 

4+1=5 and 4×5=20

Step 2: Add (25) at the end to the number obtained from step 1.

= 2,025  Ans

Type 2: Squaring of numbers less than 50 and numbers not ending with 5. 

Example: Find the square of 44

Solution:   We can write 44=50-6. 

Step 1: Square the first digit (5) of the first part (50), then add the part (-6)

5² = 25
= 25 + (-6) 
= 19

Step 2: Square the second part of the number (6) 

6² = 36

Step 3: Add the answers obtained in step 1 (19) and step 2 (36) 

19+36=55 (But keep last 2 digits of 55 → 55 and carry 0)

19+0=19

Combine step 3 and step 2

=1936 Ans

Example: Find the square of 39

Solution: We can write 39 = 50 – 11
Step 1: Square the first digit (5) of the first part (50), then add the part (–11)
5² = 25
= 25 + (–11)
= 14
Step 2: Square the second part of the number (11)
11² = 121
Step 3: Add the answers obtained in step 1 (14) and step 2 (121)
14 + 121 = 135
(But keep the last 2 digits of 135 → 35 and carry 1.)
14 + 1 = 15
Therefore, 1535 Ans

Example: Find the square of 46

Solution: We can write 46 = 50 – 4
Step 1: Square the first digit (5) of the first part (50), then add the part (–4)
5² = 25
= 25 + (–4)
= 21
Step 2: Square the second part of the number (4)
4² = 16
Step 3: Add the answers obtained in step 1 (21) and step 2 (16)
21 + 16 = 37

(But keep the last 2 digits of 37 → 37 and carry 0)

21+0=21

Combine steps 3 and 2

=2116 Ans

Type 3: Squaring of numbers greater than 50 and numbers not ending with 5. 

Example: Find the Square of 58.

Step 1: Subtract  50 from 58
 58−50=8  So our difference from the base is 8.

 Step 2: Square the Result of Step 1

  8² = 64

Step 3: Add the difference to 25

25 + 8 = 33 

Step 4: Combine the Results of Step 2 and Step 3.

 = 3364

So the square of 58 is 3364.

Example: Find the square of 72.

Step 1: Subtract 50 from 72

72 − 50 = 22. So our difference from the base is 22.

Step 2: Square the Result of Step 1

22² = 484. Write 84, carry 4. 

Step 3: Add the Difference to 25

25 + 22 = 47

Add carry 47+4=51

Step 4: Combine the Results of Step 2 and Step 3

 = 5184

So the square of 72 is 5184.

Example: Find the square of 67.

Step 1: Subtract  50 from 67

67 − 50 = 17. So our difference from the base is 17.

Step 2: Square the Result of Step 1

17² = 289. Write 89, carry 2. 

Step 3: Add the Difference to 25

25 + 17 = 42

Add carry 42+2=44

Step 4: Combine the Results of Step 2 and Step 3

 = 4,489

So the square of 67 is 4489.

Type 4: Squaring of numbers near their base 10, 100, 1000, and so on:

Example:  Find the square of 102

Step 1: Subtract  100 from 102

102 − 100 = 2. So our difference from the base is 2.

Step 2: Square the Result of Step 1

2² = 4; write 04 

Step 3: Add the Difference to the original number

102 + 2 = 104

Step 4: Combine the Results of Step 2 and Step 3

 = 10,404

So the square of 102 is 10,404.

Example: Find the square of 98.

Step 1: Subtract the number from the nearest base

100 − 98 = 2, so our difference from the base is 2.

Step 2: Subtract the difference from the original number

98 − 2 = 96  → First part of answer

Step 3: Square the difference

2² = 4 → write as 04 (2 digits)

Step 4: Combine the results

= 9604

So the square of 98 is 9604.

Example: Find the square of 998.

Step 1: Subtract the number from the nearest base

1000 − 998 = 2.  So our difference from the base is 2.

Step 2: Subtract the difference from the original number

998 − 2 = 996. First part of the answer

r

Step 3: Square the difference

2² = 4; write as 004 (3 digits because base = 1000)

Step 4: Combine the results

996 + 004 = 996004

So the square of 998 is 996004.

Type 5: Duplex method for any 2-digit number

The duplex of a single digit is the square of the digit itself. For a two-digit number, the duplex is twice the product of the digits. For a three-digit number, the duplex is twice the product of the first and third digits plus the square of the middle digit, and so on.

Duplex combination is a term used in terms of squaring and multiplication, denoted as D. The following are the basic methods used in duplex:

• D (a) = a²
• D (ab) = 2(ab)
• D (abc) = 2 (ac) + b²
• D (abcd) = 2 (ad) + 2 (bc)
• D (abcde) = 2 (ae) + 2 (bd) + c²

Example: Find the square of 12021.

Solution:

Step 1: Write the number 12021 in the form of digits: 
1, 2, 0, 2, 1.

Step 2: Calculate the duplex of the first digit: 
$D(1)=$1²=1
Step 3: Calculate the duplex of the first two digits:
$D(12)=2\cdot (1\cdot 2)=4$.
Step 4: Calculate the duplex of the first three digits: 
$D(120)=2\cdot (1\cdot 0)+$2²=4$=4$.

Step 5: Calculate the duplex of the first four digits:
$D(1202)=2\cdot (1\cdot 2)+2\cdot (2\cdot 0)=4$.
Step 6: Calculate the duplex of all five digits: 
$D(12021)=2\cdot (1\cdot 1)+2\cdot (2\cdot 2)+$0²=10

Step 7: Combine all the duplex values and sum them up to get the square of 12021:

$1+4+4+4+10=144504441$.

The square of 12021 using the duplex method is 144504441.

Example: Find the square of 23456.

Step 1: Write the number 23456 in the form of digits: 
2, 3, 4, 5, 6.

Step 2: Calculate the duplex of the first digit: 

D(2) = 2² = 4. Step 3: Calculate the duplex of the first two digits:

D(23) = 2⋅(2⋅3) = 12. Step 4: Calculate the duplex of the first three digits: 

D(234) = 2⋅(2⋅4) + 3² = 16 + 9 = 25. Step 5: Calculate the duplex of the first four digits: 

D(2345) = 2⋅(2⋅5) + 2⋅(3⋅4) = 20 + 24 = 44. Step 6: Calculate the duplex of all five digits: 

D(23456) = 2⋅(2⋅6) + 2⋅(3⋅5) + 4² = 24 + 30 + 16 = 70. Step 7: Combine all the duplex values (with the duplex method carry/placement) to get the square: D values = 4, 12, 25, 44, 70. 23456² = 550183936

Type 6: Squaring of a number near to their sub-base:

Example 1: Find the square of 308.

Solution: 308=300+8

Step 1: Add the second part of number 8 to the given number (308) and multiply it by 3

3 × (308 + 8) 

= 3 × 316

= 948

Step 2: Square the second part of the 8²

^{82 = 64}

Step 3: Combine the numbers from steps 1 and 2

=94864

Example: Find the square of 414.

Solution: 414=400+14

Step 1: Add the second part of number 14 to the given number (414) and multiply it by 4

4 × (414 + 14) 

= 4 × 428

= 1712

Step 2: Square the second part of the 14

^{142 = 196}

^{write 96, carry 1}

^{Add carry 1712+1=1713}

Step 3: Combine the numbers

=171,396

Example: Find the square of 703.

Solution: 703=700+3

Step 1: Add the second part of number 3 to the given number (703) and multiply it by 7

7 × (703 + 3) 

= 7 × 706

= 4942

Step 2: Square the second part of the 3

^{32 = 9 Write 09}

Step 3: Combine the numbers

=494209 Ans

Conclusion 

It is very important to make children learn some of the Vedic math tricks and concepts at an early stage to build a strong foundation for the child. It is one of the most refined and efficient mathematical systems possible. Math has specific principles for performing various calculations.