Square Numbers

What is a Square Number and why do we call it a Square Number?

What is a square number? Square number definition can be defined as numbers that are multiplied by itself. In other words, if a natural number is multiplied by itself it is known as a natural number. For example, 2 multiplied to itself (2x2) is a square number.  

Square numbers are named after a square because it forms an area of a square. Imagine that you have 9 items of 3 different categories. If we arrange it into a grid of 3 columns and 3 rows(3*3=9), placed vertically and horizontally. This grid thus will take the form of a square.

Squaring Decimals

Just like squaring whole numbers (integers) is easy, it is also very easy to square decimals and both are done in the same way.

For example, 1.53 = 1.53 x 1.53 = 2.3409

                      7.19 x 7.19 = 51.6961

Properties of Square Numbers

Property 1: A number that has 2, 3, 7, or 8 at its unit's place can never be a perfect square. In other words, square numbers never end in 2, 3, 7, or 8.

Example: 152, 7693, 14357, 88888, 798328 can never be perfect square as the numbers in their unit digit ends with 2,3,7 or 8

Property 2: The number of zeros at the end of a number determines if it is a perfect square or not. If a number ends with an even number of zeros then it can be a perfect square but if a number ends with an odd number of zeros then it might not be a perfect square. 

Example: 250000 is a perfect square as it has an even number of zeros.

 25000 is not a perfect square as it ends with an odd number of zeros.

Property 3: Squares of even numbers results in even numbers and squares of odd numbers result in odd numbers always.

Example : 8 2 = 8 x 8 = 64. (both 8 and 64 are even numbers)

7 2 = 7 x 7 = 49 (both 7 and 49 are odd numbers)

Property 4: On Squaring a natural number other than one the multiple will either be a multiple of 3 or will exceed a multiple of 3 by 1.

Example: 635,98,122 are not perfect squares because they leave the remainder 2 when divided by 3.

Property 5: On Squaring of a natural number other than one the multiple will either be a multiple of 4 or exceeds a multiple of 4 by 1.

Example: 67,146,10003 are not perfect squares because they leave the remainder 3,2,3 respectively when divided by 4.

Property 6: The unit’s number of the square of a natural number is the unit’s digit of the square of the digit at the unit's place of the given natural number.

Example :

1) Unit digit of square of 146.

Solution: Unit digit of 6 2 = 36 and also the unit digit of 36 is 6, therefore, the unit digit of square of 146 is 6.

2) Unit digit of square of 321.

Solution: Unit digit of 1 2 = 1, therefore, the unit digit of square of 321 is also 1.

Property 7: There are n natural numbers p and q so that p 2 = 2q 2.

Property 8: For each natural number n, (n + 1)2- n2 is equal to ( n + 1) + n.

Property 9: The square of a number n = to the sum of first n odd natural numbers.

1 2 = 1

2 2 = 1 + 3

3 2 = 1 + 3 + 5

4 2 = 1 + 3 + 5 + 7 and so on.

Property 10: If a natural number m is greater than 1,

(2m, m 2 - 1, m 2 + 1) is a Pythagorean triplet.

Table of Square Numbers-

1 x 1

1


21 x 21

441


41 x 41

1681

2 x 2

4


22 x 22

484


42 x 42

1764

3 x 3

9


23 x 23

529


43 x 43

1849

4 x 4

16


24 x 24

576


44 x 44

1936

5 x 5

25


25 x 25

625


45 x 45

2025

6 x 6

36


26 x 26

676


46 x 46

2116

7 x 7

49


27 x 27

729


47 x 47

2209

8 x 8

64


28 x 28

784


48 x 48

2304

9 x 9

81


29 x 29

841


49 x 49

2401

10 x 10

100


30 x 30

900


50 x 50

2500

11 x 11

121


31 x 31

961


51 x 51

2601

12 x 12

144


32 x 32

1024


52 x 52

2704

13 x 13

169


33 x 33

1089


53 x 53

2809

14 x 14

196


34 x 34

1156


54 x 54

2916

15 x 15

225


35 x 35

1225


55 x 55

3025

16 x 16

256


36 x 36

1296


56 x 56

3136

17 x 17

289


37 x 37

1369


57 x 57

3249

18 x 18

324


38 x 38

1444


58 x 58

3364

19 x 19

361


39 x 39

1521


59 x 59

3481

20 x 20

400


40 x 40

1600


60 x 60

3600

 

61 x 61

3721


81 x 81

6561

62 x 62

3844


82 x 82

6724

63 x 63

3969


83 x 83

6889

64 x 64

4096


84 x 84

7056

65 x 65

4225


85 x 85

7225

66 x 66

4356


86 x 86

7396

67 x 67

4489


87 x 87

7569

68 x 68

4624


88 x 88

7744

69 x 69

4761


89 x 89

7921

70 x 70

4900


90 x 90

8100

71 x 71

5041


91 x 91

8281

72 x 72

5184


92 x 92

8464

73 x 73

5329


93 x 93

8649

74 x 74

5476


94 x 94

8836

75 x 75

5625


95 x 95

9025

76 x 76

5776


96 x 96

9216

77 x 77

5929


97 x 97

9409

78 x 78

6084


98 x 98

9604

79 x 79

6241


99 x 99

9801

80 x 80

6400


100 x 100

10000

 

We can also make a square numbers list for easier reference. 


FAQ (Frequently Asked Questions)

Q1. What is the Relation Between Square Numbers and Square Root?

Ans. When a square root is multiplied, it produces a square number. Thus, a square root has an inverse operation than the square number. Let’s take an example, the square root of 36 is 6 because 6 x 6 = 36. Finding the square root of a number can be much trickier than calculating the square number. Therefore, the modern calculator has the square root button to make the calculation easier.

Q2. How do we Square Negative Numbers?

Ans. We all know that when two negative or two positive numbers are multiplied to each other then the result will always be a positive number. For example, if we multiple -5 to -5 then the result will be 10. If 5 will be multiplied to 5, the result will be 10 as well. But what if a negative number is multiplied to a positive number or vice versa? The result would be a negative number. For example, if -3 is multiplied to 3, the result would be -9. A number cannot be a square number if it is a negative number because -3 and 3 are very different from each other.