Table of Contents

In this article we are going to discuss **square numbers**.

A couple of lines down you can find our calculator which can square any real number.

The terms **square number** and **perfect square** are synonym.

It’s called *perfect* square because the number is the square of an integer (whole number).

## Calculator

## Square Numbers Definition

A square number is the product of some integer with itself.

Therefore, it is expressed as n^{2} = n × n; **n** is a **whole number**.

As you can see, a square number is an **exponentiation** involving the **base n** and the** exponent **2, sometimes written as n^2.

In other words, it means **n to the power of 2**.

It follows from that that n can be positive or negative (If you square a negative number the result will be a positive), but usually the meaning is n > 0.

If n represents the length of a side, then n^2 expresses the area of the shape of a square with side length n.

This answers the question *how do you square a number*?

### Square Numbers Formula

Every square number equals the sum of the first n odd numbers:

Next, let’s look at some examples.

## Square Numbers Examples

- Example 1 – The square number of 3: 3 × 3 = 9.
- Example 2 – The square number of 4: 4 × 4 = 16.
- Example 3 – The square number of 12: 12 × 12 = 144.

If n^{2} is the square number, then n is the principal square root.

- Example 4: The square root of 169 (13 × 13) is ± 13.

To find out if a given integer is a perfect square, calculate its square roots.

If the square root is a whole number (has no decimal places), then the number is indeed a square number.

- Example 5: Is 18 a square number? No, √18 = ±4.2426406871 (has decimals).
- Example 6: Is 25 a square number? Yes, √25 = ±5 (has no decimals).
- Example 7: If the square of a number is y, then what is the original number x? By definition, the product y = x
^{2}, so x = √y.

This table is a shortcut to the most searched items in Google:

Square of 3 | Square of 8 |

Square of 50 | Square of 125 |

Square of 64 | Square of 9 |

Square of 26 | Square of 16 |

Square of 48 | Square of 72 |

Square of 65 | Square of 52 |

Square of 80 | Square of 12 |

Square of 4 | Square of 11 |

If something remains unclear do not hesitate getting in touch with us.

We are constantly trying to improve our site, and truly appreciate your feedback.

Ahead is the table in the context of this site.

## Square Numbers List

The table below contains the square numbers up to 100:

Number | Square Number | Square Root |
---|---|---|

0 | 0 | 0 |

1 | 1 | ± 1 |

2 | 4 | ± 1.4142135624 |

3 | 9 | ± 1.7320508076 |

4 | 16 | ± 2 |

5 | 25 | ± 2.2360679775 |

6 | 36 | ± 2.4494897428 |

7 | 49 | ± 2.6457513111 |

8 | 64 | ± 2.8284271247 |

9 | 81 | ± 3 |

10 | 100 | ± 3.1622776602 |

11 | 121 | ± 3.3166247904 |

12 | 144 | ± 3.4641016151 |

13 | 169 | ± 3.6055512755 |

14 | 196 | ± 3.7416573868 |

15 | 225 | ± 3.8729833462 |

16 | 256 | ± 4 |

17 | 289 | ± 4.1231056256 |

18 | 324 | ± 4.2426406871 |

19 | 361 | ± 4.3588989435 |

20 | 400 | ± 4.472135955 |

21 | 441 | ± 4.582575695 |

22 | 484 | ± 4.6904157598 |

23 | 529 | ± 4.7958315233 |

24 | 576 | ± 4.8989794856 |

25 | 625 | ± 5 |

26 | 676 | ± 5.0990195136 |

27 | 729 | ± 5.1961524227 |

28 | 784 | ± 5.2915026221 |

29 | 841 | ± 5.3851648071 |

30 | 900 | ± 5.4772255751 |

31 | 961 | ± 5.5677643628 |

32 | 1024 | ± 5.6568542495 |

33 | 1089 | ± 5.7445626465 |

34 | 1156 | ± 5.8309518948 |

35 | 1225 | ± 5.9160797831 |

36 | 1296 | ± 6 |

37 | 1369 | ± 6.0827625303 |

38 | 1444 | ± 6.164414003 |

39 | 1521 | ± 6.2449979984 |

40 | 1600 | ± 6.3245553203 |

41 | 1681 | ± 6.4031242374 |

42 | 1764 | ± 6.4807406984 |

43 | 1849 | ± 6.5574385243 |

44 | 1936 | ± 6.6332495807 |

45 | 2025 | ± 6.7082039325 |

46 | 2116 | ± 6.7823299831 |

47 | 2209 | ± 6.8556546004 |

48 | 2304 | ± 6.9282032303 |

49 | 2401 | ± 7 |

50 | 2500 | ± 7.0710678119 |

51 | 2601 | ± 7.1414284285 |

52 | 2704 | ± 7.2111025509 |

53 | 2809 | ± 7.2801098893 |

54 | 2916 | ± 7.3484692283 |

55 | 3025 | ± 7.4161984871 |

56 | 3136 | ± 7.4833147735 |

57 | 3249 | ± 7.5498344353 |

58 | 3364 | ± 7.6157731059 |

59 | 3481 | ± 7.6811457479 |

60 | 3600 | ± 7.7459666924 |

61 | 3721 | ± 7.8102496759 |

62 | 3844 | ± 7.874007874 |

63 | 3969 | ± 7.9372539332 |

64 | 4096 | ± 8 |

65 | 4225 | ± 8.0622577483 |

66 | 4356 | ± 8.1240384046 |

67 | 4489 | ± 8.1853527719 |

68 | 4624 | ± 8.2462112512 |

69 | 4761 | ± 8.3066238629 |

70 | 4900 | ± 8.3666002653 |

71 | 5041 | ± 8.4261497732 |

72 | 5184 | ± 8.4852813742 |

73 | 5329 | ± 8.5440037453 |

74 | 5476 | ± 8.602325267 |

75 | 5625 | ± 8.6602540378 |

76 | 5776 | ± 8.7177978871 |

77 | 5929 | ± 8.7749643874 |

78 | 6084 | ± 8.8317608663 |

79 | 6241 | ± 8.8881944173 |

80 | 6400 | ± 8.94427191 |

81 | 6561 | ± 9 |

82 | 6724 | ± 9.0553851381 |

83 | 6889 | ± 9.1104335791 |

84 | 7056 | ± 9.1651513899 |

85 | 7225 | ± 9.2195444573 |

86 | 7396 | ± 9.2736184955 |

87 | 7569 | ± 9.3273790531 |

88 | 7744 | ± 9.3808315196 |

89 | 7921 | ± 9.4339811321 |

90 | 8100 | ± 9.4868329805 |

91 | 8281 | ± 9.5393920142 |

92 | 8464 | ± 9.5916630466 |

93 | 8649 | ± 9.643650761 |

94 | 8836 | ± 9.6953597148 |

95 | 9025 | ± 9.7467943448 |

96 | 9216 | ± 9.7979589711 |

97 | 9409 | ± 9.8488578018 |

98 | 9604 | ± 9.8994949366 |

99 | 9801 | ± 9.9498743711 |

100 | 10000 | ± 10 |

As follows, the **sequence** of square numbers is 0 (0 x 0), 1 (1 x 1), 4 (2 x 2), 9 (3 x 3), 16 (4 x 4), 25 (5 x 5), 36 (6 x 6), 49 (7 x 7), 64 (8 x 8), 81 (9 x 9), …

## Square Numbers Calculator

Here we inform you that our tool at the top of this page works both ways; you can either fill in the upper or the lower input field.

Our app works with all real numbers, and you don’t need to press a button unless you want to start over.

You may use the up and down arrows (called spinners) to increase or decrease the input value.

Frequently calculated terms include, for example:

We now show you two recursive **formulas** plus an identity which are sometimes useful to come up with n^2.

## Additional Information

In addition to the formula discussed above, any product n^{2} can be produced recursively:

- n
^{2}= (n − 1)^{2}+ (n − 1) + n = (n − 1)^{2}+ (2n − 1) - n
^{2}= 2 x (n − 1)^{2}− (n − 2)^{2}+ 2

This is the difference between two perfect squares:

- n
^{2}− (n − 1)^{2}= 2n − 1

Squares of even integers are even:

- (2n)
^{2}= 4n^{2}

Squares of odd integers are odd:

- (2n + 1) = 4(n
^{2}+ n) + 1

Last digit:

- In base 10, a square number cannot end in digits 2, 3, 7, 8.

Note that you can always employ the search form located in the menu and in the sidebar of this site to locate information about a particular term.

Or simply consult our square numbers **chart**.

Next is the summary of our article.

## Bottom Line

A **square number** **n ^{2}** means

**n × n**, n is an

**integer**and × is the multiplication symbol.

The exponentiation form n^{2} or n^2 is mostly used to express a square number.

For a number to be a perfect square it’s last digit must be 0, 1, 4, 5, 6 or 9; else it is an imperfect square.

**If the second root of a squared number has no decimal places, then the squared number is a square number!**

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Websites related to squares can be found in the “recommended sites” section in the sidebar.

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