## What are Square Numbers?

When a number is multiplied by itself, the resultant is called a â€˜Square Numberâ€™.

In geometry, a square has all its sides equal. Therefore the area of the square is equal to the square of its side.

Area of a square = Side \(\times\) Side

Therefore, we can say;

Square number = a \(\times\; a \; = a^{2}\)

Hence, area of square is best example of square of a number.

Side of square (in cm) |
Area of square (in cm^{2}) |

3 |
3 Ã— 3 = \(3^{2}\) = 9 |

5 |
5 Ã— 5 = \(5^{2}\) = 25 |

7 |
7 Ã— 7 = \(7^{2}\) = 49 |

8 |
8 Ã— 8 = \(8^{2}\) = 64 |

Numbers such as 1, 4, 9, 16, 25, 36, 49, 64, etc. are special numbers as these are the product of a number by itself.

If we express a number (x) in terms of the square of any natural number such as a^{2}, then x is a square number. For example, 100 can be expressed as 10 Ã— 10 = 10^{2}, where 10 is a natural number, therefore 100 is a square number. Whereas, the number 45 cannot be called a square number because it is the product of numbers 9 and 5. The number is not multiplied by itself. **Square numbers** can also be called as perfect square numbers.

## List of Square Numbers

Go through the list of squares for numbers less than 60^{2} = 3600 here.

0^{2} = 0 |
10^{2} = 100 |
20^{2} = 400 |
30^{2} = 900 |
40^{2} = 1600 |
50^{2} = 2500 |

1^{2} = 1 |
11^{2} = 121 |
21^{2} = 441 |
31^{2} = 961 |
41^{2} = 1681 |
51^{2} = 2601 |

2^{2} = 4 |
12^{2} = 144 |
22^{2} = 484 |
32^{2} = 1024 |
42^{2} = 1764 |
52^{2} = 2704 |

3^{2} = 9 |
13^{2} = 169 |
23^{2} = 529 |
33^{2} = 1089 |
43^{2} = 1849 |
53^{2} = 2809 |

4^{2} = 16 |
14^{2} = 196 |
24^{2} = 576 |
34^{2} = 1156 |
44^{2} = 1936 |
54^{2} = 2916 |

5^{2} = 25 |
15^{2} = 225 |
25^{2} = 625 |
35^{2} = 1225 |
45^{2} = 2025 |
55^{2} = 3025 |

6^{2} = 36 |
16^{2} = 256 |
26^{2} = 676 |
36^{2} = 1296 |
46^{2} = 2116 |
56^{2} = 3136 |

7^{2} = 49 |
17^{2} = 289 |
27^{2} = 729 |
37^{2} = 1369 |
47^{2} = 2209 |
57^{2} = 3249 |

8^{2} = 64 |
18^{2} = 324 |
28^{2} = 784 |
38^{2} = 1444 |
48^{2} = 2304 |
58^{2} = 3364 |

9^{2} = 81 |
19^{2} = 361 |
29^{2} = 841 |
39^{2} = 1521 |
49^{2} = 2401 |
59^{2} = 3481 |

## Odd and Even square numbers

- Squares of even numbers are even, i.e, (2n)
^{2}= 4n^{2}.

- Squares of
**odd numbers**are odd, i.e, (2n + 1) = 4(n^{2}+ n) + 1.

- Since every odd square is of the form 4n + 1, the odd numbers that are of the form 4n + 3 are not square numbers.

## Properties of Square Numbers

The following are the properties of the square numbers:

- Â A number with 2, 3, 7 or 8 at unitâ€™s place should never be a perfect square. In other words, none of the square numbers ends in 2, 3, 7 or 8.
- If the number of zeros at the end is even, then the number is a perfect square number. Otherwise, we can say that number ending in an odd number of zeros is never a perfect square
- Â If the even numbers are squared, it always gives even numbers. Also, if the odd numbers are squared, it always gives odd numbers.
- If the natural numbers other than one is squared, it should be either a multiple of 3 or exceeds a multiple of 3 by 1.
- f the natural numbers other than one is squared, it should be either a multiple of 4 or exceeds a multiple of 4 by 1.
- It is noted that the unitâ€™s digit of the square of a natural number is equal to the unitâ€™s digit of the square of the digit at unitâ€™s place of the given natural number.
- There are n natural numbers, say p and q such that pÂ
^{2}Â = 2qÂ^{2} - For every natural number n, we can write it as: (n + 1)Â
^{2}Â – nÂ^{2}Â = ( n + 1) + n.. - If a number n is squared, it equals to the sum of first n odd natural numbers.
- For any natural number, say”n” which is greater than 1, we can say that (2n, nÂ
^{2}Â – 1, nÂ^{2}Â + 1) should be a Pythagorean triplet.

Here we have covered square numbers and their properties, you can refer toÂ completing the square method which by following the link provided. To learn more about other topicsÂ download BYJU’S – The Learning App and learn the subjects in an interactive and innovative way.