 # 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.