What Are Square Numbers? List, Examples, Sum - Cuemath

Square Numbers

When an integer is multiplied by itself, the resultant number is known as its square number. Basically, a square number is a number that is obtained by the product of two same numbers. In geometry, the area of a square with 'n' as the side length (where n is an integer) is the finest example of a square number.

Let us learn more about square numbers and see the square numbers list.

1.	What are Square Numbers?
2.	Properties of Square numbers
3.	List of Square Numbers
4.	Sum of Square Numbers
5.	FAQs on Square Numbers

What are Square Numbers?

When an integer is multiplied by the same integer, the resultant number is known as a square number. For example, when we multiply 5 × 5 = 52, we get 25. Here, 25 is a square number. Square numbers are always positive, they cannot be negative because when a negative number is multiplied by the same negative number, it results in a positive number. For example, (-7)2 = -7 × -7 = 49 (two negative signs multiplied by each other result in a positive sign). When we multiply -7 by -7, the result is 49 and it is a positive square number. Here are some examples of square numbers.

• -8 × -8 = 64
• 12 × 12 = 144
• -2 × -2 = 4
• 34 × 34 =1156

In geometry, the area of a square = Side × Side = Side2, because all the sides of a square are equal. This means the area of a square is always a square number.

Square Numbers Up To 20

As we have already discussed that a square number is a perfect square of an integer. There are four square numbers up to 20 which are as follows: 1, 4, 9, and 16. The next number in this list will be 25 which is a perfect square of 5, but it is greater than 20.

• 12 = 1 × 1 = 1
• 22 = 2 × 2 = 4
• 32 = 3 × 3 = 9
• 42 = 4 × 4 = 16

Square Numbers Up To 50

There are seven square numbers up to 50. The list is as follows: 1, 4, 9, 16, 25, 36, and 49. The next number in this list will be 64 which is a square of 8, but it is greater than 50.

• 12 = 1 × 1 = 1
• 22 = 2 × 2 = 4
• 32 = 3 × 3 = 9
• 42 = 4 × 4 = 16
• 52 = 5 × 5 = 25
• 62 = 6 × 6 = 36
• 72 = 7 × 7 = 49

Square Numbers Up To 100

There are ten square numbers up to 100 which are as follows: 1, 4, 9, 16, 25, 36, 49, 64, 81, and 100. The next number in this list will be a square of 11, i.e, 121, which is greater than 100.

• 12 = 1 × 1 = 1
• 22 = 2 × 2 = 4
• 32 = 3 × 3 = 9
• 42 = 4 × 4 = 16
• 52 = 5 × 5 = 25
• 62 = 6 × 6 = 36
• 72 = 7 × 7 = 49
• 82 = 8 × 8 = 64
• 92 = 9 × 9 = 81
• 102 = 10 × 10 = 100

Square Numbers Up To 1000

There are 31 square numbers up to 1000 which are given as 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400, 441, 484, 529, 576, 625, 676, 729, 784, 841, 900 and 961. These are the perfect squares of whole numbers 1 to 31.

Now, let us learn some important properties of square numbers in the section below.

Properties of Square Numbers

The properties of square numbers are listed below which helps to identify them easily.

• Square numbers always end with the digits 0, 1, 4, 5, 6, and 9. For example, 25, 49, 81, 100, etc are perfect squares, whereas, 37, 48, 22, etc are not considered to be perfect square numbers.
• The number of zeros at the end of a square number is always even. This means if a number has an odd number of zeros at the end, then the number is not a square number. For example, 400, 3600 are square numbers, whereas, 10, 250, and 360 are non-square numbers.
• If a number has 1 or 9 as its last digit, then its square number ends with 1. For example, the square of 9 is 81, and the square of 11 is 121.
• If a number has 4 or 6 as its last digit, then its square number ends with 6. For example, the square of 4 is 16, and the square of 26 is 676.
• The square of even numbers is always even, and the square of odd numbers is always odd. For example, the square of 12 is 144, and the square of 13 is 169.
• Square numbers are always positive because two negative signs multiplied by each other result in a positive sign. For example, (-6)2 = -6 × -6 = 36 . Here, the product is 36 and it is a positive square number.
• The square root of a square number is always an integer. For example, the square root of 441 is 21, so 441 is a square number. This means if the square root of a number is a fraction or a decimal number, then that number is NOT a perfect square number. For example, √0.25 = 0.5, therefore, 0.25 is NOT a square number.

Two-Digit Square Numbers and Three-Digit Square Numbers

There are a total of 6 'two digit square numbers', which can be listed as, 16, 25, 36, 49, 64, and 81. There are 22 'three digit square numbers' that can be listed as, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400, 441, 484, 529, 576, 625, 676, 729, 784, 841, 900, 961.

Odd and Even Square Numbers

The following points are some facts about odd and even square numbers.

• Square numbers of even numbers are always even. For example, the square of 4 is 42 = 4 × 4 =16.
• Square numbers of odd numbers are always odd. For example, the square of 3 is 32 = 3 × 3 = 9.

List of Square Numbers

Here is a list of square numbers from 1 to 100 which helps to solve problems related to square numbers.

Sum of Square Numbers

The square numbers are 1, 4, 9, 16, ... The sum of first 'n' square numbers (starting from 1) is represented by Σ n2 and can be calculated using the formula [n(n+1)(2n+1)] / 6. i.e.,

• Σ n2 = [n(n+1)(2n+1)] / 6

For example, let us calculate the sum of first 6 square numbers manually, then we get 1 + 4 + 9 + 16 + 25 + 36 = 91. Now, let us calculate the same sum using the above formula, then

Sum of first 6 square numbers = [6(6+1)(2(6)+1)] / 6 = [ 6(7)(13)] / 6 = 91. We have got the same answer in both the methods.

Important Notes on Square Numbers:

• The square numbers are always whole numbers.
• The square root of a square number is always an integer.
• The square numbers of even numbers are even.
• The square numbers of odd numbers are odd.

☛ Related Topics:

Check out the interesting topics related to square numbers in math.

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• Square 1 to 30
• Square 1 to 25
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