Square Numbers: Multiplying With The Same Number

When a number or integer (not a fraction) is multiplied by itself, the resultant is called a ‘Square Number’. For example, 3 multiplied by 3 is equal to 3-squared or 3 x 3 = 3². So, basically, the exponential form of multiplication of a number or integer by itself is called a square number. Also, if we again multiply the number by itself, then we get a cube of the integer., a x a x a = a³.

Square numbers are always positive. If negative sign is multiplied by itself, it results in positive sign (+). For example, (-4)² = 16. So, we can say here 16 is a positive square number, whose square root is an integer again, i.e.√16 = 4.

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

Area of a square = Side x Side = Side²
Therefore, we can say;
Square number = a x a = a²

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

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 a², then x is a square number. For example, 100 can be expressed as 10 × 10 = 10², 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 perfect square numbers.

Examples of Square numbers

Check some examples of square numbers here.

• 1 x 1 = 1²
• 2 x 2 = 2² = 4
• 3 x 3 = 3² = 9
• 4 x 4 = 4² = 16
• 5 x 5 = 5² = 25
• 6 x 6 = 6² = 36

List of Square Numbers (1 to 100)

A list of square numbers from 1 to 100 is provided here. Use the table given below to solve problems related to square numbers.

Squares of 1 to 50

Squares of 51 to 100

Two Digit Square numbers

The list of two-digit square numbers is 16, 25, 36, 49, 64 and 81.

Odd and Even square numbers

• Squares of even numbers are even, i.e, (2n)² = 4n².

• Squares of odd numbers are odd, i.e, (2n + 1)² = 4(n² + 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:

1.  A number with 2, 3, 7 or 8 at unit’s place should never be a perfect square. In other words, none of the square numbers ends in 2, 3, 7 or 8.
2. 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
3.  If the even numbers are squared, it always gives even numbers. Also, if the odd numbers are squared, it always gives odd numbers.
4. 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.
5. 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.
6. It is noted that the unit’s digit of the square of a natural number is equal to the unit’s digit of the square of the digit at unit’s place of the given natural number.
7. There are n natural numbers, say p and q such that p ² = 2q ²
8. For every natural number n, we can write it as: (n + 1)² – n ² = ( n + 1) + n..
9. If a number n is squared, it equals the sum of first n odd natural numbers.
10. For any natural number, say”n” which is greater than 1, we can say that (2n, n ² – 1, n ² + 1) should be a Pythagorean triplet.

Square Roots

As we have already discussed in the introduction, the square numbers are generated when an integer is multiplied by itself. Now, to get back the original number, we have to find the square root of the square number. 

For example, 

7 x 7 = 49

√49 = 7

So the square root of any number gives us the value, that can be squared to get the original number. 

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