When a number is multiplied by itself, the resulting number is called as a square number.
For example, when we multiply 5 by 5, we get 5^{2 }= 25. Here, 25 is a square number.
In geometry, the area of a square is the finest example of a square number. Area of a square = Side × Side = Side^{2}, because all the sides of a square are equal. This means the area of a square is always a square number.
Square numbers are always positive and they can never 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 x -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.
2 x 2 = 4
-8 x -8 = 64
15 x 15 = 225
We observe the following properties through the patterns of square numbers.
Property 1 :
In square numbers, the digits at the unit’s place are always 0, 1, 4, 5, 6 or 9. The numbers having 2, 3, 7 or 8 at its units' place are not perfect square numbers.
Property 2 :
If a number has 1 or 9 in the unit's place then its square ends in 1.
Property 3 :
If a number has 2 or 8 in the unit's place then its square ends in 4.
Property 4 :
If a number has 3 or 7 in the unit's place then its square ends in 9.
Property 5 :
If a number has 4 or 6 in the unit's place then its square ends in 6.
Property 6 :
If a number has 5 in the unit's place then its square ends in 5.
Property 7 :
Consider the following square numbers :
From the square numbers given above, we infer that
Property 8 :
Consider the following square numbers :
Therefore, 100 is a perfect square and 81000 is not a perfect square.
Property 9 :
Square of even numbers is always even.
It has been illustrated in the table given below.
Property 10 :
Square of odd numbers is always odd.
It has been illustrated in the table given below.
From properties 9 and 10, we infer that
Addition of consecutive odd numbers :
The above figure illustrates the result that the sum of the first n natural odd numbers is n^{2}.
And, square of a rational number a/b is given by
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