When a number is multiplied by itself, we say that the number is squared.
It is denoted by a number raised to the power 2.
For example,
3 x 3 = 3^{2} = 9
5 x 5 = 5^{2} = 25
In the example above, 5^{2} is read as 5 to the power of 2 or 5 raised to the power 2 or 5 squared. 25 is the square of 5.
Similarly, 49 and 81 are the squares of 7 and 9 respectively.
The numbers 1, 4, 9, 16, 25, g are called perfect squares or square numbers as 1 = 1^{2}, 4 = 2^{2}, 9 = 3^{2}, 16 = 4^{2} and so on.
A number is called a perfect square, if it is expressed as the square of a number.
We observe the following properties through the patterns of perfect squares.
Property 1 :
In perfect squares, the digits at the one’s place are always 0, 1, 4, 5, 6 or 9. The numbers having 2, 3, 7 or 8 at its one' place are not perfect square numbers.
Property 2 :
If a number has 1 or 9 in the one's place then its square ends in 1.
Property 3 :
If a number has 2 or 8 in the one's place then its square ends in 4.
Property 4 :
If a number has 3 or 7 in the one's place then its square ends in 9.
Property 5 :
If a number has 4 or 6 in the one's place then its square ends in 6.
Property 6 :
If a number has 5 in the one's place then its square ends in 5.
Property 7 :
Consider the following square numbers :
From the perfect squares given above, we infer that
Property 8 :
A perfect square number followed by even number of zeros will be a perfect square and a perfect square number followed by odd number of zeros will not be a perfect square.
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.
Property 10 :
Square of odd numbers is always odd.
From property 11 and property 12, 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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