Perfect Squares
Definitions
In mathematics, a perfect square or square number is a positive integer that is square of an integer. In other words, when we multiply two same numbers together, then the product that we get is called the perfect square. In short, it is the product of two positive equal integers or product of an integer with itself.
Notation of Perfect Square
The perfect square is denoted by x^{2} (pronounced as squared) where x is called the base, and 2 is called the power. It means multiply two times, i.e., x×x.
For example, when we multiply 5 by 5, we get 25 as the result. It is called the perfect square.
Perfect Squares Table
0^{2} 
0 
10^{2} 
100 
20^{2} 
400 
30^{2} 
900 
40^{2} 
1600 
50^{2} 
2500 
1^{2} 
1 
11^{2} 
121 
21^{2} 
441 
31^{2} 
961 
41^{2} 
1681 
51^{2} 
2601 
2^{2} 
4 
12^{2} 
144 
22^{2} 
484 
32^{2} 
1024 
42^{2} 
1764 
52^{2} 
2704 
3^{2} 
9 
13^{2} 
169 
23^{2} 
529 
33^{2} 
1089 
43^{2} 
1849 
53^{2} 
2809 
4^{2} 
16 
14^{2} 
196 
24^{2} 
576 
34^{2} 
1156 
44^{2} 
1936 
54^{2} 
2916 
5^{2} 
25 
15^{2} 
225 
25^{2} 
625 
35^{2} 
1225 
45^{2} 
2025 
55^{2} 
3025 
6^{2} 
36 
16^{2} 
256 
26^{2} 
676 
36^{2} 
1296 
46^{2} 
2116 
56^{2} 
3136 
7^{2} 
49 
17^{2} 
289 
27^{2} 
729 
37^{2} 
1369 
47^{2} 
2209 
57^{2} 
3249 
8^{2} 
64 
18^{2} 
324 
28^{2} 
784 
38^{2} 
1444 
48^{2} 
2304 
58^{2} 
3364 
9^{2} 
81 
19^{2} 
361 
29^{2} 
841 
39^{2} 
1521 
49^{2} 
2401 
59^{2} 
3481 
We can conclude from the above table that the perfect square of a number can be calculated by adding the previous number, current number (for which calculating the square), and the square of the previous number. Suppose, we want to find the square x, then:
x^{2}= (x1)^{2}+(x1)+x
Square of a Negative Number
We can also find the square of negative numbers. But remember that:
()×()=+
It means the square of a negative number is always a positive number. For example, the square of 7^{2} is 49.
Square of a Decimal Number
We can also find the square of a decimal number. To achieve the same, ignore the decimal for a moment and write the square of the number. After that, count the decimal digits in the number from the right. In the result, count the same number of digits from the right and put a decimal point there.
For example, a square of 2.5 is 6.25. Similarly, a square of 1.1 is 1.21.
Square of The Fraction
To find the square of the fraction is easy. In this, we find the square of the numerator and the denominator, separately and simplify the fraction if necessary. For example, the square ofis.
Square of an Irrational Number
The number that cannot be expressed by an integer is called an irrational number. The example of irrational numbers is √7,√3, etc.
The square of an irrational number is the number itself without the root sign. For example, the square of √2 is 2. Similarly, the square of √15 is 15.
Properties of Square
 A number x will be the perfect square if and only if the square can arrange in a square, perfectly.
For example, the square of 2 is 4. We can arrange the four points in a square perfectly, as shown below.
Similarly, the square of 6 is 36. We can arrange 36 points in a square perfectly, as shown below.
In base 10, the square has 0, 1, 4, 5, 6 at unit places.
 If the number ends with 0, its square also ends with 0. The last two digits end with 00. For example, the square of 10 is 100.
 If the number ends with 1 or 9, its square ends with 1. For example, the square of 11 is 121, and the square of 9 is 81.
 If the number ends with 2 or 8, its square ends with 4. For example, the square of 12 is 144, and the square of 8 is 64.
 If the number ends with 3 or 7, its square ends with 9. For example, the square of 13 is 169, and the square of 7 is 49.
 If the number ends with 4 or 6, its square ends with 6. For example, the square of 4 is 16, and the square of 16 is 256.
 If the number ends with 5, its square also ends with 5. The last two digits end with 25. For example, the square of 5 is 25, and the square of 15 is 225.
In base 12 and prime numbers, the square of a number always ends with square digits (0, 1, 4, 9).
 If the number is divisible by both 2 and 3, the square of the number will have 0 at the unit place.
 If the number is not divisible by both 2 and 3, the square of the number will have 1 at the unit place.
 If the number is divisible by 2 only, its square will have 4 at the unit place.
 If the number is divisible by 3 only, its square will have 9 at the unit place.
Some other properties are:
 The square of even numbers is even number, and it is divisible by 4. Therefore,
(2x)^{2}=4x^{2}
 In the above point, we saw that even square is divisible by 4. But the even numbers having the form 4x+2 are not a perfect square.
 The square of odd numbers is odd.
(2x+1)^{2}=4(x^{2}+x)+1
 All the odd square numbers are the form of 4x+1. But the odd numbers having the form 4x+3 are not a perfect square.
