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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 x² (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

02	0	102	100	202	400	302	900	402	1600	502	2500
12	1	112	121	212	441	312	961	412	1681	512	2601
22	4	122	144	222	484	322	1024	422	1764	522	2704
32	9	132	169	232	529	332	1089	432	1849	532	2809
42	16	142	196	242	576	342	1156	442	1936	542	2916
52	25	152	225	252	625	352	1225	452	2025	552	3025
62	36	162	256	262	676	362	1296	462	2116	562	3136
72	49	172	289	272	729	372	1369	472	2209	572	3249
82	64	182	324	282	784	382	1444	482	2304	582	3364
92	81	192	361	292	841	392	1521	492	2401	592	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:

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 -7² 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,

• 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.

• 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.

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