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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:

x2= (x-1)2+(x-1)+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 -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 ofPerfect SquaresisPerfect Squares.

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.
    Perfect Squares
    Similarly, the square of 6 is 36. We can arrange 36 points in a square perfectly, as shown below.
    Perfect Squares

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=4x2
  • 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(x2+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.

Next TopicNatural Numbers




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