Sunday, August 14

Perfect Squares in Maths

In Mathematics, a perfect square is a number that can be represented as the product of an integer to itself. With the name, a perfect square represents the whole number. Hence, the square root of a perfect square is always a natural number and not a fraction. The examples of perfect squares are 4, 9, 16, 25, etc., such that:

4 = 2 x 2 = 22

9 = 3 x 3 = 32

16 = 4 x 4 = 42

25 = 5 x 5 = 52

But other numbers such as 15, 21, 35, 40, etc., are not perfect squares since their roots are not a whole number.

Let us find more details on perfect squares and the related concepts.

Perfect Squares Formula

As discussed, a perfect square is a number whose root is always an integer and not a fraction. Thus, it also confirms that a perfect squares is always a whole number. Say, y is a perfect square such that y = x2. Hence, the square root of y is equal to x. Thus, the formula of the perfect square becomes:

 

Y = X2 only if √Y = X

 

List of Perfect Squares

Now let us put some whole numbers in place of x to get the perfect squares list.

 

Natural Number Perfect Square Natural Number Perfect Square
1 1 11 121
2 4 12 144
3 9 13 169
4 16 14 196
5 25 15 225
6 36 16 256
7 49 17 289
8 64 18 324
9 81 19 361
10 100 20 400

 

How to Find Perfect Squares?

To find the perfect squares, we need to take the square root of the given number. It is easy to take the root of smaller numbers and find the values. There are certain methods such as long division and prime factorisation by which we can easily find if a number is a perfect square. But for the larger numbers, we cannot quickly find the roots since it can be a long process by dividing decimals method or long division method. Here are a few tricks to find if the number is a perfect square or not.

 

  • A perfect square will always end with 9 at its unit place, if the squared number is ending with 3 or 7 at its unit place
  • A perfect square will always end with 5 at its unit place, if the squared number is ending with 5 at units place
  • A perfect square will always end with 6 at its unit place, if the squared number is ending with 4 or 6 at units place
  • A perfect square will always end with 4 at its unit place, if the squared number is ending with 2 or 8 at units place
  • A perfect square will always end with 1 at its unit place, if the squared number is ending with 1 or 9 at units place
  • A perfect square will always end with 0 at its unit place if the square number is a multiple of 10
  • A perfect square will always be positive since squaring a negative number will always result in a positive number

Is 8 a perfect square?

8 is not a perfect square because the square root of 8 results in a fraction. You cannot express 8 as a product of two same whole numbers.

Is 0 a perfect square?

0 is a perfect square since it satisfies the definition of a perfect square, according to which it can be expressed as a product of two same whole numbers.