How Squares and Square Roots are Interconnected with Each Other

In mathematics, concepts begin with simple numbers such as 1,2,3, and so on. It is interesting to know when the square of these numbers or the square root of these numbers are greatly connected. 

The square number is given the name square due to geometrical square because 3×3 = 9 and if we arrange 9 items of 3 various categories. we arrange them into a grid of 3 rows and 3 columns placed horizontally and vertically, the grid will form a square.

Square

Definition: If we multiply any digit by itself then we get the square of a digit. For example, if we multiply 79 with 79 we get 6241, so 6241 is the square of 79. The square of any digit is always positive; it cannot be negative because if we multiply two negative numbers the result is always positive.

Square of 1 to 10

1² = 1×1 = 1

2² = 2×2 = 4

3² = 3×3 = 9

4² = 4×4 = 16

5² = 5×5 = 25

6² = 6×6 = 36

7² = 7×7 = 49

8² = 8×8 = 64

9² = 9×9 = 81

10² = 10×10 = 100

Square of Decimal No.

(1.1)² = 1.1x 1.1 = 1.21

(1.2)² = 1.2×1.2 = 1.44

(1.3)² = 1.3×1.3 = 1.69

(1.4)² = 1.4×1.4 = 1.96

Properties of a Square

• The square of an integer (either positive or negative) is always positive, it cannot be negative. This can be illustrated by an example:

(-3)² = -3x-3 = 9

• The square of the even number is always even and the square of the odd number is always odd. This can be illustrated by an example:

7² = 49 (odd no.)

8² = 64 (even no.)

• Square of any integer always ends with no. 0,1 ,4, 5, 6 or 9. They never end with no. 2, 3, 7, 8

Example: 15² = 225

11² = 121

22² = 484

• If a number ends with 2 or 8, the square ends with 4 .

Example: 12² = 144

• If a number ends with 3 or 7, the square ends with 9.

Example: 13² = 169

• If a number ends with 4 or 6, the square ends with 6.

Example: 6² = 36

• If a number ends with 5, the square also ends with 5.

25² = 625

• The square of any integer has always even no. of zeros, if they have odd no. of zeros it will not be a perfect square.

Examples: 

10² = 100

400² = 160000

Square root

Definition: It is a factor of a number that, when multiplied by itself, gives the original number. For example, the square root of 9 = 3 because 3 x 3 = 9.

It is represented by and is known by radicals and it has two values positive and negative but mostly we take positive value.

Square roots of 1 to 10

1 = 1

2 = 1.414

3 = 1.732

4 = 2

5 = 2.236

6 = 2.449

7 = 2.646

8 = 2.828

9 = 3

10 = 3.162

Properties of Square Roots

• The square root of any negative integer is not possible

• The square root of an even number is even and the square root of an odd number is odd. example- 1 = 1, 25 = 5,4 = 2, 36 = 6

• If a number ends with an odd number of zeros, its square root will always be in decimals. Example: 25000 = 158.11, 120 = 10.95

• If a number ends with 2,3,7 or 8 its square root will be always in decimals

• A perfect square root exists for a perfect square number only

How Square and Square Root are Interconnected

If we calculate the square of any integer like 5 x 5 = 25, and if we calculate the square root of 25 we get 5. That means square and square roots are inverse to each other. This means if we calculate the square and square root of any number at the same time then we get only that number because they will oppose each other and the number will remain the same. It is also applicable for decimal numbers, fractions, etc.

We can see them by a table:

In this table, it is clear that square and square roots are inverses.

Conclusion

The square of any number is derived by multiplying the number by itself. If n is any number then the square of n=nxn=n².

The square root of any number is derived by a factor of the number that, when multiplied by itself, gives the original number. If n is any number then the square root of n=n

The relation between the square and the square root is that they are inverse of each other.