Squares and Square Roots – Definition with Examples

Squares and Square Roots : The concept of squares and square roots is very widely used in mathematics and in our daily lives, to simplify the repetitive task of multiplying a number by itself. In fact, both concepts are interrelated to each other. We will see in this article how?

It is possible to arrive at the square of both positive and negative numbers, by simply multiplying the number with itself. It is recommended that students even memorise them at least for a few single and doubt-digit numbers. This can save a lot of time in the exam. Let us understand the definition of square and square roots, along with a few examples on how to find them?

Squares and Square Roots- Definition

Squares are the resulting numbers when any number is multiplied by itself. It is denoted by the number superscript with the same number. For example, when 2 is multiplied by itself, it becomes 2 multiplied by 2, which is 4. The operation is denoted as 2^2.

A square root of a number ‘x’ can be defined as that number which when multiplied by itself, gives ‘x’. If we consider the above example, we say the square root of 4 is 2 because when 2 is multiplied by 2, it gives 4. Think of another example, say 9. The square root of 9 is 3 because 3 when multiplied by 3, gives 9. It is represented as $\sqrt{9}$ .

Squares and Square Roots of Numbers From 1-25

As we already mentioned at the beginning, remembering squares of numbers from 1-25 helps you to solve the problems quickly and save time in your exam.

Check out the squares of numbers from 1-25 below:

$1^{2} = 1$	$6^{2} = 36$	$11^{2 =} 121$	$16^{2} = 256$	$21^{2} = 441$
$2^{2} = 2$	$7^{2} = 49$	$12^{2} = 144$	$17^{2} = 289$	$22^{2} = 484$
$3^{2} = 9$	$8^{2} = 64$	$13^{2} = 169$	$18^{2} = 324$	$23^{2} = 529$
$4^{2} = 16$	$9^{2} = 81$	$14^{2} = 196$	$19^{2} = 361$	$24^{2} = 576$
$5^{2} = 25$	$10^{2} = 100$	$15^{2} = 225$	$20^{2} = 400$	$25^{2} = 625$

Square of a Negative Number

Always remember, the square of a negative number is always positive. For example, the square of -5 is 25 and not -25. This is because when we are multiplying -5 with -5, the negative sign also gets multiplied. Two negatives make a positive. Therefore, it is true.

Properties of Square Numbers

Now we know how to calculate the square and square roots of numbers. But knowing some quick facts which we are going to talk about soon, will help you in quickening your calculation speeds.

Here are some of them.

$12 = 1$
$02 = 0$
Square of a positive number is always positive. Square of a negative number is also always a positive number.
Square of the root of a number is equal to the number itself. For example, $\sqrt{42}$ = 4
The number in the unit place of the square of any even number is even. For example, Four square is Sixteen, 12 square is 144 and 8 square is 64. Observe the unit place of all these numbers, 6 and 4 are both even.
The square of any number with 1 or 9 in the unit’s place, ends in 1. For example, 11 square = 121, 19 square= 361 and 29 square= 841. Observe the squares of all these numbers and as you can see, they all end in 1
The square of any number with 4 or 6 in the unit’s place, ends in 6. For example, 14 square = 196, 16 square= 256 and 36 square= 1296

Square Root of a Number

The definition of the square root of a number has already been given at the beginning of the article. The square root of a number, say ‘n’ is x if x2=n. It is denoted as n.

The square root of any number can be calculated using this formula. Remember, the square root of a number can be a fraction also. We are going to learn about this in the upcoming paragraphs.

What are Perfect Squares?

A number, say ‘p’ is said to be a ‘perfect square’ if its square root ‘n’ is a whole number and not a fraction. Let’s take the example of square root of 16. The answer is 4, which is a whole number. Let’s look at another example, the square root of 25, which is 5. This is also whole number. In both examples, we call 16 and 25 as perfect squares. We use what is called the prime factorisation method, in order to find the square root of a perfect square.

Perfect Squares from 1 to 100

The table given below shows the perfect square from 1 to 100.

Perfect Square	Square Root
1	1
4	2
9	3
25	5
36	6
49	7
64	8
81	9
100	10

What are Imperfect Squares?

A number ‘p’ is said to be an ‘imperfect square’ if its square root ‘n’ is a not a whole number, in other words, it is a fraction. A few examples of imperfect squares are given below.