You know that the area of a square = side × side (where ‘side’ means ‘the length of a side’).From the above table, can we enlist the square numbers between 1 and 100? Are there any natural square numbers upto 100 left out? You will find that the rest of the numbers are not square numbers.

Study the square numbers in the above table. What are the ending digits (that is, digits in the one’s place) of the square numbers? All these numbers end with 0, 1, 4, 5, 6 or 9 at unit’s place. None of these end with 2, 3, 7 or 8 at unit’s place.

1.1. Adding triangular numbers.
Do you remember triangular numbers (numbers whose dot patterns can be arranged
as triangles)?

2. Numbers between square numbers
Let us now see if we can find some interesting pattern between two consecutive
square numbers.

3. Adding odd numbers

So we can say that the sum of first n odd natural numbers is n2.
Looking at it in a different way, we can say: ‘If the number is a square number, it has
to be the sum of successive odd numbers starting from 1.

4. A sum of consecutive natural numbers

Squares of small numbers like 3, 4, 5, 6, 7, ... etc. are easy to find. But can we find the square of 23 so quickly? The answer is not so easy and we may need to multiply 23 by 23. There is a way to find this without having to multiply 23 × 23.

Study the following situations.
(a) Area of a square is 144 cm2. What could be the side of the square?

6.5.1 Finding square roots

The inverse (opposite) operation of addition is subtraction and the inverse operation
of multiplication is division. Similarly, finding the square root is the inverse operation
of squaring.

To find the square root of a decimal number we put bars on the integral part 17 of the number in the usual manner. And place bars on the decimal part on every pair of digits beginning with the first decimal place. Proceed as usual.Now proceed in a similar manner. The left most bar is on 17 and 42 < 17 < 52. Take this number as the divisor and the number under the left-most bar as the dividend, i.e., 17. Divide and get the remainder.

Consider the following situations:

1. Deveshi has a square piece of cloth of area 125 cm2. She wants to know whether
she can make a handkerchief of side 15 cm. If that is not possible she wants to
know what is the maximum length of the side of a handkerchief that can be made
from this piece.

2. Meena and Shobha played a game. One told a number and other gave its square
root. Meena started first. She said 25 and Shobha answered quickly as 5. Then
Shobha said 81 and Meena answered 9. It went on, till at one point Meena gave the
number 250. And Shobha could not answer. Then Meena asked Shobha if she
could atleast tell a number whose square is closer to 250.