Squares and Square Roots

We have learnt about the square that has the equal sides. Side represent the length of surface of shape. We can find the area of square like (side x side). In similarly the square of any number is (number x number or a x a = a2) in number square system 2 is the square of a.

The square root of a number n is a number that, when multiplied by itself, equals n. 

6.2 Properties of Square Numbers

Every number has own property of the square.

If a number has 1 or 9 in the units place, then it's square ends in 1.

When a square number ends in 6, the number whose square it is, will have either 4 or 6 in unit's place.

6.3 Some More Interesting Patterns

There are 2n non perfect square numbers between the squares of the numbers n and (n + 1).

Sum of first n odd natural numbers is n2.

1 [one odd number] = 1 = 12

1 + 3 [sum of first two odd numbers] = 4 = 22

1 + 3 + 5 [sum of first three odd numbers] = 9 = 32

If a natural number cannot be expressed as a sum of successive odd natural numbers starting with 1, then it is not a perfect square.

6.4 Finding the Square of a Number

Finding the squares of small numbers is easy, but for the large number it's difficult to find, so there are various way to find the square of that.

For any natural number m > 1, we have (2m)2 + (m2 −1)2 = (m2 + 1)2. So, 2m, m2 − 1 and m2 + 1 forms a Pythagorean triplet.

6.5 Square Roots

As 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.

12 = 1, therefore square root of 1 is 1

22 = 4, therefore square root of 4 is 2

32 = 9, therefore square root of 9 is 3

Positive square root of a number is denoted by the symbol √.

Number

Square

 

102

100

which is the smallest 3-digit perfect square

312

961

which is the greatest 3-digit perfect square

322

1024

which is the smallest 4-digit perfect square

992

9801

which is the greatest 4-digit perfect square

 

 

 

6.6 Square Roots of Decimals

Consider a number and it can be find by division rule of square root.

6.7 Estimating Square Root

In this cases we need to estimate the square root and find the nearest number of square root to be closed with that number which we are finding.

100 < 250 < 400 and √100 = 10 and √400 = 20.

So 10 < √250 < 20

But still we are not very close to the square number.

We know that 152 = 225 and 162 = 256

Therefore, 15 < √250 < 16 and 256 is much closer to 250 than 225.

So, √250 is approximately 16.