Chapter notes: Square and Square roots - Notes | Study Mathematics (Maths) Class 8 - Class 8

Introduction to Square Numbers

If a natural number m can be expressed as n², where n is also a natural number, then m is a square number.

Example: 1, 4, 9, 16, and 25.

Finding the Square of a Number

If n is a number, then its square is given as n×n = n².

For example Square of 5 is equal to  5×5=25

Properties of Square Numbers

• If a number has 0, 1, 4, 5, 6, or 9 in the unit’s place, then it may or may not be a square number. If a number has 2, 3, 7, or 8 in its unit place then it is not a square number.
• If a number has 1 or 9 in the unit’s place, then it’s square ends in 1.
• If a square number ends in 6, the number whose square it is will have either 4 or 6 in the unit’s place.
• For example, consider the number 64. The unit digit of 64 is 4 and it is a square number. Because the square of 8 is 64 and 64 is considered to be a square number. 
• Consider the number 11 (i.e., the unit’s place of 11 is 1). Thus, the square of 11 is 121. Hence, square number 121 also has 1 in the unit’s place.

Finding the square of a number with the unit’s place 5

The square of a number N5 is equal to (N(N+1))×100+25, where N can have one or more than one digit.
For example: If N = 1, then 15²=(1×2)×100+25=200+25=225
If N = 20, then 205²=(20×21)×100+25=42000+25=42025

Perfect Squares

A number that is obtained from the square of the other number is called a perfect square. For example, 81 is a perfect square number, which is obtained by taking the square of the number 9.

Interesting Patterns

There exist interesting patterns in:

• Adding triangular numbers
• Numbers between square numbers
• Adding odd numbers
• A sum of consecutive natural numbers
• Product of two consecutive even or odd natural numbers

Adding Triangular Numbers

Triangular numbers: It is a sequence of the numbers 1, 3, 6, 10, 15, etc. It is obtained by continued summation of the natural numbers. The dot pattern of a triangular number can be arranged as triangles.

If we add two consecutive triangular numbers, we get a square number.
Example: 1+3=4=22 and 3+6=9=32. 

Numbers between Square Numbers

There are 2n non-perfect square numbers between squares of the numbers n and (n + 1), where n is any natural number.

Example:

• There are two non-perfect square numbers (2, 3) between12=1 and 22=4.
• There are four non-perfect square numbers (5, 6, 7, 8) between 22=4 and 32=9.

Addition of Odd Numbers

The sum of first n odd natural numbers is n2.

Example:

1+3=4=22

1+3+5=9=32

Product of Two Consecutive Even or Odd Natural Numbers

The product of two even or odd natural number can be calculated as, (a+1)×(a−1)=(a2−1), where a is a natural number, and a−1, a+1, are the consecutive odd or even numbers.

For example:

11×13=(12−1)×(12+1)=122−1=144−1=143

Random Interesting Patterns Followed by Square Numbers

Patterns in numbers like 1, 11, 111, … :
12=                        1
112=                   1 2 1
1112=               1 2 3 2 1
11112=          1 2 3 4 3 2 1
111112=     1 2 3 4 5 4 3 2 1
111111112=1 2 3 4 5 6 7 8 7 6 5 4 3 2 1
Patterns in numbers like 6, 67, 667, … :
72=49
672=4489
6672=444889
66672=44448889
666672=4444488889
6666672=444444888889

Pythagorean Triplets

For any natural number m>1, we have (2m)2+(m2−1)2=(m2+1)2.
2m, (m2−1) and (m2+1) forms a Pythagorean triplet.
For m=2, 2m=4, m2−1=3 and m2+1=5.
So, 3, 4, 5 is the required Pythagorean triplet.

Square Root of a Number

Finding the number whose square is known is known as finding the square root. Finding square root is inverse operation of finding the square of a number.
For example:
12=1,  square root of 1 is 1.
22=4, square root of 4 is 2.
32=9, square root of 9 is 3.

Estimating Square Roots

Estimating the square root of 247:
Since: 100 < 247 < 400
i.e. 10<√247<20
But it is not very close.
Also, 152=225<247 and 162=256>247
15<√247<16.
256 is much closer to 247 than 225.
Therefore, √247 is approximately equal to 16.

Finding square of a number using identity

Squares of numbers having two or more digits can easily be found by writing the number as the sum of two numbers.
For example:
232=(20+3)2 =20(20+3)+3(20+3)
=202+20×3+20×3+32
=400+60+60+9
=529

To know more about Squares and Square Roots, visit here.

Square Roots of Perfect Squares

Finding square root through repeated subtraction

Every square number can be expressed as a sum of successive odd natural numbers starting from one.

The square root can be found through repeated subtraction. To find the square root of a number n:
Step 1: subtract successive odd numbers starting from one.
Step 2: stop when you get zero.

The number of successive odd numbers that are subtracted gives the square root of that number. Suppose we want to find the square root of 36.

• 36−1=35
• 35−3=32
• 32−5=27
• 27−7=20
• 20−9=11
• 11−11=0

Here 6 odd numbers (1, 3, 5, 7, 9, 11) are subtracted to from 36 to get 0.
So, the square root of 36 is 6.

Finding Square Root by Long Division Method

Steps involved in finding the square root of 484 by Long division method:
Step 1: Place a bar over every pair of numbers starting from the digit at units place. If the number of digits in it is odd, then the left-most single-digit too will have a bar.
Step 2: Take the largest number as divisor whose square is less than or equal to the number on the extreme left. Divide and write quotient.

Finding Pythagorean Triplets for Any Given Number

If we are given any member of a Pythagorean triplet, then we can find the Pythagorean triplet by using general form 2m, m2–1, m2+1.
For example, If we want to find the Pythagorean triplet whose smallest number is 8.
Let, m2−1=8⇒m=3
2m=6 and m2+1=10
The triplet is 6, 8 and 10.
But 8 is not the smallest number of this triplet.
So, we substitute 2m=8⇒m=4
m2−1=15 and m2+1=17.
Therefore, 8, 15, 17 is the required triplet.