Squares and Square Roots Class 8 Notes available here is specially designed by math experts who have taken extra care to use simple language and style so that students can understand concepts easily and quickly. Chapter 6 notes are a great reference tool and will help students to quickly go through or revise all the important topics before the exams as well as score better marks.

## Perfect Squares

#### For More Information On Perfect Squares, Watch The Below Video.

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### Properties of Square Numbers

Properties of square numbers are:

- 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 units place then it is not aÂ square number.
- If a number has 1 or 9 inÂ unitâ€™s place, then itâ€™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 unitâ€™s place.

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

The square of a number **N5** is equal toÂ (N(N+1))Ã—100+25, where **N** can have one or more than one digit.

For example: 152=(1Ã—2)Ã—100+25=200+25=225

2052=(20Ã—21)Ã—100+25=42000+25=42025

## Square Roots

#### For More Information On Square Roots, Watch The Below Video.

### 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,Â square root of 1 is 1.

22=4, square root of 4 is 2.

32=9, square root of 9 is 3.

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### Estimating the number of digits in the square root of a number

If a perfect square hasÂ *n* digits, then its square root will have \(\frac{n}{2}\)Â digits if *n* is **even** and \(\frac{(n+1)}{2}\)Â digits if *n* is **odd**.

For example: 100 has 3 digits, and its square root(10) has \(\frac{(3+1)}{2}\)
=2 digits.

## Estimating Square Roots

### Estimating the Square Root

Estimating the square root of 247:

Since: 100 < 247 < 400

i.e. 10<âˆš247<20

But it is not very close.

Also, 152=225<247 and 162=256>247

15<âˆš247<16.

256 is much closer to 247 than 225.

Therefore, âˆš247 is approximately equal to 16.

## Introduction to Squares and Square Roots

### Introduction to Square Numbers

If a natural number *m*Â 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 nÃ—n=n2.

For example: Square of 5 is equal toÂ 5Ã—5=25

#### For More Information On Finding The Square Of A Number, Watch The Below Video.

### 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)2Â =20(20+3)+3(20+3)

=202+20Ã—3+20Ã—3+32

=400+60+60+9

=529

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## Interesting Patterns

### Interesting Patterns

There exists 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 **dotÂ patternÂ **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.

To know more about Number Patterns, visit here.

### Numbers between Square Numbers

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

Example:

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

### Addition of Odd Numbers

The sum of first nÂ **odd natural numbers** is n2.

Example:

1+3=4=22

1+3+5=9=32

To know more about Addition of Odd Numbers, visit here.

### Square of an odd number as a sum

Square of an odd number *n* can be expressed as sum of two consecutive positive integers \(\frac{(n^{2}-1)}{2}\) and \(\frac{(n^{2}+1)}{2}\).

For example: 32=9=4+5=\(\frac{(3^{2}-1)}{2}\)Â + \(\frac{(3^{2}+1)}{2}\)
Similarly, 52=25=12+13=\(\frac{(5^{2}-1)}{2}\)Â +Â \(\frac{(5^{2}+1)}{2}\)

### Product of Two Consecutive Even or Odd Natural Numbers

The product of two even or odd natural number can be calculated as,Â (a+1)Ã—(aâˆ’1)=(a2âˆ’1), where *a* is a natural number, and aâˆ’1, a+1, are the consecutive odd or even numbers.

For example:

11Ã—13=(12âˆ’1)Ã—(12+1)=122âˆ’1=144âˆ’1=143

### Random Interesting Patterns Followed by Square Numbers

Patterns in numbers like 1, 11, 111, … :

12=Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â 1

112=Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â 1Â 2Â 1

1112=Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â 1Â 2Â 3Â 2Â 1

11112=Â Â Â Â Â Â Â Â Â Â 1Â 2Â 3Â 4Â 3Â 2Â 1

111112=Â Â Â Â Â 1Â 2Â 3Â 4Â 5Â 4Â 3Â 2Â 1

111111112=1Â 2Â 3Â 4Â 5Â 6Â 7Â 8Â 7Â 6Â 5Â 4Â 3Â 2Â 1

Patterns in numbers like 6, 67, 667, … :

72=49

672=4489

6672=444889

66672=44448889

666672=4444488889

6666672=444444888889

## Pythagorean Triplets

### Pythagorean Triplets

For any natural number m>1, we have (2m)2+(m2âˆ’1)2=(m2+1)2.

2m, (m2âˆ’1) and (m2+1) forms a Pythagorean triplet.

For m=2, 2m=4, m2âˆ’1=3 and m2+1=5.

So, 3, 4, 5 is the required PythagoreanÂ triplet.

## 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 successiveÂ 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.

- 36âˆ’1=35
- 35âˆ’3=32
- 32âˆ’5=27
- 27âˆ’7=20
- 20âˆ’9=11
- 11âˆ’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.

## Square Roots – Generalised

### 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 everyÂ pair of numbers starting from the digit at units place.Â 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 asÂ divisor whose square is less than or equal to the number on the extreme left. Divide and writeÂ quotient.

Step 3: Bring down the number which is under the next bar to the right side of the remainder.

Step 4: Double the value of the quotient and enter it with a blank on the right side.

Step 5: Guess the largest possible digit to fill the blank which will also become the new digit in the quotient, such that when the new divisor is multiplied to the new

quotient the product is less than or equal to the dividend.

The remainder is 0, therefore, âˆš484=22.

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## Be More Curious!

### Finding Pythagorean Triplets for Any Given Number

If we are given any member of a Pythagorean triplet, then we can find the Pythagorean triplet byÂ using general form 2m, m2â€“1, m2+1.

For example, If we want to find the Pythagorean triplet whose smallest number is 8.

Let, m2âˆ’1=8â‡’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=8â‡’m=4

m2âˆ’1=15 and m2+1=17.

Therefore, 8, 15, 17 is the required triplet.

To know more about Pythagorean Triplets, visit here.

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