Chapter notes: Square and Square roots

# Squares and Square Roots Class 8 Notes Maths Chapter 5

 Table of contents Introduction to Square Numbers Properties of Square Numbers Some More Interesting Patterns Finding the Square of a Number Square Roots Square Roots of Decimals

## Introduction to Square Numbers

A natural number n is a square number if it can be expressed as n², where n is also a natural 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.

Example Square of 5 is equal to  5×5=25

Square Numbers

## Properties of Square Numbers

1. 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.
Example: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100 are square numbers. But 28 and 97 will not be perfect squares.
2. If a number has 1 or 9 in the unit’s place, then it’s square ends in 1.
Example: 112 = 121, 212 = 441, 192 = 361 and 292 = 841
3. If a square number ends in 6, the number whose square it is will have either 4 or 6 in the unit’s place.
Example: 142 = 196, 162 = 256 and 242 = 576
4. Unit digit of square of any number will be the unit digit of square of its last digit
Example: The square of 23 is 529
232 = 529
The unit place of 23 has 3, and unit place of 529 has 9
A square of 3 is equal to 9
5. The square root of perfect square is always a natural number
Example: √4 = 2, √9 = 3, √16 = 4 and √25 = 5
6. The perfect squares will end with even numbers of zeros
Example: 2500 is a perfect square, 100 is a perfect square. But 80 is not a perfect square, and 4000 is also not a perfect square
7. Square of even numbers are always even
Example: A Square of 2 is 4, and a Square of 6 is 36
8. Square of odd numbers are always odd
Example: Square of 7 is 49, and Square of 9 is 81

### How to Find the square of a number with the unit’s place 5?

To find the square of a number with the unit's place as 5, we can follow these steps:

Step 1. Separate the number into two parts - the last digit (5) and the remaining part (N). Hence the number becomes N5.

Step 2. Multiply N with N+1.

Step 3. Append 25 at the end of the result obtained in step 2.

Examples:

Square of 15 :

Here N = 1,

Then 152=(1×2)×100+25=200+25=225

Square of 205

Here N= 20,
Then 2052=(20×21)×100+25=42000+25=42025

### What are Perfect Squares?

A number that is obtained from the square of the other number is called a perfect square.

OR

A perfect square is a number that results from multiplying another number by itself.

Example : 81 is a perfect square number, which is obtained by taking the square of the number 9.

Question for Chapter notes: Square and Square roots
Try yourself:What is a perfect square?

## Some More Interesting Patterns

There exist interesting patterns in:

2. Numbers between square numbers
4. A sum of consecutive natural numbers
5. Product of two consecutive even or odd natural numbers

Triangular numbers: Triangular numbers are a sequence created by adding natural numbers continuously. They form a triangular shape when arranged in dots. For example:  1, 3, 6, 10, 15, etc.

1
1+2=3
1+2+3=6
1+2+3+4=10

Diagram:

••
•••

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

### 2. Numbers between Square Numbers

Numbers between square numbers are the integers that fall between two perfect squares.

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

The sum of first n odd natural numbers is n2.

Example:

1+3=4=22

1+3+5=9=32

### 4. A sum of consecutive natural numbers

The sum of two consecutive numbers is always an odd number. For example, consider a consecutive number sequence 1, 2, 3, 4, 5, 6, 7, 8, 9.
Here, the sum of the two consecutive numbers will always be an odd number. 1+2 = 3, 2+3 = 5, 3+4 = 7, and so on.

Square of an odd number n can be expressed as sum of two consecutive positive integers

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

The product of two even or odd natural number can be calculated as, (a+1)×(a1)=(a21), where is a natural number, and a1, a+1, are the consecutive odd or even numbers.

For example:

11 and 13 are two odd consecutive numbers, therefore the multiplication of 11 and 13 can be calculated as:

11×13 (121)×(12+1122144143

Question for Chapter notes: Square and Square roots
Try yourself:Which of the following expressions represents the product of two consecutive even or odd natural numbers?

### 6. Some more patterns in 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

## Finding the Square of a Number

To find the square of a number, you can use the concept of expanding the expression.
1. Let's take the example of finding the square of 23:

232 = (20 + 3)2 = (20 + 3) (20 + 3)

Expanding this expression, we get:

= 20 (20 + 3) + 3 (20 + 3)

Simplifying further:

= 20 * 20 + 20 * 3 + 3 * 20 + 3 * 3

Calculating the multiplication:

= 400 + 60 + 60 + 9

= 529

Therefore, the square of 23 is 529.

2. Let's find the square of 15 using the same method:

152 = (10 + 5)2 = (10 + 5) (10 + 5)

Expanding this expression:

= 10 (10 + 5) + 5 (10 + 5)

Simplifying further:

= 10 * 10 + 10 * 5 + 5 * 10 + 5 * 5

Calculating the multiplication:

= 100 + 50 + 50 + 25

= 225

Therefore, the square of 15 is 225

### Other patterns in squares

Let's explore other patterns in squares using the given examples:

When we look at the numbers and their corresponding squares, we can observe the following pattern:

• 252 = 625 = (2 × 3) hundreds + 25
• 352 = 1225 = (3 × 4) hundreds + 25
• 752 = 5625 = (7 × 8) hundreds + 25
• 1252 = 15625 = (12 × 13) hundreds + 25

In each case, we notice that the square can be expressed as a product of two consecutive numbers, followed by the word "hundreds," and then we add 25 at the end.

This pattern can be generalized as follows:

For any number n, where n is a positive integer:

n2 = (n × (n+1)) hundreds + 25

For example:

• 152 = 225 = (1 × 2) hundreds + 25
• 202 = 400 = (2 × 3) hundreds + 25
• 302 = 900 = (3 × 4) hundreds + 25

By following this pattern, we can easily find the squares of various numbers by multiplying two consecutive numbers, adding "hundreds," and then appending 25 at the end.

### Pythagorean Triplets

A Pythagorean triplet is a set of three positive integers a, b, and c that satisfy the Pythagorean theorem, which states that the sum of the squares of the two smaller numbers is equal to the square of the largest number. In other words, a² + b² = c².

In the given formula for generating Pythagorean triplets, we have:

a = 2m

b = (m² - 1)

c = (m² + 1)

For any natural number m > 1, these three numbers form a Pythagorean triplet.

This is because (2m)² + (m² - 1)² = (m² + 1)², as stated in the formula.

Example :

Let's say m = 3, then

a = 2m = 2(3) = 6

b = (m² - 1) = (3² - 1) = (9 - 1) = 8

c = (m² + 1) = (3² + 1) = (9 + 1) = 10

We have the triplet (6, 8, 10). Now, let's check that it satisfies the Pythagorean theorem:

6² + 8² = 36 + 64 = 100

10² = 100

Question for Chapter notes: Square and Square roots
Try yourself:Which of the following is a Pythagorean triplet?

## Square Roots

Determining the number whose square is already known is called finding the square root. The process of finding the square root is the opposite of calculating the square of a number.
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.

### Finding Square Roots

Finding the square root of 180:

1. Since: 100 < 180 < 225
2. i.e. 10 < √180 < 15
3. It is not very close, so we need to narrow down the range.
4. Also, 132 = 169 < 180 and 142 = 196 > 180
5. 13 < √180 < 14
6. 196 is much closer to 180 than 169.
7. Therefore, √180 is approximately equal to 14.

### 1. Finding square root through repeated subtraction

To find the square root of a number through repeated subtraction, follow these steps:

Step 1: Start with the given number (let's call it n) for which you want to find the square root.

Step 2: Begin subtracting successive odd numbers, starting from 1. Keep track of how many odd numbers you have subtracted.

Step 3: Continue subtracting the next odd number in the sequence until the result of the subtraction is zero.

Step 4: The number of odd numbers you subtracted in the process is the square root of the given number.

Example :

Suppose you want to find the square root of 49.

2. Begin subtracting successive odd numbers:

49 - 1 = 48 (1 odd number subtracted)

48 - 3 = 45 (2 odd numbers subtracted)

45 - 5 = 40 (3 odd numbers subtracted)

40 - 7 = 33 (4 odd numbers subtracted)

33 - 9 = 24 (5 odd numbers subtracted)

24 - 11 = 13 (6 odd numbers subtracted)

13 - 13 = 0 (7 odd numbers subtracted)

3, The result is zero after subtracting 7 odd numbers.

4. The square root of 49 is 7

### 2. Finding square root through through prime factorisation

Consider the prime factorization of various numbers and their squares:

• 6 = 2 × 3
36 = 2 × 2 × 3 × 3
• 8 = 2 × 2 × 2
64 = 2 × 2 × 2 × 2 × 2 × 2
• 12 = 2 × 2 × 3
144 = 2 × 2 × 2 × 2 × 3 × 3
• 15 = 3 × 5
225 = 3 × 3 × 5 × 5

We observe that each prime factor in the prime factorization of the square of a number occurs twice the number of times it occurs in the prime factorization of the original number.

By using this pattern, we can find the square root of a given square number. For example, let's find the square root of 324:

• The prime factorization of 324 is 2 × 2 × 3 × 3 × 3 × 3.
• Pairing the prime factors, we have 324 = (2 × 3 × 3) × (2 × 3 × 3) = (2 × 3 × 3)².
• Therefore, the square root of 324 is 2 × 3 × 3 = 18.

Similarly, we can find the square root of 256:

• The prime factorization of 256 is 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2.
• Pairing the prime factors, we have 256 = (2 × 2 × 2 × 2) × (2 × 2 × 2 × 2) = (2 × 2 × 2 × 2)².
• Therefore, the square root of 256 is 2 × 2 × 2 × 2 = 16.

If we encounter a number that is not a perfect square, like 48, we can determine the smallest multiple of 48 that is a perfect square by completing the pairs of prime factors. In this case, multiplying 48 by 3 gives 144, which is a perfect square.

To find a number by which we should divide 48 to get a perfect square, we look for the factor that doesn't have a pair in the prime factorization. In this case, the factor 3 is not paired, so dividing 48 by 3 gives us 16, which is a perfect square.

### 3. Finding square of a number using identity

Calculating the squares of two-digit numbers can be made easier by expressing the number as a sum of two smaller numbers.
For example:
Example

232= (20+3)2 = 20(20+3)+3(20+3)

=202+20×3+20×3+32

=400+60+60+9

=529

### 4. Finding Square Root by Long Division Method

To find the square root  using the Long division method, follow these steps:

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.

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, √484=22.

## Square Roots of Decimals

To find the square root of a decimal number, you can use estimation or approximation methods. Let's go through the steps:
Consider 17.64

Step 1: To find the square root of a decimal number we put bars on the integral part (i.e., 17) of the number in the usual manner. And place bars on the decimal part  (i.e., 64) on every pair of digits beginning with the first decimal place. Proceed as usual. We get bar17.64

Step 2: Now proceed in a similar manner. The left most bar is on 17 and 42 < 17 < 52

Step 3: 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.

Step 4: Double the divisor and enter it with a blank on its right. Since 64 is the decimal part so put a decimal point in the quotient.

Step 5: We know 82 × 2 = 164, therefore, the new digit is 2. Divide and get the remainder

Step 6: Since the remainder is 0 and no bar left, therefore square root of 17.64 is 4.2

The document Squares and Square Roots Class 8 Notes Maths Chapter 5 is a part of the Class 8 Course Mathematics (Maths) Class 8.
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## Mathematics (Maths) Class 8

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## FAQs on Squares and Square Roots Class 8 Notes Maths Chapter 5

 1. What are square numbers?
Ans. Square numbers are the numbers obtained by multiplying a number by itself. For example, 4 is a square number because it can be obtained by multiplying 2 by itself (2 x 2 = 4). Similarly, 9 is a square number because it can be obtained by multiplying 3 by itself (3 x 3 = 9).
 2. What are some properties of square numbers?
Ans. Some properties of square numbers are: - Square numbers are always non-negative. They can be zero or positive. - The sum of any two consecutive square numbers is always equal to the next square number. For example, 3^2 + 4^2 = 5^2 (9 + 16 = 25). - The difference between any two consecutive square numbers is always equal to the sum of the two numbers. For example, 5^2 - 4^2 = 1^2 (25 - 16 = 9).
 3. Can a negative number be a square number?
Ans. No, square numbers are always non-negative. This means that they can be zero or positive, but not negative. The square of a negative number will always result in a positive number.
 4. How can we find the square of a number?
Ans. To find the square of a number, we multiply the number by itself. For example, to find the square of 5, we multiply 5 by itself: 5 x 5 = 25. Similarly, to find the square of -3, we multiply -3 by itself: (-3) x (-3) = 9.
 5. What is a square root?
Ans. A square root is the inverse operation of squaring a number. It is a value that, when multiplied by itself, gives the original number. For example, the square root of 25 is 5 because 5 x 5 = 25. The symbol used to represent the square root is √.

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