**Notes of chapter: Squares and Square Roots are presented below. Indepth notes along with worksheets and NCERT Solutions for Class 8.**

**(1) Square Number-**

In simple words, multiplication of a number by itself is known as **square number **of that number. If a natural number m can be expressed as n^{2}, where n is a natural number, then m is a **square number**.

m = n^{2}

Where, m and n are natural numbers.

**Eg:- **Represent 36 and 34 as a square number.

**(i)**36 is a natural number.

It can be represent as 6 × 6 or 6^{2}.

36 = 6^{2}

Hence 36 is a square of 6.

**(ii)**

34 is a natural number.

But, it can not be represent as n^{2}. [Because fraction of 32 = 2, 2, 2, 2, 2]

Hence 34 is not a square number.

**(2)Perfect squares-**

Squares numbers are also called **perfect squares**.

**(3)** **Properties of square numbers-**

**(i)** If a number has 0, 1, 4, 5, 6, or 9 at ones place, then it must be a square number.

**Eg:-**

16, 81, 64, 169, 100 are square numbers.

**(ii)** If a number has 1 or 9 at the unit place, then its square ends in 1.

**Eg:-**

Number |
Square |

1 | 1 |

9 | 81 |

11 | 121 |

19 | 361 |

21 | 441 |

**(iii)** If a number has 4 or 6 at its unit place, then its square ends in 6.

**Eg:-**

Number |
Square |

4 | 16 |

6 | 36 |

14 | 196 |

16 | 256 |

**(iv)** Square numbers can only have even numbers of zeros at the end of the number.

**Eg:-**

Number |
Square |

10 | 100 |

20 | 400 |

100 | 10000 |

20 | 40000 |

1000 | 1000000 |

2000 | 2000000 |

**(v)** Even numbers have even square numbers.

**Eg:-**

Even Number |
Square of Even Number |

2 | 4 |

6 | 36 |

14 | 144 |

10 | 100 |

22 | 484 |

26 | 676 |

**(vi)** Odd numbers have odd square number.

**Eg:-**

Odd Numbers |
Square of Odd Numbers |

3 | 9 |

5 | 25 |

9 | 81 |

11 | 121 |

25 | 625 |

27 | 729 |

**(4)** **Some interesting patterns**

**(i)Adding triangular numbers-**

**(a)Triangular numbers** are those numbers whose dot patterns can be arranged as triangles.

**Eg:-**

**(b)**If we combine two consecutive triangular numbers, we get a square number.

**Eg:-**

**(ii)Numbers between square numbers**

**Method 1-**

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

**Eg:- **Find how many non square numbers are there between the square of numbers 1 and 2.

**Ans- **

We know that there are 2n non perfect square numbers between the squares of the numbers n and (n+1).

n = 1

Non perfect square numbers between the squares of the numbers n and (n+1)

= 2n

Non perfect square numbers between the squares of the numbers 1 and 2

= 2× 1=2

Hence,non perfect square numbers between square of 1 and 2 will be 2.

**Method 2-**

Non squares numbers between two square numbers is one less than the difference between squares of those two consecutive numbers.

**Eg:- **Find how many non square numbers are there between the square of numbers 1 and 2.

**Ans-**

Square of 1 = 1^{2 }= 1

Square of 2 = 2^{2} = 4

Non square numbers between 1 and 4 are 2, 3.

Difference between squares of two consecutive numbers = 4 – 1 = 3

Non squares numbers between two square numbers is one less than the difference between squares of those two consecutive numbers.

**(iv)Subtracting odd numbers**

**If a natural number can be expressed as a subtraction of successive odd natural numbers starting with 1, then it is a perfect square.**

**Eg:-**

**(a)**Check whether 25 is a perfect square.

**Ans-**

Subtraction of the successive odd numbers starting from 1

**(a)** 25 – 1 = 24

**(b)** 24 – 3 = 21

**(c)** 21 – 5 = 16

**(d)** 16 – 7 = 9

**(e)** 9 – 9 = 0

1 + 3 + 5 + 7 + 9 = 25, a perfect square.

**(b)**Check whether 27 is a perfect square.

**Ans-**

**Eg:-**

Subtraction of the successive odd numbers starting from 1

**(a) **27 – 1 = 26

**(b)** 26 – 3 = 23

**(c)** 23 – 5 = 18

**(d)** 18 – 7 = 11

**(e)** 11 – 9 = 2

**(f)** 2 – 11 = – 9

Hence, 27 is not a perfect square.

**(9)** **Finding square root**

**(i)** **Finding square root through repeated subtraction**

Sum of first n numbers = n^{2}

or

If the number is a square number, it has to be the sum of successive odd numbers starting from 1.

**Eg:-**Find √81

**(a)** 81 – 1 = 80

**(b)** 80 – 3 = 77

**(c)** 77 – 5 = 72

**(d)** 72 – 7 = 65

**(e)** 65 – 9 = 56

**(f)** 56 – 11 = 45

**(g)** 45 – 13 = 32

**(h)** 32 – 15 = 17

**(i)** 17 – 17 = 0

Total steps to subtract successive odd numbers = 9

√81 = 9

**Step 4-**

Double the divisor and enter it with a blank on its right.

**Helping Topics**