CLASS VIII

• CHAPTER – 6

​

SQUARE AND SQUARE ROOT

​

Prepared by –

Mahesh kumar Badetri

TGT Maths

JNV Bemetara

​

​

Square

and

Square roots

We know that the area of a �square = side x side�(where side means length of the side)

INTRODUCTION

Square number

• What is special about the numbers 4,9,25,64 and other such numbers? Since, 4 can be expressed as 2×2 = 2², 9 can be expressed as 3×3=3², all such numbers can be expressed as product of the number with itself. Such numbers like1,4,9,16,25,… are known as square numbers. In general, if a natural number m can be expressed as n², where n is also a natural number, then m is a square number. is 32 a square number. We know that 5²=25 and 6²=36.If 32 is a square number, it must be the square of a natural number between 5&6.but there is no natural number between 5&6.therefore 32 is not a square number.

​

What is special about the numbers 4,9,25,64 and other such numbers? Since, 4 can be expressed as 2×2 = 2², 9can be expressed as 3×3=3², all such numbers can be expressed as product of the number with itself. Such numbers like1,4,9,16,25,… are known as square numbers.

In general, if a natural number m can be expressed as n², where n is also a natural number, then m is a square number. is 32 a square number. We know that 5²=25and 6²=36.If 32 is a square number, it must be the square of a natural number between 5&6.but there is no natural number between 5&6.therefore 32 is not a square number.

PROPERTIES OF SQUARE NUMBERS NUMBER

• Consider following numbers and their squares

From the above table, can we enlist the square numbers between 1&100 Are there any natural numbers upto 100 left out? We will find that rest of the numbers are not square numbers. The numbers 1,4,9,16…are square numbers. So these are called perfect squares.

Study the square numbers in table in the last slide .What are ending digits (that is, digits in the one᾿s place)of the square numbers?

All these numbers end with0,1,4,5,6 or 9 at unit ᾿s place. None of these end with 2,3,7 or 8 at unit ᾿s place. Can we say that if a number ends in 0,1,4,5,6 or 9,then it must be a square number? yes of course

Study the following table of some numbers and their squares and observe the one᾿s place in both.

The following square numbers end with digit 1

​

​

​

​

​

​

​

​

​

​

​

If a number contains one zero at the end, its square have two zeros and if a number contains two zero at the end, its square have four zeros.

For egg:-

10²=100 100²=10000

20²=400 200²=40000

80²=6400 900²=810000

NUMBERS BETWEEN SQUARE NUMBERS

Let us now see if we can find some interesting pattern between two consecutive square numbers. Between 1²& 2² there are two non-square numbers2,3 If we think of any natural no. n &(n+1),then, (n+1)²-n²=(n²+2n+1)-n²=2n+1. We can say that there are 2n non-perfect square numbers between the squares of the number n&(n+1)

Adding odd numbers:-

Consider the following:-

1+3+5 �(Sum of first 3 odd numbers)=9=3²

So, we can say that sum of first n odd natural no. is n².

NOTE:- If a natural no. can’t be expressed as a sum of successive odd natural nos. starting with 1,then it is not a perfect square.

A sum of consecutive natural number

• A sum of consecutive natural number:-Consider the following- 3² = 9 = 4+5 ,

5²= 25 =12+13

• Product of two consecutive even/odd nos.:- 11×13 = 143 = 12²-1

11×13 = (12-1)×(12+1) = 12²-1

So in general we can say that(a+1) ×(a- 1)=a²-1

Finding the square of a number

We can find the square of a number quickly by knowing the method shown below:- To find the square of 23: 23 = 20+3

23²= (20+3)² = 20(20+3) + 3(20+3) =20²+20×3+3×20+3²

=400+60+60+9=529

PYTHAGOREAN TRIPLETS

• Consider the following

3²+4² = 9+16 = 25 = 5²

The collection of nos. 3,4,5 is known as Pythagorean triplet. 6,8,10 is also a Pythagorean triplet, since

6²+ 8² = 36 + 64 =100 =10²

• FORMULA:- (2m)² + (m²-1) ²=(m²+1) ²

SQUARE ROOTS

• Square root is the inverse operation of squaring and its symbol is √. FOR EG:- 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

From the above, you may say that are two integral square roots of a perfect square number. In this chapter, we shall take up only positive square roots of a natural number. positive square root of a number is denoted by the symbol √

For example: √4 =2 (not – 2) ; √9 = 3 (not -3)

​

​

​

SQUARE ROOTS

Finding square root through prime factorisation

• Consider the prime factorisation of the following numbers and their squares . Each prime factor in prime factorisation of the square of a number,occurs twice the number of times it occurs in the prime factorisation of the number itself.

Finding square root through repeated subtraction

• Do you remember that the sum of the first n odd natural numbers is n2 ? That is every square number can be expressed as a sum of successive odd natural numbers starting from 1 consider √81.then.
• (i) 81-1 = 80 (ii) 80-3 = 77 (iii) 77-5 = 72

(iv) 72-7 = 65 (v) 65-9 = 56 (vi) 56 – 11 = 45

(vii) 45-13 = 32 (viii) 32 – 15 = 17 (ix) 17-17 = 0

From 81 we have subtracted successive odd numbers starting from 1 and obtained 0 at 9^th step

Therefore √81 = 9

​

Finding square root by division method

• When the numbers are large, even the method of finding square root by prime factorisation becomes lengthy and difficult. To overcome this problem we use Long Division Method.

​

Square Roots of Decimals by Division Method

• Consider √17.64
• Step 1: To find the square root of a decimal number we put bars on the integral part of the number in the usual manner. And place bars on the decimal part on every pair of digits beginning with the first decimal place proceed as usual.

We get 17.64

• Step 2:-Now proceed in a similar manner. The left most bar is on 17&4<17<5.Take this number as the divisor and the number under the left-most bar as the dividend. Divide and get the remainder
• Step 3:-the remainder is 1.Write the number under the next bar on the right of this remainder, to get 164.

continue

• Step 4:-Double the divisor and enter it with a blank on its right. Since is the decimal part so put a decimal point in the quotient.
• Step5:-We know 82×2=164,therefore the new digit is 2. Divide and get the remainder.
• Step6:-Since the remainder is 0 and no bar left, therefore

√17.64 = 4.2

​

Example 1:-

Find a Pythagorean triplet in which one number is 12.

​

Solution:-

If we take m²-1=12 Then, m²=12+1=13

Then the value of m will not be an integer.

So, we try to take m² +1=12.again m²=11 will not give an integer value for m.

So, let us take 2m=12 Then m=6

Thus, m²-1=36-1=35 and m²+1=36+1=37 Therefore, the required triplet is 12,35,37.

Example 2 :-

Is 90 a perfect square?

​

Solution:-

We have 90=2×3×3×5

The prime factors 2&5 do not occur in pairs. Therefore,90 is not a perfect square. That 90 is not a perfect square can also be seen from the fact that it has only one zero.

MAIN POINTS

1. If a natural number m can be expressed as n², where n is also a natural number, then m is a square number.
2. All square numbers end with 0, 1,4,5,6 or 9 at units place.
3. Square numbers can only have even number of zeros at the end.
4. Square root is the inverse proportion of square.

Thank you