WBBSE Notes For Class 6 Maths Arithmetic Chapter 11 Square Root

Arithmetic Chapter 11 Square Root

Arithmetic Chapter 11 Square and square root :

Definition:

1. If a number is multiplied by the same number then the product obtained is called the square of that number and the number is called the square root of the product.
2. For example, when 2 is multiplied by 2, the product is 4 and it is written as 2 x 2 = 4.
3. Then 4 is the square of 2 and the square root of 4 is 2.
4. In order to indicate the square of a number, a small 2 is written on the right top of that number.
5. For example, the square of 4 is written as 4² = 16 and the square of 5 is written as 5² = 25.
6. Again, in order to express the square root of a number; we write the sign √ on the left-hand side of that number.
7. For example; the square root of 4, we write it as √4 i.e., the square root of 4 = √4, and similarly, the square root of 9 = √9.
8. Remember, √4 = 2 and √9 = 3.
9. Mathematically, the square of any number a is a x a = a^2, and the square root of a² is √a=a

WBBSE Class 6 Square Root Notes

Arithmetic Chapter 11 Mathematical Significance of square and square root

1. All of you know that if you add 2 twice then the result of the addition is 4.
2. If you add 3 thrice then the result of addition is 9,
3. If you 4, four times then the result of addition is 16.
4. ∴ The square of 2 is 4, the square of 3 is 9, and the square of 4 is 16.
5. So, the meaning of doing the square of any number is to perform a mathematical process where a number is added up to at a number of times.
6. Again 2 can be subtracted from 4, two times, and the result of the final subtraction is 0; 3 can be subtracted from 9, three times, and the result of-final subtraction is 0; 4 can be subtracted from 16, four times and the result Of final subtraction is 0.
7. The square root of 4 is 2, the square root of 9 is 3 and the square root of 16 is 4, These square roots are whole numbers.
8. The number from which the subtraction is done must be a perfect square whole number. ,
9. So, the meaning of doing the square root is to obtain a whole number so that this whole number can be subtracted from the given number, the same whole number of times.

 

Arithmetic Chapter 11 Perfect Square Numbers

1. We know that if an integer is multiplied by the same integer, then the product obtained is a square number. Again it is not only a square number, but also an integer. So the square roof of this perfect square number is an integer.
2. Therefore, we can give the following definition of a perfect square number
3. An integer that is a whole number is said to be a Perfect Square Number if the square root of that integer (whole number) is an integral whole number.
4. Finally, we can give the following definition of a perfect square number
5. An integral whole number is called a Perfect Square Number if it can be expressed as the product of two same-directed integral numbers.
6. For example, the integral whole number 4 can be expressed as the product of two same-directed integers (+2), i.e., 4 = (+2) x (+2).
   ∴ 4 is a perfect square number
7. or, the square root of any integral whole number is an integer, then the integral whole number is called Perfect Square Number.
8. For example, √9 = 3 and is an integer.
9. ∴ 9 is a perfect square number.

Understanding Square Roots

Arithmetic Chapter 11 The Perfect Square Numbers from 1 to 1000 and their list

 

 

Arithmetic Chapter 11 Characteristics of Perfect Square Numbers

1. The digit in the unit’s place of a perfect square number must be any one of the digits 0, 1, 4, 5, 6, or 9.
2. The digit in the unit’s place of any perfect square number never is 2, 3, 7, or 8.
3. Any perfect square number cannot contain an odd number of zeroes at the end.
4. Any perfect square number can be expressed as the product of two equal (both magnitude and sign) integers.

5.

1. The square number of a number containing one digit is a number having 1 digit or 2 digits.
2. The square number of a number containing 2 digits is a number having 3 digits or 4 digits.
3. The square number of a number containing 3 digits is a number having 5 digits or 6 digits.
4. The square number of a number containing 4 digits is a number having 7 digits or 8 digits.
5. The square number of a number containing 5 digits is a number having 9 digits or 10 digits.
6. The square number of a number containing 6 digits is a number having 11 digits or 12 digits etc.

6.

1.  If the unit’s place digit of a number is 0, then the units place digit of the square of that number is 0.
2. If the unit’s place digit of a number is 1, then the unit’s place digit of the square of that number is 1.
3. If the unit’s place digit of a number is 2, then the unit’s place digit of the square of that number is 4.
4. If the unit’s place digit of a number is 3, then the unit’s place digit of the square of that number is 9.
5. If the unit’s place digit of a number is 4, then the unit’s place digit of the square of that number is 6.
6. If the unit’s place digit of a number is 5, then the unit’s place digit of the square of that number is 5.
7. If the unit’s place digit of a number is 6, then the unit’s place digit of the square of that number is 6.
8. If the unit’s place digit of a number is 7, then the unit’s place digit of the square of that number is 9.
9. If the unit’s place digit of a number is 8, then the unit’s place digit of the square of that number is 4.
10. If the units place digit of a number is 9, then the unit’s place digit of the square of that number is 1.

 

Arithmetic Chapter 11 Determination of two special types of the square and square root

1.

1. We know that (11)² = 11 x 11 = 121
   (111)² = 111 x 111 = 12321 .
   (1111)² = 1111 x 1111 = 1234321
2. We observe the squares of the above numbers. We can find the square of a number containing 1111….etc i.e., the number formed only by 1 at case (without multiplicities)

Step 1.

1. First, find the number 1 in the given number. Then write the consecutive natural numbers starting from 1 as many 1 are there in the given number.
2. For example, suppose you have to find the square of 1111 i.e. (1111)².
3. The number 1 in the number is 4.
4. At first, you write the first four consecutive natural numbers starting from 1 i.e., 1234.

 

Step 2.

1. Then you write the numbers in a reverse way up to 1.
2. The number so obtained is the required square of the number.
3. For the above example, after 1234 you write 321 i.e., after 1234 you put 321 so that the number obtained is 1234321. This is the square of 1111.
   (1111)² = 1234321.
4. Similarly, We get (111111)² = 12345654321.
5. Here the number 111111 contains 6 ones. So we write first 123456 and then 54321 is written in a reverse way, so that we get 12345654321 which is the square of 111111.
6. Again, the square root of 12321 is 111 because the given number 12321 contains the first 3 consecutive natural numbers 1,2,3, and then 2, 1 are written in a reverse way.
   712321 = 111.
7. Similarly, 7123454321 = 11111.

 

2.

We know that 9² = 81, (99)² = 9801, (999)² = 998001.

We observe the squares of the above numbers. We can find the square of a number of form 999…… etc i.e., the number formed by 9 only at ease (without multiplication).

The process of obtaining the square of such a number is discussed below

Step 1.

First, count the number of 9s’ i.e. first find how many 9s’ are there in the given number.

Then subtract 1 from the number obtained.

So the square number must contain one 9 less than the number of 9s that the given number.

 

Step 2.

1. Then put one 8 on the right side of the last 9 and then put as many zeroes as the number of 9 written and lastly put 1.
2. Then the required square number is obtained.
3. For example, in the square of 999999 i.e., (999999) the given number contains 6 nines, so the square number contains 6-1 = 5 nines, then put one 8 and 5 zeroes and then put 1 in the last
4. ∴(999999)² = 999998000001. .
5. In this way, we can find easily the square of any number formed by 9 only.
6. Again in a reverse way, we can find the square root of the numbers satisfying the above conditions easily.
7. For example, √99980001 = 9999,
   √999999998000000001= 999999999.

Important Definitions Related to Square Roots

Arithmetic Chapter 11 Method of finding the square root

1. Here we shall discuss two methods of finding the square root of any positive integer.
2. The methods are
3. Determination of square root by Factorisation Method
4. The working rule of this method can be discussed step-wise as follows
5. Step 1. The given perfect square number is factorized into prime factors.
6. Step 2. Write the same prime factors arranging them in pairs by multiplication.
7. Step 3. For each pair of the same prime factors, take or select only one prime factor. The selection of a single factor can be done for each pair of different factors.
8. Step 4. Then obtain the product of these selected prime factors.
9. Step 5. This required product will be the square root of the given perfect square number.
10. Observe the following example :

Short Questions on Finding Square Roots

Example 1: Find the square root of 14400.

Solution:

Step 1

Step 2

14400 = (2×2) x (2×2) x (2×2) x (3×3) x (5×5)

Step 3

∴ √14400 = 2 x 2 x 2 x 3 x 5 = 120

step 4

The required square root =120.

2. Determination of square root by division method :

The square root of each perfect square number can be determined easily by the division method. Generally, we use the division method for the calculation of the square root of large perfect square numbers.

The working rule of this method can be discussed step-wise as follows :

Step 1:

1. At first, we mark each pair of digits starting from the extreme right digit towards the left (i.e., starting from the digit that lies in the units’ place towards the left) by putting short lines over them.
2. For example,
3. If the number of digits of the given number is an odd number, we go on marking by short lines over each pair of digits then at the extreme left end, we are left with a single digit.
4. Therefore, there will be a short line marking over this single digit at the left end in this case.
5. If the number of digits of the given number is an even number, then we go on marking by short lines over each pair of digits till the end.
6. So marking will be done over all the digits.

 

Step 2.

1. Now give two division signs in two sides of the given number.
2. Then below the digit (in the case of a number containing an odd number of digits) or the pair of digits (in the case of a number containing an even number of digits), we write a perfect square number equal to or nearer to but less than the number above it.
3. Here the first digit i.e., the extreme left digit (which is put in the quotient place of this division process) in the required square root of the given number will be the square root of that perfect square number.
4. Now subtract this perfect square number from the number above it.
5. For. example

 

 

Common Questions About Perfect Squares

Step 3.

1. Put the next pair of marking digits on the right side of the result of subtraction.
2. For example,

 

 

 

Step 4.

1. Now take this number so obtained as a dividend and for the divisor, we take the number which is twice the number already put in the first place of the required square root.
2. For example,

 

 

 

Step 5.

1. Then on the right side of this number (i.e., the divisor), we put a maximum digit by our choice of selection such that when the divisor (which is obtained after putting this digit) so obtained is multiplied by that digit obtained by our selection produces the maximum number not exceeding the dividend number.
2. The product so obtained is put below the dividend and subtracted.
3. For example,

 

 

Step 6.

1. Proceed step 5 repeatedly till the last pair of marking digits in the extreme right end.
2. For example,

 

 

Step-7.

1. After the completion of the division process, the obtained number in the quotient is the required square root of the given number.
2. ∴ The square root of 55225
   = √55225
   = 235
3. And the square root of 853776
   = √853776
   = 924.

Practice Problems on Square Roots

Arithmetic Chapter 11  Some special learning things

When the given number is factorized:

1. if it is seen that after pairing the same factors one extra factor is remaining which is not paired up, then we note that :
2. the given number is not a perfect square number; if the given number is multiplied by or divided by that factor, then the number so obtained in either of the cases will be a perfect square number.
3. For example, we take the number 180
4. Here factor 5 is only one and it is remaining.
5. It is unpaired.
6. So the given number 180 is not a perfect square number
7. If the given number 180 is multiplied by 5, then the product is 180 x 5 = 900 which is a perfect square number.
8. Again if the given number 180 is divided by 5, we get 180 ÷ 5 = 36 which is also a perfect square number.

If 2 or more factors are remaining which are not paired up, then:

1. The given number is not a perfect square number.
2. If the given number is multiplied by the product of the remaining factors or if the given number is divided by the product of the remaining factors.
3. Then the numbers obtained in both cases will be perfect square numbers.
4. For example, we take the number 1260.
5. 1260 = 2x2x3x3x5x7 = (2 x 2)x(3 x 3)x5x7.
6. Here 5 and 7 these two factors are remaining unpaired.
7. So the given number 1260 is not a perfect square number.
8. When the given number 1260 is multiplied, by 5 x 7 = 35, then the product is 1260 x 35 = 44100 which is perfect or if the given number 1260 is divided by 35, we get, 1260 35 = 36 which is also a perfect square number.

 

If there is a remainder in the division process of finding the square root of a given number, then:

1.  The given number is not a perfect square number
2. If we subtract the remainder from the given number, then the number obtained after subtraction will be a perfect square number
3. The square of the next integer number obtained in the quotient will be the perfect square number next to the given number.

Examples of Square Roots in Real Life

1. For example, let us take the number 3250
2. Here the remainder is 1.
3. ∴ The given number 3250 is not a perfect square number.
4. 3250 – 1 = 3249, which is a perfect square number.
5. The quotient in the division of the square root is 57. Its next integral number is 57 + 1 = 58.
6. The square of the number 58 is 3364.
7. 3364 is the perfect square number next to the given number 3250.