21 2 = = = = = = 900. The square of 24 (which is an even number): 24 2 = 576 (an even number)

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1 1. Squares Square Numbers or Perfect Squares: The numbers, which can be expressed as the product of two identical numbers, are known as square numbers or perfect squares. Squares of numbers ending with different digits: - If a number ends with 1 or 9, its square ends with the digit 1. - If a number ends with 2 or 8, its square ends with the digit 4. - If a number ends with 3 or 7, its square ends with the digit 9. - If a number ends with 4 or 6, its square ends with the digit 6. - If a number ends with 5, its square also ends with the digit 5. - If a number ends with 0, its square also ends with the digit = 11 11, so it is a perfect square. To verify the properties for squares ending with different digits, check the examples below: 21 2 = = = = = = 900 1) Find the square of the following number: 122, 37, 781, ) What will be the ones digit in the square of the following numbers? 29, 42, 313, 44, ) 14884, 1369, , ) 1, 4, 9, 6, 0 Even and Odd Square Numbers: - The square of an even number is always an even number. - The square of an odd number is always an odd number. - The square of any number can never end with odd number of zeroes. The square of 24 (which is an even number): 24 2 = 576 (an even number) The square of 39 (which is an odd number): 39 2 = 1521 (an odd number) The squares of 20 and 300 are 400 1

2 (ending with 2 zeroes)and (ending with 4 zeroes). 2. Interesting Patterns Involving Square Numbers Finding Natural Numbers between two Consecutive Square Numbers: Between two consecutive square numbers n 2 and (n+1) 2, we have 2n non-square numbers. To find the number of natural numbers between 3 2 and 4 2 : We have, 4 2 = (3 + 1) 2. So, number of natural numbers between 3 2 and 4 2 = 2 3 = 6. 1) Find how many nonsquares are there between and ) 210 Square of an Odd Number as the Sum of Two Consecutive Positive Integers: If n is any odd number, then Square of a Number as the Sum of Odd Numbers: The sum of the first n odd natural numbers = n 2. To represent 11 2 as the sum of two consecutive positive integers: = ( = 121). Here, 60 and 61 are two consecutive positive integers. To express 6 2 as the sum of first 6 odd natural numbers: 6 2 = Finding the Square of a Number Without Actual Multiplication The square of any number can be easily calculated using its nearest 10 multiple values. 4. Pythagorean Triplets To calculate 52 2, without actual multiplication: 52 2 = (50 + 2) 2 = (50 + 2) (50 + 2) = (50) 2 + (2 50 2) = = 2704 Find the square of 54, without actual multiplication

3 If the sum of the squares of two numbers is equal to the square of a third number, then the three numbers form a Pythagorean triplet. For any natural number m > 1: 2m, m 2 1 and m form a Pythagorean triplet. 5. Square Roots of a Number = = = 100 = 10 2 So, 6, 8 and 10 form a Pythagorean triplet. Find a Pythagorean triplet whose smallest member is , 120 and 122 The Square Root of a number is a value that, when multiplied by itself, gives the number, i.e., the square root of any number is one of its equal factors. The symbol for square root is. To find the square root of 36: We know, 6 6 = 36. So, 6 is the square root of 36, i.e., Find the least number by which should be divided so as to get a perfect square. Also, find the square root of the resulting number. 6; Finding the square root of a number by Repeated Subtraction method The square root of a number can be obtained by repeated subtraction of odd numbers 1, 3, 5, 7, 9, 11 till we get 0. The number of times subtraction is done to get zero gives the square root. This method is not suitable for finding the square root of large numbers as it is very tedious and tiresome. To find the square root of 144 by repeated subtraction: = = = = = = = = = = = = 0 Find the square root of 529 by repeated subtraction. 23 3

4 So, 7. Finding the square root of a number by prime factorisation method Square root of a number can be obtained by prime factorisation. To obtain the square root of 1225 by prime factorisation: Find the square root of 3136 by prime factorisation. 56 First, we find the prime factors of the square number. Then by pairing the prime factors, we get the square root. We find the prime factors of 1225 as follows, = = Now, by pairing the prime factors, we obtain the square root, 8. Finding the square root of a number by division method Square root of numbers can be calculated by division method irrespective of the numbers being perfect squares or non-perfect squares. This method is usually To find the square root of 676 by division method: Step 1: Place a bar over every pair of digits starting from the right hand side. If there is odd number of digits, the extreme left digit is without a bar. So, we have. Step 2: The number in the extreme left is 6. We have to find the greatest number whose square is less than or equal to 6. We take this number both as the 1) Find the square root of using division method 2) What is the least number that should be added to 6082 in order to obtain a perfect square? Also, find the square root of the resulting number. 1) 109 2) 2; 78 4

5 adopted when the numbers are very large. divisor and the quotient. We have } 6 lying between 2 2 and 3 2. So, 2 is the quotient and 2 is also the divisor. 2 multiplied by 2 gives 4. Write 4 below 6 and subtract. The remainder is Step 3: Bring down 76 to the right of 2. The new dividend is _ Step 4: For the new divisor, double the quotient and write it leaving a blank space next to it. Step 5: The new divisor is 2 followed by a digit. This digit will also be the new quotient such that the new quotient multiplied by the new divisor will be less than or equal to 276. Step 6: Clearly that digit is 6 because 46 6 = 276. Write 276 below 276 and subtract. The remainder is

6 9. Square Root of Decimal Numbers Square root of decimal numbers can also be calculated by division method. To find the square root of by division method: Step 1: A decimal number has an integral part and a decimal part. (For the integral part, place a bar over every pair of digits starting from the right hand side.) Step 2: For the decimal part, place a bar over every pair of digits starting from the first decimal place. If the number of digits in the decimal part is not even, we add a zero to the extreme right and then pair up the digits in the decimal part. Step 3: So, we have. The pair in the extreme left is 65. Find the greatest number whose square is less than or equal to 65. Take the number as the divisor and the quotient. We have } 64 lying between 8 2 and 9 2. So, 8 is the quotient and 8 is also the divisor. 8 multiplied by 8 gives 64. We write 64 below 65 and subtract. The remainder is Step 4: Since the next pair of digit is of the decimal part, place decimal after 8 in the quotient and bring down the pair 61 next Find the square root of using division method

7 10. Estimating the Square Root to 1. The new dividend is Step 4: For the new divisor, double the quotient and write it leaving a blank space next to it _ 161 Step 5: The new divisor is 16 followed by a digit. This digit will also be the new quotient such that the new quotient multiplied by the new divisor will be less than or equal to 161. Step 6: Clearly that digit is 1 because = 161. Write 161 below 161 and subtract. The remainder is To find a number whose square is close to a given number. To find the estimate value of square root of 240: 240 lies between 225 and 256, i.e., Also, 240 is closer to 225 than 256. is closer to than. Hence, is approximately equal to 15. Estimate the value of the following square roots to the nearest whole number: 1) 2) 1) 36 2) 30 7

8 11. Square Root of a Non-Perfect Square Number Square root of non-perfect squares can be calculated in the same way as that of perfect squares, using division method. To find the square root of 5, up to two decimal places: We follow the same method as we did in the procedure for finding the square root by division method. Since, we are asked to find the value of up to two decimal places, we will calculate the value up to three decimal places and then take approximation Find the square root of the following up to 2 places of decimal. 1) ) 45 1) ) 6.71 (up to 2 decimal places) 8