Finding Square and Square roots
Square: A number m is called a square number if it is expressed as n2. Here n is called the square root of m. Square root is the inverse operation of squares. Every square number is a sum of the first n odd natural numbers.           152 = (1 × 2)100 + 25           252 = (2 × 3)100 + 25           352 = (3 × 4)100 + 25           452 = (4 × 5)100 + 25     If a number is of the form a5, then: a52 = a(a + 1)100 + 25 Pythagorean Triplet: A set of three numbers that satisfies the Pythagoras Theorem is called Pythagorean Triplet. Pythagorean Triplets can be found using a general expression (2m)2 + (m2 – 1)2 = (m2 + 1)2, where m is any natural number greater than one. Methods to find the square of the numbers 1) Column method 2) Visual method 3) Diagonal method Square root: Square root of a given number is a number whose square is equal to the given number. Positive square root of a number is denoted by the symbol √.  Square root of a number can be found using the following three methods      •  Repeated Subtraction Method      •  Prime Factorisation Method      •  Long Division Method. The following steps to find the square root of a given number using Repeated subtraction method 1. Subtract successive odd numbers from the given number starting from 1 till the difference becomes zero. 2. Count the number of steps it took to get the difference as zero. The following steps to find the square root by prime factorization method 1. Find the prime factors of the given number. 2. Form pairs of prime factors. 3. Take one prime factor from each pair of prime factors of the given number. 4. Find the product of these prime factors to get the square root of the given number. If prime factors of a number do not occur in pairs, the given number is not a perfect square. The following steps to find the square root by long division method 1. Draw lines over pairs of digits from right to left. 2. Find the greatest number whose square is less than or equal to the digits in the first group. 3. Take this number as the divisor and quotient of the first group and find the remainder. 4. Move the digits from the second group besides the remainder to get the new dividend. 5. Double the first divisor and bring it down as the new divisor. 6. Complete the divisor and continue the division. 7. Repeat the process till the remainder becomes zero. The following steps to find the square root of desimals 1) Assume the number in the decimal form. 2) Place bars on the integral part as we do in the process of finding the square root of a perfect square of some natural number. 3) Make even number of decimal places by affixing a zero on the extreme right of decimal part, if neccessary. 4) Place bars on the decimal part on every pair of digits beginning with the first decimal place. 5) Start finding the square root by the long division method and put the decimal point in the square root as soon as the integral part is exhausted. The following steps to find the estimating square root by long division method 1) Assume the number whose square root is to be computed. 2) Determine the number of decimal places to which the square root of the number is to be computed. Suppose the square root of the given number is to be computed correct to n places of decimal. 3) Count the number of digits in the decimal part. If the number of digits is less than 2n, then affix a suitable number of zeros at the extreme right of the decimal part so that the number of digits in decimal part becomes 2n. 4) By using the method of long division to find the square root upto (n+1) places of decimal. 5) Check the digit at (n+1)th decimal place, if it is less than 5, then delete it to get the answer correct to n decimal places. If the digit at (n+1)th decimal place is 5 or morethan, then increse the digit at nth decimal place by one and delete the digit at (n+1)th place to obtain the square root correct upto n decimal places.

#### Summary

Square: A number m is called a square number if it is expressed as n2. Here n is called the square root of m. Square root is the inverse operation of squares. Every square number is a sum of the first n odd natural numbers.           152 = (1 × 2)100 + 25           252 = (2 × 3)100 + 25           352 = (3 × 4)100 + 25           452 = (4 × 5)100 + 25     If a number is of the form a5, then: a52 = a(a + 1)100 + 25 Pythagorean Triplet: A set of three numbers that satisfies the Pythagoras Theorem is called Pythagorean Triplet. Pythagorean Triplets can be found using a general expression (2m)2 + (m2 – 1)2 = (m2 + 1)2, where m is any natural number greater than one. Methods to find the square of the numbers 1) Column method 2) Visual method 3) Diagonal method Square root: Square root of a given number is a number whose square is equal to the given number. Positive square root of a number is denoted by the symbol √.  Square root of a number can be found using the following three methods      •  Repeated Subtraction Method      •  Prime Factorisation Method      •  Long Division Method. The following steps to find the square root of a given number using Repeated subtraction method 1. Subtract successive odd numbers from the given number starting from 1 till the difference becomes zero. 2. Count the number of steps it took to get the difference as zero. The following steps to find the square root by prime factorization method 1. Find the prime factors of the given number. 2. Form pairs of prime factors. 3. Take one prime factor from each pair of prime factors of the given number. 4. Find the product of these prime factors to get the square root of the given number. If prime factors of a number do not occur in pairs, the given number is not a perfect square. The following steps to find the square root by long division method 1. Draw lines over pairs of digits from right to left. 2. Find the greatest number whose square is less than or equal to the digits in the first group. 3. Take this number as the divisor and quotient of the first group and find the remainder. 4. Move the digits from the second group besides the remainder to get the new dividend. 5. Double the first divisor and bring it down as the new divisor. 6. Complete the divisor and continue the division. 7. Repeat the process till the remainder becomes zero. The following steps to find the square root of desimals 1) Assume the number in the decimal form. 2) Place bars on the integral part as we do in the process of finding the square root of a perfect square of some natural number. 3) Make even number of decimal places by affixing a zero on the extreme right of decimal part, if neccessary. 4) Place bars on the decimal part on every pair of digits beginning with the first decimal place. 5) Start finding the square root by the long division method and put the decimal point in the square root as soon as the integral part is exhausted. The following steps to find the estimating square root by long division method 1) Assume the number whose square root is to be computed. 2) Determine the number of decimal places to which the square root of the number is to be computed. Suppose the square root of the given number is to be computed correct to n places of decimal. 3) Count the number of digits in the decimal part. If the number of digits is less than 2n, then affix a suitable number of zeros at the extreme right of the decimal part so that the number of digits in decimal part becomes 2n. 4) By using the method of long division to find the square root upto (n+1) places of decimal. 5) Check the digit at (n+1)th decimal place, if it is less than 5, then delete it to get the answer correct to n decimal places. If the digit at (n+1)th decimal place is 5 or morethan, then increse the digit at nth decimal place by one and delete the digit at (n+1)th place to obtain the square root correct upto n decimal places.

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