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Ganit Learning Guides Basic Arithmetic Number Sets Author: Raghu M.D. Contents NUMBER SETS ........................................................................................................... 2 SQUARES AND SQUARE NUMBERS.................................................................................................... 2 SQUARE ROOTS .................................................................................................................................. 8 CUBES................................................................................................................................................ 13 CUBE ROOTS ..................................................................................................................................... 16 Basic-Arithmetic 1 of 19 ©2014, www.learningforkowledge.com/glg NUMBER SETS Introduction: Numbers can be classified as Integers, Whole numbers and Natural number sets. These sets of numbers are discrete, meaning each number in a set is more or one less than the next number in the set. Quotients are ratios of any two discrete numbers and are elements of another set. Number sets can be represented by a line, generally called a number line. A number line has all the other numbers sets as subsets. I N -5 0 5 W Fig 1.1 An integer line showing (a) Natural numbers (b) Whole numbers (c) Integers N W I 1, 2, 3…….. 0, 1, 2, 3……… -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5…………. Quotients can be any point on the line examples 3 7 −4 …….. , ,......... 7 5 2 SQUARES AND SQUARE NUMBERS An integer n is said to be a square number if it is equal to the product of an integer multiplied by itself. ∴n = m × m = m 2 (n and m are integers) A square root is the inverse of a square If n = m2 then n = m If m is an integer then n is a square number 1, 4, 9, 16, 25……….are all square numbers or perfect squares. Properties of square numbers: 1. Unit digit of a square number can be 0, 1, 4, 5, 6 or 9, in other words, a square number can only end in 0, 1, 4, 5, 6 or 9 and not in 2, 3, 7, 8. 2. A square number can not be negative 3. If m2 = n, where n is a square number and if p is a prime number then p×n is not a square number. If p×n = m2, m = p × n and p cannot be exactly square rooted as it has no factors. Basic-Arithmetic 2 of 19 ©2014, www.learningforkowledge.com/glg 4. Product of two square numbers is also a square number. For example, 4 and 9 are square numbers. Hence 4 × 9 = 36, where 36 is also a square number having a square root of 6. 5. A square number or a perfect square when divided by 3 leaves a reminder of 1 or 0 For example: Given 49 = 72 16 We have 3) 49 48 r1 6. A square number can be expressed as a set of dots arranged in equal number of rows and columns. Every column or row has the same number of dots For example 9 which is 32 can be drawn as a pattern or 9 dots arranged in 3 rows and 3 columns with 3 dots in each row or columns 7. Square of an even number is even and an odd number is odd. For example 62 = 36 92 = 81 Column method: A simpler method of finding the square of a number is to use the formula (a+b)2 = a2 + 2ab + b2 For example 272 = (20 + 7)2 = 202 + 2 × 20 × 7 + 72 = 400 + 280 + 49 = 729 This method can be improved to find 312 302 2 × 30 × 1 12 900 62 1 312 = 963 Square diagonal method This method is useful for finding squares of numbers with 3 or more digits. In this method a square table is drawn. This square is divided into square cells further divided into two parts by a diagonal. Each part represents a digit. For example 4562 is represented below. 4 4 5 6 5 1 2 6 2 2 0 2 0 2 4 3 5 3 4 Basic-Arithmetic 6 0 3 0 6 3 of 19 ©2014, www.learningforkowledge.com/glg 4562 is calculated as shown below add numbers enclosed by diagonals as shown. 1, 2+6+2, 2+0+2+0+2, 4+3+5+3+4, 0+3+0, 6 Adding these numbers within a pair of diagonals or diagonal and a corner is shown below. 1, 10, 6, 19, 3, 6 Please note that number 10 is written as 1 followed by 0 underlined to facilitate easy addition. 1 +1 2 0 6 0 1 0 7 9 3 6 0 0 0 9 3 6 ∴ 4562 = 207936 Add and separate the unit digit as shown by an underline, write the tens digits one space moved to the left in the second line. Add the two lines of numbers to get the answer as square of 456 which is equal to 207936. Example 1: Find the squares of 21, 411 and 78 Working: a) 212 = 21 × 21 Answer: 212 = 441 b) 4112 = 411 × 411 411 × 411 411 411 1644 168921 Answer: 4112 = 168921 c) 782 = 78 × 78 78 × 78 624 546 6084 Basic-Arithmetic 4 of 19 ©2014, www.learningforkowledge.com/glg Answer: 782 = 6084 Example 2: Observe the pattern and fill up the missing number 112 = 121 1112 = 12321 11112 = 1234321 111112 = ______________ Ans: 123454321 Example 3: Find 672 using (a + b)2 or column method 672 = (60+7)2 672 = a2 + 2ab + b2 602 + 2x6x7 + 72 840 + 49 3600 + = 4489 Answer: 672 = 4489 Column method a2 2 × a × b b2 6x6 2×6×7 7×7 36 84 49 8 4 44 8 9 Note: 8 is carried over from 84 to the first column and 4 is carried over from 49 to the second column. Answer: 672 = 4489 Example 4: Find 352 using the square diagonal method 3 0 5 2 0, 1+9+2, 5+2+5, 5 0, 12, 12, 5 9 1 5 2 00 01 01 5 5 3 5 Basic-Arithmetic 12 01 3 12 00 2 5 of 19 05 05 5 ©2014, www.learningforkowledge.com/glg Note: 1 is carried over from 12 to the first column and 1 is carried over from 12 to the second column. Answer: 352 = 1325 Example 5: Using the remainder method to identify the numbers which are perfect squares a) 81 b) 7396 c) 65 d) 4226 Working: a) Divide 81 by 3 27 3) 81 81 γ 00 Answer: Since the remainder is 0, the number 81 is a perfect square. b) Divide 7396 by 3 213 3) 7396 6 396 3 96 9 6 6 γ0 Answer: Since the remainder is 0, the number 7396 is a perfect square. c) Divide 65 by 3 22 3) 65 6 05 3 γ2 Answer: Since the remainder is 2, the number 65 is not a square. Basic-Arithmetic 6 of 19 ©2014, www.learningforkowledge.com/glg Note: For a perfect square, the remainder has to be 0 or 1. d) Divide 4226 by 3 1408 3) 4226 3 12 12 002 0006 0026 0024 γ 2 Answer: Since the remainder is 2, the number 4226 is not a square. EXERCISE 1.1 1. Find the squares of the following numbers a) 31 b) 97 c) 415 d) 100 e) 1278 2. Using the remainder method identify square numbers from the list a) 625 b) 97 c) 12321 d) 10,000 e) 148 3. State True or False a) Square of an odd number is odd number b) Square of an even number is odd number c) Square of a negative number is a negative number d) Square of a prime number is a prime number e) Square of the length of a square is equal to its area 4. Observe the following pattern and find the missing square numbers 112 = 121 1112 = ________ 11112 = 1234321 5. Observe the following pattern and find the missing numbers. 12 + 22 + 22 = 32………………… (1×2=1 and 2+1=3) 22 + 32 + 62 = 72………………… (2×3=6 and 6+1=7) 32 + 42 + 122 = 52 + 62 + 302 = 6. Using the column method find the squares of a) 79 b) 31 c) 53 d) 22 e) 17 7. List all the square numbers in between the following numbers a) 50 to 60 b) 60 to 90 c) 90 to160 d) 160 to 250 e) 250 to 350 Basic-Arithmetic 7 of 19 ©2014, www.learningforkowledge.com/glg 8. Using the identity (a+b)2 = a2+2ab+b2 find the square of a) 21 b) 12 c) 32 d) 101 e) 501 9. Using the identity (a—b)2 = a2 – 2ab + b2, find the squares of a) 99 b) 28 c) 38 d) 998 e) 199 10. Using square diagonal method find the squares of a) 586 b) 35 c) 21 d) 145 e) 236 Answers to EXERCISE 1.1 1. a) 961 b) 9409 c) 172225 2. a), c), d) are square numbers, others are not square numbers. 3. a) True 4. 12321 5. The missing numbers are: 132 and 312 6. a) 6241 7. a) Nil b) 64, 81 8. a) 441 b) 144 9. a) 9801 10. a) 343396 b) False c) False b) 1961 b) 784 c) 2809 d) 10000 d) False d) 484 e) True e) 289 c) 100, 121, 144 c) 1024 d) 10201 c) 1444 b) 1225 c) 441 e) 1633284 d) 169, 225 e) 256, 289, 324 e) 251001 d) 996004 d) 21025 e) 39601 e) 55696 SQUARE ROOTS If n=m2, then n = m, or more precisely n = ± m , because –m×–m=m2 as well as m×m=m2. n is called the square root of n or a radical. Only perfect squares have integers as square roots. Properties: 1) If n is not a perfect square, then its square root is not an integer. For example, 10 is not a perfect square, hence 10 cannot be an integer. Its value can only be found approximately. Basic-Arithmetic 8 of 19 ©2014, www.learningforkowledge.com/glg 2) Only numbers with unit digits 0, 1, 4, 5, 6 and 9 can have integers as square roots, for example, 25 =5 but 23 is not an integer. However all numbers not ending with 0, 1, 4, 5, 6 and 9 do not have integer square roots. 3) Square root of an even number is even and an odd number is odd. Square roots by inspection: Square roots of 2 or 3 digit numbers can be found by inspection or trial and error. For example, 81 is 9, because 9×9 = 81. Table of Squares Number Square 1 1 2 4 3 9 4 16 5 25 6 36 7 49 8 64 9 81 10 100 ∴ 81 = 9 Square roots by progression rule: A square root of a number n=m2 is given by the number of terms, which will be in fact m terms, in the progression of successive odd numbers, when added is n = m2 = 1+3+………. (2m-1) is equal to m2, where (2m-1) is the last term of the progression. For example 16 can be calculated by the progression 16 = 1+3+5+7 which has 4 terms 16 = m = 4 The above method can be modified for calculating the square roots of numbers only. For example 25 can be obtained by reducing 25 by successive odd numbers till it becomes zero. The number of times the subtraction is carried out gives the square root. 25-1=24, 24-3=21, 21-5=16, 16-7=9, 9-9=0 It has taken 5 steps to reduce 25 to 0, hence 25 = 5 Factorization method: Any composite number can be written as a product of its prime factors. For example 36 = 2 × 2 × 3 × 3 Basic-Arithmetic 9 of 19 ©2014, www.learningforkowledge.com/glg Or = (2 × 3) × (2 × 3) (rearranging the factors as similar pairs) Or = 6 × 6 Hence 36 = 6 In a composite number, if the prime factors cannot be paired, then that number is not a perfect square. Division method: In this method the square root of any number can be found. Answers can be integers or decimals. For example, consider 99856 which is split as 9 98 56 and the following steps are carried out. Working: Step 1: The first digit (or pair if the number has even digits) is to be divided by a number whose square is less than or equal to 9. 1) 3 3 99856 9 0 Step 2: Drop the next pair 98 and for the divisor, add 3 to itself to give 6 which becomes the first digit of the divisor. The second digit is the smallest number in this case 1, because 61 × 1 is just less then 98. (Note: 61 × 2 = 122 > 98). As it is the case, use 61 to divide 98. Digit 1 is the quotient from this operation and 37 the remainder. 316 3 99856 9 61 098 61 626 3756 3756 0000 Step 3: Drop 56 down and 3756 is the number to be divided. Add the last digit of 61 to itself, that is, 61+1=62. Now 6 and 2 are the first two digits of the divisor. For the third digit select a number in this case 6, such that 626 x 6 ≤ 3756. Since 626 x 6 = 3756, there is no remainder. The digit 6 is included in the quotient. Now 316 is the square root of the number. Answer: 99856 = 316 Example 1: Find the square roots of the following numbers by inspection or referring to the table of squares. a) 36 b) 64 c) 144 Answers from table of squares: ∴ √36 = 6 62 = 36 Basic-Arithmetic 10 of 19 ©2014, www.learningforkowledge.com/glg 82 = 64 By inspection 122 = 12×12 = 144 ∴ √64 = 8 ∴ √144 = 8 Example 2: State if the square roots of the following numbers are odd or even a) 400 b) 729 c) 169 Answers: a) 400 is even, hence 400 is even ( 400 =20 ) b) 729 is odd, hence 729 is odd ( 729 =27 ) c) 169 is odd, hence 169 is odd ( 169 = 13 ) Example 3: Find the square root of 1089 using factorization method 3 1089 3 363 11 121 11 1089=3 × 3 × 11 × 11 1089 = √3 × 3 × 11 × 11 ∴ 1089 = 3 x 11 Answer: 1089 = 33 Example 4: Using the division method find the square root of 841 29 2 841 4 49 441 441 000 Answer: 841 =29 Example 5: Find the square root of 6.25 625 100 625 25 = 100 10 6.25 = ∴ √6.25 = Answer: √6.25 = 2.5 Basic-Arithmetic 11 of 19 ©2014, www.learningforkowledge.com/glg EXERCISE 1.2 1. Find by inspection, or from the table, the square roots of the following: a) 121 b) 625 c) 1225 2. If the unit digit of a number is 5, what will be the unit digit of its square root? 3. Using the progression method (1+3+……… (2m-1)) = m2 = n find the square roots of a) 49 b) 144 4. Using prime factorization method find the smallest number that divides 2700 to give a quotient that has an integer as the square root. 5. If the product of 16 and 49 is 784, find the square root of 784. 6. Area of a square is 49m2. (Area a = l2, where l is the length of side). Find the length of its side. 7. Find the square root of the following numbers using division method: a) 1849 b) 2209 c) 361 8. Find the square roots of following rational numbers: 49 17 a) b) 9.61 c) 1 36 64 9. Find the nearest perfect square to the following numbers: a) 1448 b) 126 c) 3481 (For example, 9 is the nearest perfect square to 10) 10. Find the square roots of the following numbers correct to two decimal places: a) 10 b) 2 c) 312 Answers to EXERCISE 1.2 1. a) 11 2. 5 3. a) 7 4. 3 5. 28 6. 7m 7. a) 43 Basic-Arithmetic b) 25 c) 35 b) 12 b) 47 c) 19 12 of 19 ©2014, www.learningforkowledge.com/glg 8. a) b) 3.1 9. a) 4 b) 5 10. a) 3.16 c) c) 59 b) 1.41 c) 17.66 CUBES If n = m³ = m × m × m then n is said to be the cube of m. In fact m3 is the volume of a cube of edge length m. The cube of a number can be simply obtained by multiplying the number by itself twice. For example 63 = (6 x 6) x 6 = 36 x 6 = 216 Properties 1: Cube of an even number is even and odd number is odd. 2: Cube of a negative number is negative and positive number is positive. 3: Cube of a number is the product of cubes of its factors. For example 63 = (2 x 3)3 = 23 x 33 = 8 x 27 = 216 Identity Method Cube of a two-digit number can be found by using the identity: (a+b)3 = a3 + 3a2b + 3ab2 + b3 For example: 313 Working: (31)3 = (30 +1)3 = 303 + 3 x 302 x 1 + 3 x 30 x 12 + 13 = 27000 + 2700 + 90 + 1 = 29791 Column Method For example: 563 Number 56 can be split as 50 + 6, t=50 and u=6 Working: t3 3 t2 u 3 t u2 u3 503 3×502×6 3×50×62 63 125000 45000 5400 216 Hence (56)3 = 125000 + 45000 + 5400 + 216 = 175616 Basic-Arithmetic 13 of 19 ©2014, www.learningforkowledge.com/glg Example 1: Write the unit digit of 2173 Unit digit of 2173 is the last digit of 7 × 7 × 7 = 343 Answer: 3 Example 2: Find 113 by multiplication 113 = 11 × 11 × 11 = (11 × 11) × 11 = 121 × 11 = 1331 Answer: 113 = 1331 Example 3: Find 423 using identity method. 423 = (40+2)2, a=40 and b=2 (a + b) 3 = a3 + 3a2b + 3ab2 + b3 (40 + 2)3 = 403 + 3 × 402 × 2 + 3 x 40 x 22 +23 = (40 × 40 × 40) + 3 × (40 × 40) × 2 + 3 × 40 × (2 × 2) + (2 × 2 × 2) = 64000 + 9600 + 480 + 8 Answer: 423 = 74088 Example 4: Find 683 using column method Here t=60, u=8 t3 3 t2 u 3 t u2 u3 603 3×602×8 3×60×82 83 216000 86400 11520 512 Hence (68)3 = 216000 + 86400 + 11520 + 512 = 314432 Answer: 682=314432 Basic-Arithmetic 14 of 19 ©2014, www.learningforkowledge.com/glg Exercise 1.3 1. Find the cubes of the following numbers by multiplication. a) 8 b) 13 c) 20 2. Find the cubes of the following numbers using the identity: (a+b) 3 =a 3 +3a 2 b + 3ab 2 + b 3 a) 101 b) 72 c) 23 3. Find the cubes of the following numbers using column method a) 31 b) 42 c) 53 4. A cube of ice has length of its edge = 8cms. Find the volume of the ice cube. 5. State true or false a) 343 is perfect cube b) Cube of a number with 2 as the unit digit is 6 c) Cube of a negative number is positive d) 73 > 63 + 13 6. Given 63 =216 and 33 = 27, find 183 7. 11 Find 5 3 8. Find the cube of the following numbers: a) -6 b) +5 and c) -30 9. Write the unit digits of the cube of the following numbers: a) 137137 b) 24 c) 398 10. 1729 is called Ramanujam number. Given 1729=123+b3 and 1729=c3+93 find b and c. Answers to EXERCISE 1.3 1. a) 512 2. a) 1030301 3. a) 29791 4. 512 cm3 5. a) true 6. 5832 Basic-Arithmetic b) 2197 c) 8000 b) 343248 b) 74088 b) false c) 12167 c) 148877 c) false d) true 15 of 19 ©2014, www.learningforkowledge.com/glg 7. 8. a) -216 b) 125 9. a) 3 10. b = 1, c = 10 b) 4 c) -27000 c) 2 CUBE ROOTS If n=m3 , then m = √ A perfect cube has an integer as its cube root. Properties 1: Cube root of an odd number is odd and even number is even 2: Cube root of a positive number is positive and negative number is negative 3: Cube root of the product of two numbers is equal to the product of the cube roots of the individual numbers. = √ × Inspection method Cube root can be found by inspection or by trial and error. Perfect cubes are only a few. These are only 4 perfect two digit cubes and10 perfect three digit cubes. A table of perfect cubes can be used for obtaining the cube root. Table of Cubes Number Cube 1 1 2 8 3 27 4 64 5 125 6 216 7 343 8 512 9 729 10 1000 Basic-Arithmetic 16 of 19 ©2014, www.learningforkowledge.com/glg Prime factorization method Any composite number can be factorized into a set of prime numbers. If the number is a perfect cube, it will contain 3 (or multiples of 3) sets of prime numbers. The example below illustrates this method. For example: √216 Factorize 216 2 216 2 108 2 54 3 27 3 9 3 Hence 216 = 2 × 2 × 2 × 3 × 3 × 3 √216 = 2 × 3 = 6 Answer: √216 = 6 Example1: Find the cube root of 125 Unit digit of 125 is 5, hence 5 is the possible cube root. (Note: This assumption may not be valid for all cubes). Verification: 5 × 5 × 5 = 125 Answer: √125 = 5 Example 2: Find √5832 by prime factorization 2 5832 2 2966 2 1458 3 729 3 243 3 81 3 27 3 9 3 Hence 5832 = 2 × 2 × 2 × 3 × 3 × 3 × 3 × 3 × 3 √5832 = 2 × 3 × 3 = 18 Answer: √5832 = 18 Basic-Arithmetic 17 of 19 ©2014, www.learningforkowledge.com/glg Example 3: Find √8 × 216 √8 × 216 = √8 × √216 = 2 × 6 = 12 Answer: √8 × 216 = 12 Example 4: Find the smallest number needed to divide 1024 to get an integer as the cube root. 1024 = 2 × 512 From the table by inspection, 512 is perfect cube and 8 is its cube root. Hence 2 is the smallest number needed to divide 1024. Answer: 2 Example 5: Find √−0.343 √−0.343 = √ √ −7 + 10 = - 0.7 = (from the table by inspection) Answer: - 0.7 EXERCISE 1.4 1. Find the cube roots of a) 8 b) 729 c) 1000 2. By prime factorization find the cube roots of a) 343 b) 9261 c) 21952 3. By prime factorization check if the following numbers are perfect cubes a) 64 b) 72 c) 140 4. Volume of a cube is 216 cm2. Find length of its edge 5. Find the cube root of following negative numbers a) -1728 b) -125 c) -3375 Basic-Arithmetic 18 of 19 ©2014, www.learningforkowledge.com/glg 6. If 63 = 216 and 83=512, find the cube root of 110592 7. Find the unit digits of the cube roots of following perfect cubes a) 12167 b) 8 c) 6859 8. Find the cube roots of following decimals a) 0.125 b) 0.008 c)-0.729 9. Find the cube roots of following Quotients 216 512 16 a) b) c) 343 1331 1024 10. Given a3 – b3 = (a-b) (a2 + ab + b2), find 43 - 33 Answers to EXERCISE 1.4 1. a) 2 b) 9 c) 10 2. a) 7 b) 21 c) 28 3. a) Yes 4. 6 cm 5. a) - 12 6. 48 7. a) 23 b) 2 c) 19 8. a) 0.5 b) 0.2 c) -0.9 9. a) 10. 37 Basic-Arithmetic b) No c) No b) - 5 b) c) - 15 c) or 19 of 19 ©2014, www.learningforkowledge.com/glg