Page 1 SQUARES AND SQUARE ROOTS 89 6.1 Introduction You know that the area of a square = side × side (where ‘side’ means ‘the length of a side’). Study the following table. Side of a square (in cm) Area of the square (in cm 2 ) 1 1 × 1 = 1 = 1 2 2 2 × 2 = 4 = 2 2 3 3 × 3 = 9 = 3 2 5 5 × 5 = 25 = 5 2 8 8 × 8 = 64 = 8 2 a a × a = a 2 What is special about the numbers 4, 9, 25, 64 and other such numbers? Since, 4 can be expressed as 2 × 2 = 2 2 , 9 can be expressed as 3 × 3 = 3 2 , all such numbers can be expressed as the product of the number with itself. Such numbers like 1, 4, 9, 16, 25, ... are known as square numbers. In general, if a natural number m can be expressed as n 2 , where n is also a natural number, then m is a square number. Is 32 a square number? W e know that 5 2 = 25 and 6 2 = 36. If 32 is a square number, it must be the square of a natural number between 5 and 6. But there is no natural number between 5 and 6. Therefore 32 is not a square number. Consider the following numbers and their squares. Number Square 1 1 × 1 = 1 2 2 × 2 = 4 Squares and Square Roots CHAPTER 6 Page 2 SQUARES AND SQUARE ROOTS 89 6.1 Introduction You know that the area of a square = side × side (where ‘side’ means ‘the length of a side’). Study the following table. Side of a square (in cm) Area of the square (in cm 2 ) 1 1 × 1 = 1 = 1 2 2 2 × 2 = 4 = 2 2 3 3 × 3 = 9 = 3 2 5 5 × 5 = 25 = 5 2 8 8 × 8 = 64 = 8 2 a a × a = a 2 What is special about the numbers 4, 9, 25, 64 and other such numbers? Since, 4 can be expressed as 2 × 2 = 2 2 , 9 can be expressed as 3 × 3 = 3 2 , all such numbers can be expressed as the product of the number with itself. Such numbers like 1, 4, 9, 16, 25, ... are known as square numbers. In general, if a natural number m can be expressed as n 2 , where n is also a natural number, then m is a square number. Is 32 a square number? W e know that 5 2 = 25 and 6 2 = 36. If 32 is a square number, it must be the square of a natural number between 5 and 6. But there is no natural number between 5 and 6. Therefore 32 is not a square number. Consider the following numbers and their squares. Number Square 1 1 × 1 = 1 2 2 × 2 = 4 Squares and Square Roots CHAPTER 6 90 MATHEMATICS TRY THESE 3 3 × 3 = 9 4 4 × 4 = 16 5 5 × 5 = 25 6  7  8  9  10  From the above table, can we enlist the square numbers between 1 and 100? Are there any natural square numbers upto 100 left out? Y ou will find that the rest of the numbers are not square numbers. The numbers 1, 4, 9, 16 ... are square numbers. These numbers are also called perfect squares. 1. Find the perfect square numbers between (i) 30 and 40 (ii) 50 and 60 6.2 Properties of Square Numbers Following table shows the squares of numbers from 1 to 20. Number Square Number Square 1 1 11 121 2 4 12 144 3 9 13 169 4 16 14 196 5 25 15 225 6 36 16 256 7 49 17 289 8 64 18 324 9 81 19 361 10 100 20 400 Study the square numbers in the above table. What are the ending digits (that is, digits in the one’s place) of the square numbers? All these numbers end with 0, 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? Think about it. 1. Can we say whether the following numbers are perfect squares? How do we know? (i) 1057 (ii) 23453 (iii) 7928 (iv) 222222 (v) 1069 (vi) 2061 TRY THESE Can you complete it? Page 3 SQUARES AND SQUARE ROOTS 89 6.1 Introduction You know that the area of a square = side × side (where ‘side’ means ‘the length of a side’). Study the following table. Side of a square (in cm) Area of the square (in cm 2 ) 1 1 × 1 = 1 = 1 2 2 2 × 2 = 4 = 2 2 3 3 × 3 = 9 = 3 2 5 5 × 5 = 25 = 5 2 8 8 × 8 = 64 = 8 2 a a × a = a 2 What is special about the numbers 4, 9, 25, 64 and other such numbers? Since, 4 can be expressed as 2 × 2 = 2 2 , 9 can be expressed as 3 × 3 = 3 2 , all such numbers can be expressed as the product of the number with itself. Such numbers like 1, 4, 9, 16, 25, ... are known as square numbers. In general, if a natural number m can be expressed as n 2 , where n is also a natural number, then m is a square number. Is 32 a square number? W e know that 5 2 = 25 and 6 2 = 36. If 32 is a square number, it must be the square of a natural number between 5 and 6. But there is no natural number between 5 and 6. Therefore 32 is not a square number. Consider the following numbers and their squares. Number Square 1 1 × 1 = 1 2 2 × 2 = 4 Squares and Square Roots CHAPTER 6 90 MATHEMATICS TRY THESE 3 3 × 3 = 9 4 4 × 4 = 16 5 5 × 5 = 25 6  7  8  9  10  From the above table, can we enlist the square numbers between 1 and 100? Are there any natural square numbers upto 100 left out? Y ou will find that the rest of the numbers are not square numbers. The numbers 1, 4, 9, 16 ... are square numbers. These numbers are also called perfect squares. 1. Find the perfect square numbers between (i) 30 and 40 (ii) 50 and 60 6.2 Properties of Square Numbers Following table shows the squares of numbers from 1 to 20. Number Square Number Square 1 1 11 121 2 4 12 144 3 9 13 169 4 16 14 196 5 25 15 225 6 36 16 256 7 49 17 289 8 64 18 324 9 81 19 361 10 100 20 400 Study the square numbers in the above table. What are the ending digits (that is, digits in the one’s place) of the square numbers? All these numbers end with 0, 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? Think about it. 1. Can we say whether the following numbers are perfect squares? How do we know? (i) 1057 (ii) 23453 (iii) 7928 (iv) 222222 (v) 1069 (vi) 2061 TRY THESE Can you complete it? SQUARES AND SQUARE ROOTS 91 Write five numbers which you can decide by looking at their one’s digit that they are not square numbers. 2. Write five numbers which you cannot decide just by looking at their unit’s digit (or one’s place) whether they are square numbers or not. • Study the following table of some numbers and their squares and observe the one’s place in both. Table 1 Number Square Number Square Number Square 1 1 11 14 21 441 2 4 12 144 22 484 3 9 13 169 23 529 4 16 14 196 24 576 5 25 15 225 25 625 6 36 16 256 30 900 7 49 17 289 35 1225 8 64 18 324 40 1600 9 81 19 361 45 2025 10 100 20 400 50 2500 The following square numbers end with digit 1. Square Number 1 1 81 9 121 11 361 19 441 21 Write the next two square numbers which end in 1 and their corresponding numbers. Y ou will see that if a number has 1 or 9 in the unit’ s place, then it’ s square ends in 1. • Let us consider square numbers ending in 6. Square Number 16 4 36 6 196 14 256 16 TRY THESE Which of 123 2 , 77 2 , 82 2 , 161 2 , 109 2 would end with digit 1? TRY THESE Which of the following numbers would have digit 6 at unit place. (i) 19 2 (ii) 24 2 (iii) 26 2 (iv) 36 2 (v) 34 2 Page 4 SQUARES AND SQUARE ROOTS 89 6.1 Introduction You know that the area of a square = side × side (where ‘side’ means ‘the length of a side’). Study the following table. Side of a square (in cm) Area of the square (in cm 2 ) 1 1 × 1 = 1 = 1 2 2 2 × 2 = 4 = 2 2 3 3 × 3 = 9 = 3 2 5 5 × 5 = 25 = 5 2 8 8 × 8 = 64 = 8 2 a a × a = a 2 What is special about the numbers 4, 9, 25, 64 and other such numbers? Since, 4 can be expressed as 2 × 2 = 2 2 , 9 can be expressed as 3 × 3 = 3 2 , all such numbers can be expressed as the product of the number with itself. Such numbers like 1, 4, 9, 16, 25, ... are known as square numbers. In general, if a natural number m can be expressed as n 2 , where n is also a natural number, then m is a square number. Is 32 a square number? W e know that 5 2 = 25 and 6 2 = 36. If 32 is a square number, it must be the square of a natural number between 5 and 6. But there is no natural number between 5 and 6. Therefore 32 is not a square number. Consider the following numbers and their squares. Number Square 1 1 × 1 = 1 2 2 × 2 = 4 Squares and Square Roots CHAPTER 6 90 MATHEMATICS TRY THESE 3 3 × 3 = 9 4 4 × 4 = 16 5 5 × 5 = 25 6  7  8  9  10  From the above table, can we enlist the square numbers between 1 and 100? Are there any natural square numbers upto 100 left out? Y ou will find that the rest of the numbers are not square numbers. The numbers 1, 4, 9, 16 ... are square numbers. These numbers are also called perfect squares. 1. Find the perfect square numbers between (i) 30 and 40 (ii) 50 and 60 6.2 Properties of Square Numbers Following table shows the squares of numbers from 1 to 20. Number Square Number Square 1 1 11 121 2 4 12 144 3 9 13 169 4 16 14 196 5 25 15 225 6 36 16 256 7 49 17 289 8 64 18 324 9 81 19 361 10 100 20 400 Study the square numbers in the above table. What are the ending digits (that is, digits in the one’s place) of the square numbers? All these numbers end with 0, 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? Think about it. 1. Can we say whether the following numbers are perfect squares? How do we know? (i) 1057 (ii) 23453 (iii) 7928 (iv) 222222 (v) 1069 (vi) 2061 TRY THESE Can you complete it? SQUARES AND SQUARE ROOTS 91 Write five numbers which you can decide by looking at their one’s digit that they are not square numbers. 2. Write five numbers which you cannot decide just by looking at their unit’s digit (or one’s place) whether they are square numbers or not. • Study the following table of some numbers and their squares and observe the one’s place in both. Table 1 Number Square Number Square Number Square 1 1 11 14 21 441 2 4 12 144 22 484 3 9 13 169 23 529 4 16 14 196 24 576 5 25 15 225 25 625 6 36 16 256 30 900 7 49 17 289 35 1225 8 64 18 324 40 1600 9 81 19 361 45 2025 10 100 20 400 50 2500 The following square numbers end with digit 1. Square Number 1 1 81 9 121 11 361 19 441 21 Write the next two square numbers which end in 1 and their corresponding numbers. Y ou will see that if a number has 1 or 9 in the unit’ s place, then it’ s square ends in 1. • Let us consider square numbers ending in 6. Square Number 16 4 36 6 196 14 256 16 TRY THESE Which of 123 2 , 77 2 , 82 2 , 161 2 , 109 2 would end with digit 1? TRY THESE Which of the following numbers would have digit 6 at unit place. (i) 19 2 (ii) 24 2 (iii) 26 2 (iv) 36 2 (v) 34 2 92 MATHEMATICS TRY THESE TRY THESE We can see that when a square number ends in 6, the number whose square it is, will have either 4 or 6 in unit’ s place. Can you find more such rules by observing the numbers and their squares (T able 1)? What will be the “one’s digit” in the square of the following numbers? (i) 1234 (ii) 26387 (iii) 52698 (iv) 99880 (v) 21222 (vi) 9106 • Consider the following numbers and their squares. 10 2 = 100 20 2 = 400 80 2 = 6400 100 2 = 10000 200 2 = 40000 700 2 = 490000 900 2 = 810000 If a number contains 3 zeros at the end, how many zeros will its square have ? What do you notice about the number of zeros at the end of the number and the number of zeros at the end of its square? Can we say that square numbers can only have even number of zeros at the end? • See Table 1 with numbers and their squares. What can you say about the squares of even numbers and squares of odd numbers? 1. The square of which of the following numbers would be an odd number/an even number? Why? (i) 727 (ii) 158 (iii) 269 (iv) 1980 2. What will be the number of zeros in the square of the following numbers? (i) 60 (ii) 400 6.3 Some More Interesting Patterns 1. Adding triangular numbers. Do you remember triangular numbers (numbers whose dot patterns can be arranged as triangles)? * * * * * ** * ** * ** *** * *** * ** *** **** * **** 1 3 6 10 15 But we have four zeros But we have two zeros We have one zero We have two zeros Page 5 SQUARES AND SQUARE ROOTS 89 6.1 Introduction You know that the area of a square = side × side (where ‘side’ means ‘the length of a side’). Study the following table. Side of a square (in cm) Area of the square (in cm 2 ) 1 1 × 1 = 1 = 1 2 2 2 × 2 = 4 = 2 2 3 3 × 3 = 9 = 3 2 5 5 × 5 = 25 = 5 2 8 8 × 8 = 64 = 8 2 a a × a = a 2 What is special about the numbers 4, 9, 25, 64 and other such numbers? Since, 4 can be expressed as 2 × 2 = 2 2 , 9 can be expressed as 3 × 3 = 3 2 , all such numbers can be expressed as the product of the number with itself. Such numbers like 1, 4, 9, 16, 25, ... are known as square numbers. In general, if a natural number m can be expressed as n 2 , where n is also a natural number, then m is a square number. Is 32 a square number? W e know that 5 2 = 25 and 6 2 = 36. If 32 is a square number, it must be the square of a natural number between 5 and 6. But there is no natural number between 5 and 6. Therefore 32 is not a square number. Consider the following numbers and their squares. Number Square 1 1 × 1 = 1 2 2 × 2 = 4 Squares and Square Roots CHAPTER 6 90 MATHEMATICS TRY THESE 3 3 × 3 = 9 4 4 × 4 = 16 5 5 × 5 = 25 6  7  8  9  10  From the above table, can we enlist the square numbers between 1 and 100? Are there any natural square numbers upto 100 left out? Y ou will find that the rest of the numbers are not square numbers. The numbers 1, 4, 9, 16 ... are square numbers. These numbers are also called perfect squares. 1. Find the perfect square numbers between (i) 30 and 40 (ii) 50 and 60 6.2 Properties of Square Numbers Following table shows the squares of numbers from 1 to 20. Number Square Number Square 1 1 11 121 2 4 12 144 3 9 13 169 4 16 14 196 5 25 15 225 6 36 16 256 7 49 17 289 8 64 18 324 9 81 19 361 10 100 20 400 Study the square numbers in the above table. What are the ending digits (that is, digits in the one’s place) of the square numbers? All these numbers end with 0, 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? Think about it. 1. Can we say whether the following numbers are perfect squares? How do we know? (i) 1057 (ii) 23453 (iii) 7928 (iv) 222222 (v) 1069 (vi) 2061 TRY THESE Can you complete it? SQUARES AND SQUARE ROOTS 91 Write five numbers which you can decide by looking at their one’s digit that they are not square numbers. 2. Write five numbers which you cannot decide just by looking at their unit’s digit (or one’s place) whether they are square numbers or not. • Study the following table of some numbers and their squares and observe the one’s place in both. Table 1 Number Square Number Square Number Square 1 1 11 14 21 441 2 4 12 144 22 484 3 9 13 169 23 529 4 16 14 196 24 576 5 25 15 225 25 625 6 36 16 256 30 900 7 49 17 289 35 1225 8 64 18 324 40 1600 9 81 19 361 45 2025 10 100 20 400 50 2500 The following square numbers end with digit 1. Square Number 1 1 81 9 121 11 361 19 441 21 Write the next two square numbers which end in 1 and their corresponding numbers. Y ou will see that if a number has 1 or 9 in the unit’ s place, then it’ s square ends in 1. • Let us consider square numbers ending in 6. Square Number 16 4 36 6 196 14 256 16 TRY THESE Which of 123 2 , 77 2 , 82 2 , 161 2 , 109 2 would end with digit 1? TRY THESE Which of the following numbers would have digit 6 at unit place. (i) 19 2 (ii) 24 2 (iii) 26 2 (iv) 36 2 (v) 34 2 92 MATHEMATICS TRY THESE TRY THESE We can see that when a square number ends in 6, the number whose square it is, will have either 4 or 6 in unit’ s place. Can you find more such rules by observing the numbers and their squares (T able 1)? What will be the “one’s digit” in the square of the following numbers? (i) 1234 (ii) 26387 (iii) 52698 (iv) 99880 (v) 21222 (vi) 9106 • Consider the following numbers and their squares. 10 2 = 100 20 2 = 400 80 2 = 6400 100 2 = 10000 200 2 = 40000 700 2 = 490000 900 2 = 810000 If a number contains 3 zeros at the end, how many zeros will its square have ? What do you notice about the number of zeros at the end of the number and the number of zeros at the end of its square? Can we say that square numbers can only have even number of zeros at the end? • See Table 1 with numbers and their squares. What can you say about the squares of even numbers and squares of odd numbers? 1. The square of which of the following numbers would be an odd number/an even number? Why? (i) 727 (ii) 158 (iii) 269 (iv) 1980 2. What will be the number of zeros in the square of the following numbers? (i) 60 (ii) 400 6.3 Some More Interesting Patterns 1. Adding triangular numbers. Do you remember triangular numbers (numbers whose dot patterns can be arranged as triangles)? * * * * * ** * ** * ** *** * *** * ** *** **** * **** 1 3 6 10 15 But we have four zeros But we have two zeros We have one zero We have two zeros SQUARES AND SQUARE ROOTS 93 If we combine two consecutive triangular numbers, we get a square number, like 1 + 3 = 4 3 + 6 = 9 6 + 10 = 16 = 2 2 = 3 2 = 4 2 2. Numbers between square numbers Let us now see if we can find some interesting pattern between two consecutive square numbers. 1 (= 1 2 ) 2, 3, 4 (= 2 2 ) 5, 6, 7, 8, 9 (= 3 2 ) 10, 11, 12, 13, 14, 15, 16 (= 4 2 ) 17, 18, 19, 20, 21, 22, 23, 24, 25 (= 5 2 ) Between 1 2 (=1) and 2 2 (= 4) there are two (i.e., 2 × 1) non square numbers 2, 3. Between 2 2 (= 4) and 3 2 (= 9) there are four (i.e., 2 × 2) non square numbers 5, 6, 7, 8. Now, 3 2 = 9, 4 2 = 16 Therefore, 4 2 – 3 2 = 16 – 9 = 7 Between 9(=3 2 ) and 16(= 4 2 ) the numbers are 10, 11, 12, 13, 14, 15 that is, six nonsquare numbers which is 1 less than the difference of two squares. W e have 4 2 = 16 and 5 2 = 25 Therefore, 5 2 – 4 2 = 9 Between 16(= 4 2 ) and 25(= 5 2 ) the numbers are 17, 18, ... , 24 that is, eight non square numbers which is 1 less than the difference of two squares. Consider 7 2 and 6 2 . Can you say how many numbers are there between 6 2 and 7 2 ? If we think of any natural number n and (n + 1), then, (n + 1) 2 – n 2 = (n 2 + 2n + 1) – n 2 = 2n + 1. We find that between n 2 and (n + 1) 2 there are 2n numbers which is 1 less than the difference of two squares. Thus, in general we can say that there are 2n non perfect square numbers between the squares of the numbers n and (n + 1). Check for n = 5, n = 6 etc., and verify. Two non square numbers between the two square numbers 1 (=1 2 ) and 4(=2 2 ). 4 non square numbers between the two square numbers 4(=2 2 ) and 9(3 2 ). 8 non square numbers between the two square numbers 16(= 4 2 ) and 25(=5 2 ). 6 non square numbers between the two square numbers 9(=3 2 ) and 16(= 4 2 ).Read More
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