Page 1 Counting things is easy for us now. We can count objects in large numbers, for example, the number of students in the school, and represent them through numerals. We can also communicate large numbers using suitable number names. It is not as if we always knew how to convey large quantities in conversation or through symbols. Many thousands years ago, people knew only small numbers. Gradually, they learnt how to handle larger numbers. They also learnt how to express large numbers in symbols. All this came through collective efforts of human beings. Their path was not easy, they struggled all along the way. In fact, the development of whole of Mathematics can be understood this way. As human beings progressed, there was greater need for development of Mathematics and as a result Mathematics grew further and faster. We use numbers and know many things about them. Numbers help us count concrete objects. They help us to say which collection of objects is bigger and arrange them in order e.g., first, second, etc. Numbers are used in many different contexts and in many ways. Think about various situations where we use numbers. List five distinct situations in which numbers are used. W e enjoyed working with numbers in our previous classes. W e have added, subtracted, multiplied and divided them. We also looked for patterns in number sequences and done many other interesting things with numbers. In this chapter, we shall move forward on such interesting things with a bit of review and revision as well. 1.1 Introduction Chapter 1 Chapter 1 Chapter 1 Chapter 1 Chapter 1 Knowing our Knowing our Knowing our Knowing our Knowing our Numbers Numbers Numbers Numbers Numbers 202021 Page 2 Counting things is easy for us now. We can count objects in large numbers, for example, the number of students in the school, and represent them through numerals. We can also communicate large numbers using suitable number names. It is not as if we always knew how to convey large quantities in conversation or through symbols. Many thousands years ago, people knew only small numbers. Gradually, they learnt how to handle larger numbers. They also learnt how to express large numbers in symbols. All this came through collective efforts of human beings. Their path was not easy, they struggled all along the way. In fact, the development of whole of Mathematics can be understood this way. As human beings progressed, there was greater need for development of Mathematics and as a result Mathematics grew further and faster. We use numbers and know many things about them. Numbers help us count concrete objects. They help us to say which collection of objects is bigger and arrange them in order e.g., first, second, etc. Numbers are used in many different contexts and in many ways. Think about various situations where we use numbers. List five distinct situations in which numbers are used. W e enjoyed working with numbers in our previous classes. W e have added, subtracted, multiplied and divided them. We also looked for patterns in number sequences and done many other interesting things with numbers. In this chapter, we shall move forward on such interesting things with a bit of review and revision as well. 1.1 Introduction Chapter 1 Chapter 1 Chapter 1 Chapter 1 Chapter 1 Knowing our Knowing our Knowing our Knowing our Knowing our Numbers Numbers Numbers Numbers Numbers 202021 MATHEMATICS 2 1.2 Comparing Numbers As we have done quite a lot of this earlier, let us see if we remember which is the greatest among these : (i) 92, 392, 4456, 89742 (ii) 1902, 1920, 9201, 9021, 9210 So, we know the answers. Discuss with your friends, how you find the number that is the greatest. Can you instantly find the greatest and the smallest numbers in each row? 1. 382, 4972, 18, 59785, 750. Ans. 59785 is the greatest and 18 is the smallest. 2. 1473, 89423, 100, 5000, 310. Ans. ____________________ 3. 1834, 75284, 111, 2333, 450 . Ans. ____________________ 4. 2853, 7691, 9999, 12002, 124. Ans. ____________________ Was that easy? Why was it easy? We just looked at the number of digits and found the answer. The greatest number has the most thousands and the smallest is only in hundreds or in tens. Make five more problems of this kind and give to your friends to solve. Now, how do we compare 4875 and 3542? This is also not very difficult.These two numbers have the same number of digits. They are both in thousands. But the digit at the thousands place in 4875 is greater than that in 3542. Therefore, 4875 is greater than 3542. Next tell which is greater, 4875 or 4542? Here too the numbers have the same number of digits. Further, the digits at the thousands place are same in both. What do we do then? We move to the next digit, that is to the digit at the hundreds place. The digit at the hundreds place is greater in 4875 than in 4542. Therefore, 4875 is greater than 4542. Find the greatest and the smallest numbers. (a) 4536, 4892, 4370, 4452. (b) 15623, 15073, 15189, 15800. (c) 25286, 25245, 25270, 25210. (d) 6895, 23787, 24569, 24659. 202021 Page 3 Counting things is easy for us now. We can count objects in large numbers, for example, the number of students in the school, and represent them through numerals. We can also communicate large numbers using suitable number names. It is not as if we always knew how to convey large quantities in conversation or through symbols. Many thousands years ago, people knew only small numbers. Gradually, they learnt how to handle larger numbers. They also learnt how to express large numbers in symbols. All this came through collective efforts of human beings. Their path was not easy, they struggled all along the way. In fact, the development of whole of Mathematics can be understood this way. As human beings progressed, there was greater need for development of Mathematics and as a result Mathematics grew further and faster. We use numbers and know many things about them. Numbers help us count concrete objects. They help us to say which collection of objects is bigger and arrange them in order e.g., first, second, etc. Numbers are used in many different contexts and in many ways. Think about various situations where we use numbers. List five distinct situations in which numbers are used. W e enjoyed working with numbers in our previous classes. W e have added, subtracted, multiplied and divided them. We also looked for patterns in number sequences and done many other interesting things with numbers. In this chapter, we shall move forward on such interesting things with a bit of review and revision as well. 1.1 Introduction Chapter 1 Chapter 1 Chapter 1 Chapter 1 Chapter 1 Knowing our Knowing our Knowing our Knowing our Knowing our Numbers Numbers Numbers Numbers Numbers 202021 MATHEMATICS 2 1.2 Comparing Numbers As we have done quite a lot of this earlier, let us see if we remember which is the greatest among these : (i) 92, 392, 4456, 89742 (ii) 1902, 1920, 9201, 9021, 9210 So, we know the answers. Discuss with your friends, how you find the number that is the greatest. Can you instantly find the greatest and the smallest numbers in each row? 1. 382, 4972, 18, 59785, 750. Ans. 59785 is the greatest and 18 is the smallest. 2. 1473, 89423, 100, 5000, 310. Ans. ____________________ 3. 1834, 75284, 111, 2333, 450 . Ans. ____________________ 4. 2853, 7691, 9999, 12002, 124. Ans. ____________________ Was that easy? Why was it easy? We just looked at the number of digits and found the answer. The greatest number has the most thousands and the smallest is only in hundreds or in tens. Make five more problems of this kind and give to your friends to solve. Now, how do we compare 4875 and 3542? This is also not very difficult.These two numbers have the same number of digits. They are both in thousands. But the digit at the thousands place in 4875 is greater than that in 3542. Therefore, 4875 is greater than 3542. Next tell which is greater, 4875 or 4542? Here too the numbers have the same number of digits. Further, the digits at the thousands place are same in both. What do we do then? We move to the next digit, that is to the digit at the hundreds place. The digit at the hundreds place is greater in 4875 than in 4542. Therefore, 4875 is greater than 4542. Find the greatest and the smallest numbers. (a) 4536, 4892, 4370, 4452. (b) 15623, 15073, 15189, 15800. (c) 25286, 25245, 25270, 25210. (d) 6895, 23787, 24569, 24659. 202021 KNOWING OUR NUMBERS 3 9 8 6 7 1 0 2 7 4 4 9 9 1 If the digits at hundreds place are also same in the two numbers, then what do we do? Compare 4875 and 4889 ; Also compare 4875 and 4879. 1.2.1 How many numbers can you make? Suppose, we have four digits 7, 8, 3, 5. Using these digits we want to make different 4digit numbers in such a way that no digit is repeated in them. Thus, 7835 is allowed, but 7735 is not. Make as many 4digit numbers as you can. Which is the greatest number you can get? Which is the smallest number? The greatest number is 8753 and the smallest is 3578. Think about the arrangement of the digits in both. Can you say how the largest number is formed? Write down your procedure. 1. Use the given digits without repetition and make the greatest and smallest 4digit numbers. (a) 2, 8, 7, 4 (b) 9, 7, 4, 1 (c) 4, 7, 5, 0 (d) 1, 7, 6, 2 (e) 5, 4, 0, 3 (Hint : 0754 is a 3digit number.) 2. Now make the greatest and the smallest 4digit numbers by using any one digit twice. (a) 3, 8, 7 (b) 9, 0, 5 (c) 0, 4, 9 (d) 8, 5, 1 (Hint : Think in each case which digit will you use twice.) 3. Make the greatest and the smallest 4digit numbers using any four different digits with conditions as given. (a) Digit 7 is always at Greatest ones place Smallest (Note, the number cannot begin with the digit 0. Why?) (b) Digit 4 is always Greatest at tens place Smallest (c) Digit 9 is always at Greatest hundreds place Smallest (d) Digit 1 is always at Greatest thousands place Smallest 1 202021 Page 4 Counting things is easy for us now. We can count objects in large numbers, for example, the number of students in the school, and represent them through numerals. We can also communicate large numbers using suitable number names. It is not as if we always knew how to convey large quantities in conversation or through symbols. Many thousands years ago, people knew only small numbers. Gradually, they learnt how to handle larger numbers. They also learnt how to express large numbers in symbols. All this came through collective efforts of human beings. Their path was not easy, they struggled all along the way. In fact, the development of whole of Mathematics can be understood this way. As human beings progressed, there was greater need for development of Mathematics and as a result Mathematics grew further and faster. We use numbers and know many things about them. Numbers help us count concrete objects. They help us to say which collection of objects is bigger and arrange them in order e.g., first, second, etc. Numbers are used in many different contexts and in many ways. Think about various situations where we use numbers. List five distinct situations in which numbers are used. W e enjoyed working with numbers in our previous classes. W e have added, subtracted, multiplied and divided them. We also looked for patterns in number sequences and done many other interesting things with numbers. In this chapter, we shall move forward on such interesting things with a bit of review and revision as well. 1.1 Introduction Chapter 1 Chapter 1 Chapter 1 Chapter 1 Chapter 1 Knowing our Knowing our Knowing our Knowing our Knowing our Numbers Numbers Numbers Numbers Numbers 202021 MATHEMATICS 2 1.2 Comparing Numbers As we have done quite a lot of this earlier, let us see if we remember which is the greatest among these : (i) 92, 392, 4456, 89742 (ii) 1902, 1920, 9201, 9021, 9210 So, we know the answers. Discuss with your friends, how you find the number that is the greatest. Can you instantly find the greatest and the smallest numbers in each row? 1. 382, 4972, 18, 59785, 750. Ans. 59785 is the greatest and 18 is the smallest. 2. 1473, 89423, 100, 5000, 310. Ans. ____________________ 3. 1834, 75284, 111, 2333, 450 . Ans. ____________________ 4. 2853, 7691, 9999, 12002, 124. Ans. ____________________ Was that easy? Why was it easy? We just looked at the number of digits and found the answer. The greatest number has the most thousands and the smallest is only in hundreds or in tens. Make five more problems of this kind and give to your friends to solve. Now, how do we compare 4875 and 3542? This is also not very difficult.These two numbers have the same number of digits. They are both in thousands. But the digit at the thousands place in 4875 is greater than that in 3542. Therefore, 4875 is greater than 3542. Next tell which is greater, 4875 or 4542? Here too the numbers have the same number of digits. Further, the digits at the thousands place are same in both. What do we do then? We move to the next digit, that is to the digit at the hundreds place. The digit at the hundreds place is greater in 4875 than in 4542. Therefore, 4875 is greater than 4542. Find the greatest and the smallest numbers. (a) 4536, 4892, 4370, 4452. (b) 15623, 15073, 15189, 15800. (c) 25286, 25245, 25270, 25210. (d) 6895, 23787, 24569, 24659. 202021 KNOWING OUR NUMBERS 3 9 8 6 7 1 0 2 7 4 4 9 9 1 If the digits at hundreds place are also same in the two numbers, then what do we do? Compare 4875 and 4889 ; Also compare 4875 and 4879. 1.2.1 How many numbers can you make? Suppose, we have four digits 7, 8, 3, 5. Using these digits we want to make different 4digit numbers in such a way that no digit is repeated in them. Thus, 7835 is allowed, but 7735 is not. Make as many 4digit numbers as you can. Which is the greatest number you can get? Which is the smallest number? The greatest number is 8753 and the smallest is 3578. Think about the arrangement of the digits in both. Can you say how the largest number is formed? Write down your procedure. 1. Use the given digits without repetition and make the greatest and smallest 4digit numbers. (a) 2, 8, 7, 4 (b) 9, 7, 4, 1 (c) 4, 7, 5, 0 (d) 1, 7, 6, 2 (e) 5, 4, 0, 3 (Hint : 0754 is a 3digit number.) 2. Now make the greatest and the smallest 4digit numbers by using any one digit twice. (a) 3, 8, 7 (b) 9, 0, 5 (c) 0, 4, 9 (d) 8, 5, 1 (Hint : Think in each case which digit will you use twice.) 3. Make the greatest and the smallest 4digit numbers using any four different digits with conditions as given. (a) Digit 7 is always at Greatest ones place Smallest (Note, the number cannot begin with the digit 0. Why?) (b) Digit 4 is always Greatest at tens place Smallest (c) Digit 9 is always at Greatest hundreds place Smallest (d) Digit 1 is always at Greatest thousands place Smallest 1 202021 MATHEMATICS 4 Ramhari (160 cm) Dolly (154 cm) Mohan (158 cm) Shashi (159 cm) ` 2635 ` 1897 ` 2854 ` 1788 ` 3975 4. Take two digits, say 2 and 3. Make 4digit numbers using both the digits equal number of times. Which is the greatest number? Which is the smallest number? How many different numbers can you make in all? Stand in proper order 1. Who is the tallest? 2. Who is the shortest? (a) Can you arrange them in the increasing order of their heights? (b) Can you arrange them in the decreasing order of their heights? Which to buy? Sohan and Rita went to buy an almirah. There were many almirahs available with their price tags. (a) Can you arrange their prices in increasing order? (b) Can you arrange their prices in decreasing order? Ascending order Ascending order means arrangement from the smallest to the greatest. Descending order Descending order means arrangement from the greatest to the smallest. Think of five more situations where you compare three or more quantities. 202021 Page 5 Counting things is easy for us now. We can count objects in large numbers, for example, the number of students in the school, and represent them through numerals. We can also communicate large numbers using suitable number names. It is not as if we always knew how to convey large quantities in conversation or through symbols. Many thousands years ago, people knew only small numbers. Gradually, they learnt how to handle larger numbers. They also learnt how to express large numbers in symbols. All this came through collective efforts of human beings. Their path was not easy, they struggled all along the way. In fact, the development of whole of Mathematics can be understood this way. As human beings progressed, there was greater need for development of Mathematics and as a result Mathematics grew further and faster. We use numbers and know many things about them. Numbers help us count concrete objects. They help us to say which collection of objects is bigger and arrange them in order e.g., first, second, etc. Numbers are used in many different contexts and in many ways. Think about various situations where we use numbers. List five distinct situations in which numbers are used. W e enjoyed working with numbers in our previous classes. W e have added, subtracted, multiplied and divided them. We also looked for patterns in number sequences and done many other interesting things with numbers. In this chapter, we shall move forward on such interesting things with a bit of review and revision as well. 1.1 Introduction Chapter 1 Chapter 1 Chapter 1 Chapter 1 Chapter 1 Knowing our Knowing our Knowing our Knowing our Knowing our Numbers Numbers Numbers Numbers Numbers 202021 MATHEMATICS 2 1.2 Comparing Numbers As we have done quite a lot of this earlier, let us see if we remember which is the greatest among these : (i) 92, 392, 4456, 89742 (ii) 1902, 1920, 9201, 9021, 9210 So, we know the answers. Discuss with your friends, how you find the number that is the greatest. Can you instantly find the greatest and the smallest numbers in each row? 1. 382, 4972, 18, 59785, 750. Ans. 59785 is the greatest and 18 is the smallest. 2. 1473, 89423, 100, 5000, 310. Ans. ____________________ 3. 1834, 75284, 111, 2333, 450 . Ans. ____________________ 4. 2853, 7691, 9999, 12002, 124. Ans. ____________________ Was that easy? Why was it easy? We just looked at the number of digits and found the answer. The greatest number has the most thousands and the smallest is only in hundreds or in tens. Make five more problems of this kind and give to your friends to solve. Now, how do we compare 4875 and 3542? This is also not very difficult.These two numbers have the same number of digits. They are both in thousands. But the digit at the thousands place in 4875 is greater than that in 3542. Therefore, 4875 is greater than 3542. Next tell which is greater, 4875 or 4542? Here too the numbers have the same number of digits. Further, the digits at the thousands place are same in both. What do we do then? We move to the next digit, that is to the digit at the hundreds place. The digit at the hundreds place is greater in 4875 than in 4542. Therefore, 4875 is greater than 4542. Find the greatest and the smallest numbers. (a) 4536, 4892, 4370, 4452. (b) 15623, 15073, 15189, 15800. (c) 25286, 25245, 25270, 25210. (d) 6895, 23787, 24569, 24659. 202021 KNOWING OUR NUMBERS 3 9 8 6 7 1 0 2 7 4 4 9 9 1 If the digits at hundreds place are also same in the two numbers, then what do we do? Compare 4875 and 4889 ; Also compare 4875 and 4879. 1.2.1 How many numbers can you make? Suppose, we have four digits 7, 8, 3, 5. Using these digits we want to make different 4digit numbers in such a way that no digit is repeated in them. Thus, 7835 is allowed, but 7735 is not. Make as many 4digit numbers as you can. Which is the greatest number you can get? Which is the smallest number? The greatest number is 8753 and the smallest is 3578. Think about the arrangement of the digits in both. Can you say how the largest number is formed? Write down your procedure. 1. Use the given digits without repetition and make the greatest and smallest 4digit numbers. (a) 2, 8, 7, 4 (b) 9, 7, 4, 1 (c) 4, 7, 5, 0 (d) 1, 7, 6, 2 (e) 5, 4, 0, 3 (Hint : 0754 is a 3digit number.) 2. Now make the greatest and the smallest 4digit numbers by using any one digit twice. (a) 3, 8, 7 (b) 9, 0, 5 (c) 0, 4, 9 (d) 8, 5, 1 (Hint : Think in each case which digit will you use twice.) 3. Make the greatest and the smallest 4digit numbers using any four different digits with conditions as given. (a) Digit 7 is always at Greatest ones place Smallest (Note, the number cannot begin with the digit 0. Why?) (b) Digit 4 is always Greatest at tens place Smallest (c) Digit 9 is always at Greatest hundreds place Smallest (d) Digit 1 is always at Greatest thousands place Smallest 1 202021 MATHEMATICS 4 Ramhari (160 cm) Dolly (154 cm) Mohan (158 cm) Shashi (159 cm) ` 2635 ` 1897 ` 2854 ` 1788 ` 3975 4. Take two digits, say 2 and 3. Make 4digit numbers using both the digits equal number of times. Which is the greatest number? Which is the smallest number? How many different numbers can you make in all? Stand in proper order 1. Who is the tallest? 2. Who is the shortest? (a) Can you arrange them in the increasing order of their heights? (b) Can you arrange them in the decreasing order of their heights? Which to buy? Sohan and Rita went to buy an almirah. There were many almirahs available with their price tags. (a) Can you arrange their prices in increasing order? (b) Can you arrange their prices in decreasing order? Ascending order Ascending order means arrangement from the smallest to the greatest. Descending order Descending order means arrangement from the greatest to the smallest. Think of five more situations where you compare three or more quantities. 202021 KNOWING OUR NUMBERS 5 1. Arrange the following numbers in ascending order : (a) 847, 9754, 8320, 571 (b) 9801, 25751, 36501, 38802 2. Arrange the following numbers in descending order : (a) 5000, 7500, 85400, 7861 (b) 1971, 45321, 88715, 92547 Make ten such examples of ascending/descending order and solve them. 1.2.2 Shifting digits Have you thought what fun it would be if the digits in a number could shift (move) from one place to the other? Think about what would happen to 182. It could become as large as 821 and as small as 128. Try this with 391 as well. Now think about this. Take any 3digit number and exchange the digit at the hundreds place with the digit at the ones place. (a) Is the new number greater than the former one? (b) Is the new number smaller than the former number? Write the numbers formed in both ascending and descending order. Before 7 9 5 Exchanging the 1st and the 3rd tiles. After 5 9 7 If you exchange the 1st and the 3rd tiles (i.e. digits), in which case does the number become greater? In which case does it become smaller? Try this with a 4digit number. 1.2.3 Introducing 10,000 We know that beyond 99 there is no 2digit number. 99 is the greatest 2digit number. Similarly, the greatest 3digit number is 999 and the greatest 4digit number is 9999. What shall we get if we add 1 to 9999? Look at the pattern : 9 + 1 = 10 = 10 × 1 99 + 1 = 100 = 10 × 10 999 + 1 = 1000 = 10 × 100 We observe that Greatest single digit number + 1 = smallest 2digit number Greatest 2digit number + 1 = smallest 3digit number Greatest 3digit number + 1 = smallest 4digit number 202021Read More
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