Knowing Our Numbers

Introduction

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.

        We enjoyed working with numbers in our previous classes. We 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.

Knowing Our Numbers

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.

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.

       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.

Knowing Our Numbers

How many numbers can you make?

Suppose, we have four digits 7, 8, 3, 5. Using these digits we want to make different 4-digit numbers in such a way that no digit is repeated in them. Thus, 7835 is allowed, but 7735 is not. Make as many 4-digit 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.

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.

Knowing Our Numbers

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 3-digit 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 4-digit number.

Knowing Our Numbers

Introducing 10,000

We know that beyond 99 there is no 2-digit number. 99 is the greatest 2-digit number. Similarly, the greatest 3-digit number is 999 and the greatest 4-digit 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 2-digit number

Greatest 2-digit number + 1 = smallest 3-digit number

Greatest 3-digit number + 1 = smallest 4-digit number

          We should then expect that on adding 1 to the greatest 4-digit number, we would get the smallest 5-digit number, that is 9999 + 1 = 10000.

          The new number which comes next to 9999 is 10000. It is called ten thousand. Further, 10000 = 10 × 1000.

Knowing Our Numbers

Revisiting place value

You have done this quite earlier, and you will certainly remember the expansion of a 2-digit number like 78 as

      78 = 70 + 8 = 7 × 10 + 8

      Similarly, you will remember the expansion of a 3-digit number like 278 as

      278 = 200 + 70 + 8 = 2 × 100 + 7 × 10 + 8

      We say, here, 8 is at ones place, 7 is at tens place and 2 at hundreds place.

       Later on we extended this idea to 4-digit numbers.

      For example, the expansion of 5278 is

       5278 = 5000 + 200 + 70 + 8 = 5 × 1000 + 2 × 100 + 7 × 10 + 8

       Here, 8 is at ones place, 7 is at tens place, 2 is at hundreds place and 5 is at thousands place.

       With the number 10000 known to us, we may extend the idea further. We may write 5-digit numbers like

45278 = 4 × 10000 + 5 × 1000 + 2 × 100 + 7 × 10 + 8

       We say that here 8 is at ones place, 7 at tens place, 2 at hundreds place, 5 at thousands place and 4 at ten thousands place. The number is read as forty five thousand, two hundred seventy eight. Can you now write the smallest and the greatest 5-digit numbers?

Knowing Our Numbers

Introducing 1,00,000

Which is the greatest 5-digit number?

         Adding 1 to the greatest 5-digit number, should give the smallest 6-digit number : 99,999 + 1 = 1,00,000   

         This number is named one lakh. One lakh comes next to 99,999.

        10 × 10,000 = 1,00,000

        We may now write 6-digit numbers in the expanded form as

        2,46,853 = 2 × 1,00,000 + 4 × 10,000 + 6 × 1,000 + 8 × 100 + 5 × 10 +3 × 1

        This number has 3 at ones place, 5 at tens place, 8 at hundreds place, 6 at thousands place, 4 at ten thousands place and 2 at lakh place. Its number name is two lakh forty six thousand eight hundred fifty three.

Knowing Our Numbers

Larger numbers

          If we add one more to the greatest 6-digit number we get the smallest 7-digit number. It is called ten lakh. Write down the greatest 6-digit number and the smallest 7-digit number.

         Write the greatest 7-digit number and the smallest 8-digit number. The smallest 8-digit number is called one crore.

Complete the pattern : 9 + 1 = 10

99 + 1 = 100

999 + 1 = _______

9,999 + 1 = _______

99,999 + 1 = _______

9,99,999 + 1 = _______

99,99,999 + 1 = 1,00,00,000

Remember

1 hundred = 10 tens

1 thousand = 10 hundreds

                 = 100 tens

1 lakh = 100 thousands

         = 1000 hundreds

1 crore = 100 lakhs

           = 10,000 thousands

We come across large numbers in many different situations. For example, while the number of children in your class would be a 2-digit number, the number of children in your school would be a 3 or 4-digit number.

The number of people in the nearby town would be much larger.

Is it a 5 or 6 or 7-digit number?

Do you know the number of people in your state?

How many digits would that number have?

What would be the number of grains in a sack full of wheat? A 5-digit number, a 6-digit number or more?

Knowing Our Numbers

An aid in reading and writing large numbers

Try reading the following numbers :

(a) 279453      (b) 5035472

(c) 152700375 (d) 40350894

Was it difficult?

Did you find it difficult to keep track?

Sometimes it helps to use indicators to read and write large numbers. Shagufta uses indicators which help her to read and write large numbers. Her indicators are also useful in writing the expansion of numbers. For example, she identifies the digits in ones place, tens place and hundreds place in 257 by writing them under the tables O, T and H as

H  T  O            Expansion

2  5  7             2 × 100 + 5 × 10 + 7 × 1

Similarly, for 2902,

Th H T O       Expansion

2   9 0  2        2 × 1000 + 9 × 100 + 0 × 10 + 2 × 1

One can extend this idea to numbers upto lakh as seen in the following table. (Let us call them placement boxes). Fill the entries in the blanks left.

Similarly, we may include numbers upto crore as shown below :

You can make other formats of tables for writing the numbers in expanded form.

Use of commas

You must have noticed that in writing large numbers in the sections above, we have often used commas. Commas help us in reading and writing large numbers. In our Indian System of Numeration we use ones, tens, hundreds, thousands and then lakhs and crores. Commas are used to mark thousands, lakhs and crores. The first comma comes after hundreds place (three digits from the right) and marks thousands. The second comma comes two digits later (five digits from the right). It comes after ten thousands place and marks lakh. The third comma comes after anothertwodigits(seven digitsfrom the right).Itcomes aftertenlakh place and marks crore.

For example, 5, 08, 01, 592

                   3, 32, 40, 781

                   7, 27, 05, 062

    Try reading the numbers given above. Write five more numbers in this form and read them.

International System of Numeration

In the International System of Numeration, as it is being used we have ones, tens, hundreds, thousands and then millions. One million is a thousand thousands. Commas are used to mark thousands and millions. It comes after every three digits from the right. The first comma marks thousands and the next comma marks millions. For example, the number 50,801,592 is read in the International System as fifty million eight hundred one thousand five hundred ninety two. In the Indian System, it is five crore eight lakh one thousand five hundred ninety two.

How many lakhs make a million?

How many millions make a crore?

Take three large numbers. Express them in both Indian and International Numeration systems. Interesting fact :

To express numbers larger than a million, a billion is used in the International System of Numeration: 1 billion = 1000 million.

How much was the increase in population during 1991-2001? Try to find out.

Do you know what is India’s population today? Try to find this too.

Do you know?

India’s population increased by about

27 million during 1921-1931;

37 million during 1931-1941;

44 million during 1941-1951;

78 million during 1951-1961!

Can you help me write the numeral?

To write the numeral for a number you can follow the boxes again.

(a) Forty two lakh seventy thousand eight.

(b) Two crore ninety lakh fifty five thousand eight hundred.

(c) Seven crore sixty thousand fifty five.

Knowing Our Numbers

Large Numbers in Practice

In earlier classes, we have learnt that we use centimetre (cm) as a unit of length. For measuring the length of a pencil, the width of a book or notebooks etc., we use centimetres. Our ruler has marks on each centimetre. For measuring the thickness of a pencil, however, we find centimetre too big. We use millimetre (mm) to show the thickness of a pencil.

(a) 10 millimetres = 1 centimetre To measure the length of the classroom or the school building, we shall find centimetre too small. We use metre for the purpose.

(b) 1 metre = 100 centimetres

                  = 1000 millimetres

Even metre is too small, when we have to state distances between cities, say, Delhi and Mumbai, or Chennai and Kolkata. For this we need kilometres (km).

(c) 1 kilometre = 1000 metres

How many millimetres make 1 kilometre?

Since 1 m = 1000 mm

1 km = 1000 m = 1000 × 1000 mm = 10,00,000 mm

We go to the market to buy rice or wheat; we buy it in kilograms (kg). But items like ginger or chillies which we do not need in large quantities, we buy in grams (g). We know 1 kilogram = 1000 grams.

Have you noticed the weight of the medicine tablets given to the sick? It is very small. It is in milligrams (mg).

1 gram = 1000 milligrams.

What is the capacity of a bucket for holding water? It is usually 20 litres (l). Capacity is given in litres. But sometimes we need a smaller unit, the millilitres. A bottle of hair oil, a cleaning liquid or a soft drink have labels which give the quantity of liquid inside in millilitres (ml).

1 litre = 1000 millilitres.

Note that in all these units we have some words common like kilo, milli and centi. You should remember that among these kilo is the greatest and milli is the smallest; kilo shows 1000 times greater, milli shows 1000 times smaller, i.e. 1 kilogram = 1000 grams, 1 gram = 1000 milligrams.

Similarly, centi shows 100 times smaller, i.e. 1 metre = 100 centimetres

         We have done a lot of problems that have addition, subtraction, multiplication and division. We will try solving some more here. Before starting, look at these examples and follow the methods used.

Example  : Population of Sundarnagar was 2,35,471 in the year 1991. In the year 2001 it was found to be increased by 72,958. What was the population of the city in 2001?

Solution : Population of the city in 2001

= Population of the city in 1991 + Increase in population

= 2,35,471 + 72,958

Now, 235471

     + 72958

    =  308429

         Salma added them by writing 235471 as 200000 + 35000 + 471 and 72958 as 72000 + 958. She got the addition as 200000 + 107000 + 1429 = 308429. Mary added it as 200000 + 35000 + 400 + 71 + 72000 + 900 + 58 = 308429

Answer : Population of the city in 2001 was 3,08,429.

All three methods are correct.

Example 2 : In one state, the number of bicycles sold in the year 2002-2003 was 7,43,000. In the year 2003-2004, the number of bicycles sold was 8,00,100. In which year were more bicycles sold? and how many more?

Solution : Clearly, 8,00,100 is more than 7,43,000. So, in that state, more bicycles were sold in the year 2003-2004 than in 2002-2003.

Now, 800100

      – 743000

      =057100

Check the answer by adding

743000

+ 57100

800100 (the answer is right)

Can you think of alternative ways of solving this problem?

Answer : 57,100 more bicycles were sold in the year 2003-2004.

Example 3 : The town newspaper is published every day. One copy has 12 pages. Everyday 11,980 copies are printed. How many total pages are printed everyday?

Solution : Each copy has 12 pages. Hence, 11,980 copies will have 12 × 11,980 pages. What would this number be? More than 1,00,000 or lesser. Try to estimate.

Now,

Answer:Everyday 1,43,760 pages are printed.

Example 4 : The number of sheets of paper available for making notebooks is 75,000. Each sheet makes 8 pages of a notebook. Each notebook contains 200 pages. How many notebooks can be made from the paper available?

Solution : Each sheet makes 8 pages. Hence, 75,000 sheets make 8 × 75,000 pages,

Now, 75000

         × 8

     =600000

Thus, 6,00,000 pages are available for making notebooks.

Now, 200 pages make 1 notebook.

Hence, 6,00,000 pages make 6,00,000 ÷ 200 notebooks.

Now,

The answer is 3,000 notebooks.