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Knowing our Numbers
1.1 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.
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 :
 92, 392, 4456, 89742 (I am the greatest)
 1902, 1920, 9201, 9021, 9210 (I am the greatest)
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  Smallest: 18 Greatest: 59785 
2  1473, 89423, 100, 5000, 310  Smallest:Greatest: 
3  1834, 75284, 111, 2333, 450  Smallest:Greatest: 
4  2853, 7691, 9999, 12002, 124  Smallest:Greatest: 
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.
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.
Find the greatest and the smallest numbers.
(a)  4536, 4892, 4370, 4452.  Smallest:Greatest: 
(b)  15623, 15073, 15189, 15800.  Smallest:Greatest: 
(c)  25286, 25245, 25270, 25210.  Smallest:Greatest: 
(d)  6895, 23787, 24569, 24659.  Smallest:Greatest: 
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 such that no digits are repeated in a number. 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  Smallest:Greatest: 
(b)  9, 7, 4, 1  Smallest:Greatest: 
(c)  4, 7, 5, 0  Smallest:Greatest: 
(d)  1, 7, 6, 2  Smallest:Greatest: 
(e)  5, 4, 0, 3  Smallest:Greatest: 
(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  Smallest:Greatest: 
(b)  9, 0, 5  Smallest:Greatest: 
(c)  0, 4, 9  Smallest:Greatest: 
(d)  8, 5, 1  Smallest:Greatest: 
(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.
(Note, the number cannot begin with the digit 0. Why?)
(a)  Digit 7 is always at ones place  Smallest:
Greatest: 
(b)  Digit 4 is always at tens place  Smallest:
Greatest: 
(c)  Digit 9 is always at hundreds place  Smallest:
Greatest: 
(d)  Digit 1 is always at thousands place  Smallest:
Greatest: 
Stand in proper order
 Who is the tallest?
 Who is the shortest?
 Can you arrange them in the increasing order of their heights?
 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 ormore quantities.
Ascending order Ascending order means arrangement from the smallest to the greatest.
Descending order Descending order means arrangement from the greatest to the smallest.
1. Arrange the following numbers in ascending order :
 847, 9754, 8320, 571
 9801, 25751, 36501, 38802
2. Arrange the following numbers in descending order :
 5000, 7500, 85400, 7861
 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
Should not we then expect that on adding 1 to the greatest 4digit number, we would get the smallest 5digit number, that is
9999 + 1 = 10000.
The new number which comes next to 9999 is 10000. It is called ten thousand.
Further, we expect 10000 = 10 × 1000.
1.2.4 Revisiting place value
You have done this quite earlier, and you will certainly remember the expansion
of a 2digit number like 78 as
78 = 70 + 8 = 7 × 10 + 8
Similarly, you will remember the expansion of a 3digit 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 4digit 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 5digit 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 5digit numbers?
Read and expand the numbers wherever there are blanks.
Number  Number Name  Expansion 
20000  twenty thousand  2 × 10000 
26000  twenty six thousand  2 × 10000 + 6 × 1000 
38400  thirty eight thousand four hundred  3 × 10000 + 8 × 1000 + 4 × 100 
38400  thirty eight thousand four hundred  3 × 10000 + 8 × 1000 + 4 × 100 
65740  sixty five thousand seven hundred forty  6 × 10000 + 5 × 1000 + 7 × 100 + 4 × 10 
89324  eighty nine thousand three hundred twenty four  8 × 10000 + 9 × 1000 + 3 × 100 + 2 × 10 + 4 × 1 
50000  
41000  
47300  
57630  
29485  
29085  
20085  
20005 
Write five more 5digit numbers, read them and expand them.
1.2.5 Introducing 1,00,000
Which is the greatest 5digit number?
Adding 1 to the greatest 5digit number, should give the smallest 6digit 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 6digit 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.
Read and expand the numbers wherever there are blanks.
Number  Number Name  Expansion 
3,00,000  three lakh  3 × 1,00,000 
3,50,000  three lakh fifty thousand  3 × 1,00,000 + 5 × 10,000 
38400  thirty eight thousand four hundred  3 × 10000 + 8 × 1000 + 4 × 100 
3,53,500  three lakh fifty three thousand five hundred  3 × 1,00,000 + 5 × 10,000 + 3 × 1000 + 5 × 100 
4,57,928  
4,07,928  
4,00,829  
4,00,029 
1.2.6 Larger numbers
If we add one more to the greatest 6digit number we get the smallest 7digitnumber. It is called ten lakh.
Write down the greatest 6digit number and the smallest 7digit number. Write the greatest 7digit number and the smallest 8digit number. The smallest 8digit 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 
1.  What is 10 – 1 =?  
2.  What is 100 – 1 =?  
3.  What is 10,000 – 1 =?  
3.  What is 1,00,000 – 1 =?  
3.  What is 1,00,00,000 – 1 =? 
(Hint : Use the said pattern.)
We come across large numbers in many different situations. For example, while the number of children in your class would be a 2digit number, the number of children in your school would be a 3 or 4digit number. The number of people in the nearby town would be much larger. Is it a 5 or 6 or 7digit 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 5digit number, a 6digit number or more?
 Give five examples where the number of things counted would be more than 6digit number.
 Starting from the greatest 6digit number, write the previous five numbers in descending order.
 Starting from the smallest 8digit number, write the next five numbers inascending order and read them.
1.2.7 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 bywriting 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.
Number  TLakh  Lakh  TTh  Th  H  T  O  Number Name  Expansion 

7,34,543    7  3  4  5  4  3  Seven lakh thirty four thousand five hundred forty three   
32,75,829  3  2  7  5  8  2  9    3 × 10,00,000
+ 2 × 1,00,000 + 7 × 10,000 + 5 × 1000 + 8 × 100 + 2 × 10 + 9 
Similarly, we may include numbers upto crore as shown below :
Number  TCr  Cr  TLakh  Lakh  TTh  Th  H  T  O  Number Name 

7,34,543    7  3  4  5  4  3  Seven lakh thirty four thousand five hundred forty three     
32,75,829  3  2  7  5  8  2  9    3 × 10,00,000
+ 2 × 1,00,000 + 7 × 10,000 + 5 × 1000 + 8 × 100 + 2 × 10 + 9 
 
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 another two digits (seven digits from the right). It comes after ten lakh place and marks crore.
While writing number names, we do not use commas.
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. This may interest you : 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 during 19912001? Try to find out.
Do you know what is India’s population
today? Try to find this too.
1. Read these numbers. Write them using placement boxes and then write
their expanded forms.
(i) 475320 (ii) 9847215 (iii) 97645310 (iv) 30458094
 Which is the smallest number?
 Which is the greatest number?nd read them.
 Arrange these numbers in ascending and descending orders.
2. Read these numbers.
(i) 527864 (ii) 95432 (iii) 18950049 (iv) 70002509
 Write these numbers using placement boxes and then using commas in Indian as well as International System of Numeration..?
 Arrange these numbers in ascending and descending orders.
3. Take three more groups of large numbers and do the exercise given above
Can you help me write the numeral?
To write the numeral for a number you can follow the boxes again.
 Forty two lakh seventy thousand eight.
 Two crore ninety lakh fifty five thousand eight hundred.
 Seven crore sixty thousand fifty five.
1. You have the following digits 4, 5, 6, 0, 7 and 8. Using them, make five numbers each with 6 digits.
 Put commas for easy reading.
 Arrange them in ascending and descending orders.
2. Take the digits 4, 5, 6, 7, 8 and 9. Make any three numbers each with 8 digits. Put commas for easy reading.
3. From the digits 3, 0 and 4, make five numbers each with 6 digits. Use commas.
1.3 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).
 How many centimeters make a kilometer?
 Name five large cities in India. Find their population. Also, find the distance in kilometres between each pair of these cities.
(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 . 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 centimeters.
 How many milligrams make one kilogram?
 A box contains 2,00,000 medicine tablets each weighing 20 mg. What is the total weight of all the tablets in the box in grams and in kilograms?g ,kg

A bus started its journey and reached different places with a speed of 60 km/hour. The journey is shown below.
 Find the total distance covered by the bus from A to D.
 Find the total distance covered by the bus from D to G.
 Find the total distance covered by the bus, if it starts from A and returns back to A.
 Can you find the difference of distances from C to D and D to E?
 Find out the time taken by the bus to reach
 A to B
 C to D
 E to G
 Total journey

Raman’s shop
Things Price Apples Rs 40 per kg Oranges Rs 30 per kg Combs Rs 3 for one Tooth brushes Rs 10 for one Pencils Re 1 for one Note books Rs 6 for one Soap cakes Rs 8 for one The sales during the last year
Things Price Apples 2457 kg Oranges 3004 kg Combs 22760 Tooth brushes 25367 Pencils 38530 Note books 40002 Soap cakes 20005  Can you find the total weight of apples and oranges Raman sold last year?
Weight of apples = kg
Weight of oranges = kg
Therefore, total weight =kg +kg = kg
Answer – The total weight of oranges and apples = kg.  Can you find the total money Raman got by selling apples?
 Can you find the total money Raman got by selling apples and oranges together?
 Make a table showing how much money Raman received from selling each item. Arrange the entries of amount of money received in descending order. Find the item which brought him the highest amount. How much is this amount?
 Can you find the total weight of apples and oranges Raman sold last year?
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.
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?
opulation of the city in 2001
= Population of the city in 1991 + Increase in population
= 2,35,471 + 72,958
Salma added them by writing 235471 as 200000 + 35000 + 471 and 72958 as 72000 + 958. She got the addit
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?
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
In one state, the number of bicycles sold in the year 20022003 was 7,43,000. In the year 20032004, the number of bicycles sold was 8,00,100. In which year were more bicycles sold? and how many more?
Clearly, 8,00,100 is more than 7,43,000. So, in that state, more bicycles were sold in the year 20032004 than in 20022003.
Can you think of alternative ways of solving this problem?
Answer : 57,100 more bicycles were sold in the year 20032004.
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?
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.
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?
Each sheet makes 8 pages.
Hence, 75,000 sheets make 8 × 75,000 pages,
Now,
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.
1.3.1 Estimation
News
 India drew with Pakistan in a hockey match watched by 51,000 spectators in the stadium and 40 million television viewers world wide.
 Approximately, 2000 people were killed and more than 50000 injured in a cyclonic storm in coastal areas of India and Bangladesh.
 Over 13 million passengers are carried over 63,000 kilometre route of railway track every day.
Can we say that there were exactly as many people as the numbers quoted in these news items? For example,
In (1), were there exactly 51,000 spectators in the stadium? or did exactly 40 million viewers watched the match on television?  Find which flags are closer to 260.
 Find the flags which are closer to 270.
Locate the numbers 10,17 and 20 on your ruler. Is 17 nearer to 10 or 20?
The gap between 17 and 20 is smaller when compared to the gap between 17 and 10.  87 × 313
 9 × 795
 898 × 785
 958 × 387
 Write the expressions for each of the following using brackets.
 Four multiplied by the sum of nine and two.
 Divide the difference of eighteen and six by four.
 Forty five divided by three times the sum of three and two.
 Write three different situations for (5 + 8) × 6.
(One such situation is : Sohani and Reeta work for 6 days; Sohani works 5 hours a day and Reeta 8 hours a day. How many hours do both of them work in a week?)  Write five situations for the following where brackets would be necessary.
(a) 7(8 – 3) (b) (7 + 2) (10 – 3)  7 × 109 = 7 × (100 + 9) = 7 × 100 + 7 × 9 = 700 + 63 = 763
 102 × 103 = (100 + 2) × (100 + 3) = (100 + 2) × 100 + (100 + 2) × 3
= 100 × 100 + 2 × 100 + 100 × 3 + 2 × 3
= 10,000 + 200 + 300 + 6 = 10,000 + 500 + 6
= 10,506  17 × 109 = (10 + 7) × 109 = 10 × 109 + 7 × 109
= 10 × (100 + 9) + 7 × (100 + 9)
= 10 × 100 + 10 × 9 + 7 × 100 + 7 × 9
= 1000 + 90 + 700 + 63 = 1,790 + 63
= 1,853  If a symbol is repeated, its value is added as many times as it occurs:
i.e. II is equal 2, XX is 20 and XXX is 30.  A symbol is not repeated more than three times. But the symbols V, L and D are never repeated.
 If a symbol of smaller value is written to the right of a symbol of greater value, its value gets added to the value of greater symbol.
VI = 5 + 1 = 6, XII = 10 + 2 = 12
and LXV = 50 + 10 + 5 = 65  If a symbol of smaller value is written to the left of a symbol of greater value, its value is subtracted from the value of the greater symbol.
IV = 5 – 1 = 4, IX = 10 – 1 = 9
XL= 50 – 10 = 40, XC = 100 – 10 = 90  The symbols V, L and D are never written to the left of a symbol of greater value, i.e. V, L and D are never subtracted.
The symbol I can be subtracted from V and X only.
The symbol X can be subtracted from L, M and C only.  73
 32
 Given two numbers, one with more digits is the greater number. If the number of digits in two given numbers is the same, that number is larger, which has a greater leftmost digit. If this digit also happens to be the same, we look at the next digit and so on.
 In forming numbers from given digits, we should be careful to see if the conditions under which the numbers are to be formed are satisfied. Thus, to form the greatest four digit number from 7, 8, 3, 5 without repeating a single digit, we need to use all four digits, the greatest number can have only 8 as the leftmost digit.
 The smallest four digit number is 1000 (one thousand). It follows the largest three digit number 999. Similarly, the smallest five digit number is 10,000. It is ten thousand and follows the largest four digit number 9999.
Further, the smallest six digit number is 100,000. It is one lakh and follows the largest five digit number 99,999. This carries on for higher digit numbers in a similar manner.  Use of commas helps in reading and writing large numbers. In the Indian system of numeration we have commas after 3 digits starting from the right and thereafter every 2 digits. The commas after 3, 5 and 7 digits separate thousand, lakh and crore respectively. In the International system of numeration commas are placed after every 3 digits starting from the right. The commas after 3 and 6 digits separate thousand and million respectively.
 Large numbers are needed in many places in daily life. For example, for giving number of students in a school, number of people in a village or town, money paid or received in large transactions (paying and selling), in measuring large distances say between various cities in a country or in the world and so on.
 Remember kilo shows 1000 times larger, Centi shows 100 times smaller and milli shows 1000 times smaller, thus, 1 kilometer = 1000 metres, 1 meter = 100 centimeters or 1000 millimeters etc.
 There are a number of situations in which we do not need the exact quantity but need only a reasonable guess or an estimate. For example, while stating how many spectators watched a particular international hockey match, we state the approximate number, say 51,000, we do not need to state the exact number.
 Estimation involves approximating a quantity to an accuracy required. Thus, 4117 may be approximated to 4100 or to 4000, i.e. to the nearest hundred or to the nearest thousand depending on our need.
 In number of situations, we have to estimate the outcome of number operations. This is done by rounding off the numbers involved and getting a quick, rough answer.
 Estimating the outcome of number operations is useful in checking answers.
 Use of brackets allows us to avoid confusion in the problems where we need to carry out more than one number operation.
 We use the HinduArabic system of numerals. Another system of writing numerals is the Roman system.
Obviously, not. The word approximately itself shows that the number of people were near about these numbers. Clearly, 51,000 could be 50,800 or 51,300 but not 70,000. Similarly, 40 million implies much more than 39 million but quite less than 41 million but certainly not 50 million.
The quantities given in the examples above are not exact counts, but are estimates to give an idea of the quantity. Discuss what each of these can suggest.
Where do we approximate? Imagine a big celebration at your home. The first thing you do is to find out roughly how many guests may visit you. Can you get an idea of the exact number of visitors? It is practically impossible. The finance minister of the country presents a budget annually. The minister provides for certain amount under the head ‘Education’. Can the amount be absolutely accurate? It can only be a reasonably good estimate of the expenditure the country needs for education during the year.
Think about the situations where we need to have the exact numbers and compare them with situations where you can do with only an approximately estimated number. Give three examples of each of such situations.
1.3.2 Estimating to the nearest tens by rounding off
Look at the following :
So, we round off 17 as 20, correct to the nearest tens.
Now consider 12, which also lies between 10 and 20. However, 12 is closer to 10 than to 20. So, we round off 12 to 10, correct to the nearest tens.
How would you round off 76 to the nearest tens? Is it not 80?
We see that the numbers 1,2,3 and 4 are nearer to 0 than to 10. So, we round off 1, 2, 3 and 4 as 0. Number 6, 7, 8, 9 are nearer to 10, so, we round them off as 10. Number 5 is equidistant from both 0 and 10; it is a common practice to round it off as 10.
Round these numbers to the nearest tens.
28 32 52 41 39 48 64
59 99 215 1453 2936
1.3.3 Estimating to the nearest hundreds by rounding off
Is 410 nearer to 400 or to 500?
410 is closer to 400, so it is rounded off to 400, correct to the nearest hundred.
889 lies between 800 and 900.
It is nearer to 900, so it is rounded off as 900 correct to nearest hundred.
Numbers 1 to 49 are closer to 0 than to 100, and so are rounded off to 0.
Numbers 51 to 99 are closer to 100 than to 0, and so are rounded off to 100.
Number 50 is equidistant from 0 and 100 both. It is a common practice to round it off as 100.
Check if the following rounding off is correct or not :
841 → 800; 9537 → 9500; 49730 → 49700;
2546 → 2500; 286 → 200; 5750 → 5800;
168 → 200; 149 → 100; 9870 → 9800.
Correct those which are wrong.
1.3.4 Estimating to the nearest thousands by rounding off
We know that numbers 1 to 499 are nearer to 0 than to 1000, so these numbers are rounded off as 0.
The numbers 501 to 999 are nearer to 1000 than 0 so they are rounded off a 1000.
Number 500 is also rounded off as 1000.
Check if the following rounding off is correct or not :
2573 → 3000; 53552 → 53000;
6404 → 6000; 65437 → 65000;
7805 → 7000; 3499 → 4000.
Correct those which are wrong.
Round off the given numbers to the nearest tens, hundreds and thousands.
Given Number  Approximate to Nearest  Rounded Form 
75847  Tens  
75847  Hundreds  
75847  Thousands  
75847  Ten thousands 
1.3.5 Estimating outcomes of number situations
How do we add numbers? We add numbers by following the algorithm (i.e. the given method) systematically. We write the numbers taking care that the digits in the same place (ones, tens, hundreds etc.) are in the same column. For example,
3946 + 6579 + 2050 is written as —
Th  H  T  O  
3  9  4  6  
6  5  7  9  
+  2  0  5  0 
We add the column of ones and if necessary carry forward the appropriate number to the tens place as would be in this case. We then add the tens column and this goes on. Complete the rest of the sum yourself. This procedure takes time. There are many situations where we need to find answers more quickly. For example, when you go to a fair or the market, you find a variety of attractive things which you want to buy. You need to quickly decide what you can buy. So, you need to estimate the amount you need. It is the sum of the prices of things you want to buy.
A trader is to receive money from two sources. The money he is to receive is Rs 13,569 from one source and Rs 26,785 from another. He has to pay Rs 37,000 to someone else by the evening. He rounds off the numbers to their nearest thousands and quickly works out the rough answer. He is happy that he has enough money.
Do you think he would have enough money? Can you tell without doing the exact addition/subtraction?
Sheila and Mohan have to plan their monthly expenditure. They know their monthly expenses on transport, on school requirements, on groceries, on milk, and on clothes and also on other regular expenses. This month they have to go for visiting and buying gifts. They estimate the amount they would spend on all this and then add to see, if what they have, would be enough.
Would they round off to thousands as the trader did?
Think and discuss five more situations where we have to estimate sums or remainders.
Did we use rounding off to the same place in all these?
There are no rigid rules when you want to estimate the outcomes of numbers. The procedure depends on the degree of accuracy required and how quickly the estimate is needed. The most important thing is, how sensible the guessed answer would be.
1.3.6 To estimate sum or difference
As we have seen above we can round off a number to any place. The trader rounded off the amounts to the nearest thousands and was satisfied that he had enough. So, when you estimate any sum or difference, you should have an idea of why you need to round off and therefore the place to which you would round off. Look at the following examples.
Estimate: 5,290 + 17,986.
You find 17,986 > 5,290.
Round off to thousands.
17,986 is rounds off to 18,000
5,290 is rounds off to 5,000
there Estimated sum = 23,000
Does the method work? You may attempt to find the actual answer and verify if the estimate is reasonable.
Estimate: 5,673 – 436.
To begin with we round off to thousands. (Why?)
5,673 rounds off to 6,000
436 rounds off to 0
Estimated difference = 6,000
This is not a reasonable estimate. Why is this not reasonable?
To get a closer estimate, let us try rounding each number to hundreds.
5,673 rounds off to 5,700
436 rounds off to 400
Estimated difference = 5,300
This is a better and more meaningful estimate.
1.3.7 To estimate products
How do we estimate a product?
What is the estimate for 19 × 78?
It is obvious that the product is less than 2000. Why?
If we approximate 19 to the nearest tens, we get 20 and then approximate 78 to nearest tens, we get 80 and 20 × 80 = 1600
Look at 63 × 182
If we approximate both to the nearest hundreds we get 100 × 200 = 20,000. This is much larger than the actual product. So, what do we do? To get a more reasonable estimate, we try rounding off 63 to the nearest 10, i.e. 60, and also 182 to the nearest ten, i.e. 180. We get 60 × 180 or 10,800. This is a good estimate, but is not quick enough.
If we now try approximating 63 to 60 and 182 to the nearest hundred, i.e. 200, we get 60 × 200, and this number 12,000 is a quick as well as good estimate of the product.
The general rule that we can make is, therefore, Round off each factor to its greatest place, then multiply the rounded off factors. Thus, in the above example, we rounded off 63 to tens and 182 to hundreds.
Now, estimate 81 × 479 using this rule : 479 is rounded off to 500 (rounding off to hundreds), and 81 is rounded off to 80 (rounding off to tens).
The estimated product = 500 × 80 = 40,000 An important use of estimates for you will be to check your answers. Suppose, you have done the multiplication 37 × 1889, but are not sure about your answer. A quick and reasonable estimate of the product will be 40 × 2000 i.e. 80,000. If your answer is close to 80,000, it is probably right. On the other hand, if it is close to 8000 or 8,00,000, something is surely wrong in your multiplication.
Same general rule may be followed by addition and subtraction of two or more numbers.
Estimate the following products :
Make five more such problems and solve them.
1.4 Using Brackets
Suman bought 6 notebooks from the market and the cost was Rs 10 per notebook. Her sister Sama also bought 7 notebooks of the same type. Find the total money they paid.
Seema calculated the amount like this  Meera calculated the amount like this 
6 × 10 + 7 × 10
= 60 + 70 = 130 Ans. Rs 130 
6 + 7 =13
and 13 × 10 = 130 Ans. Rs 130 
You can see that Seema’s and Meera’s ways to get the answer are a bit different.
But both give the correct result. Why?
Seema says, what Meera has done is 7 + 6 × 10.
Appu points out that 7 + 6 × 10 = 7 + 60 = 67. Thus, this is not what Meera had done. All the three students are confused.
To avoid confusion in such cases we may use brackets. We can pack the numbers 6 and 7 together using a bracket, indicating that the pack is to be treated as a single number. Thus, the answer is found by (6 + 7) × 10 = 13 × 10.
This is what Meera did. She first added 6 and 7 and then multiplied the sum by 10.
This clearly tells us : First, turn everything inside the brackets ( ) into a single number and then do the operation outside which in this case is to multiply by 10.
1.4.1 Expanding brackets
Now, observe how use of brackets allows us to follow our procedure systematically. Do you think that it will be easy to keep a track of what steps we have to follow without using brackets?
1.5 Roman Numerals
We have been using the HinduArabic numeral system so far. This is not the only system available. One of the early systems of writing numerals is the system of Roman numerals. This system is still used in many places.
For example, we can see the use of Roman numerals in clocks; it is also used for classes in the school time table etc.
Find three other examples, where Roman numerals are used.
The Roman numerals
I, II, III, IV, V, VI, VII, VIII, IX, X
denote 1,2,3,4,5,6,7,8,9 and 10 respectively. This is followed by XI for 11, XII
I  V  X  L  C  D  M 
1  5  10  50  100  500  1000 
The rules for the system are :
1 = I  10 = X  100 = C 
2 = II  20 = XX  
3 = III  30 = XXX  
4 = IV  40 = XL  
5 = V  50 = L  
6 = VI  60 = LX  
7 = VII  70 = LXX  
8 = VIII  80 = LXXX  
9 = IX  90 = XC 
(a) Write in Roman numerals the missing numbers in the table.
(b)XXXX, VX, IC, XVV are not written. Can you tell why?
Write in Roman numerals.
Write in Roman Numerals (a) 69 (b) 98.
(a) 69 = 60 + 9  (b) 98 = 90 + 8 
= (50 + 10) + 9  = (100 – 10) + 8 
= LX + IX  = XC + VIII 
= LX IX  = XCVIII 