Chapter 1: Knowing Our Numbers

Publisher Update Release -> 23/8/2021  4:45 PM

Introduction

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 have added, subtracted, multiplied and divided them. In this chapter, we shall move forward on such interesting things with a bit of review and revision as well.  

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.
Ex:    5, 08, 01, 592
         3, 32, 40, 781
         7, 27, 05, 062

Suppose a newspaper report states that Rs.2500 crore has been allotted by the government for National Highway construction. The same amount of Rs.2500 crore is sometimes expressed as Rs. 25 billion. In the Indian system, we express it as Rs. 2500 crore and in the International system, the same number is expressed as 25 billion. Hence we need to understand both the systems and their relationship.

Ascending order 
    Ascending order means arrangement from the smallest to the greatest.

Descending order 
    Descending order means arrangement from the greatest to the smallest.

Ex:    Arrange the following numbers in ascending order :
        257536, 38952, 385081, 365062
Sol.    The smallest number is 38952. Other numbers greater than 38952, in order are 257536, 365062 and 385081.
    ∴    The numbers in ascending order are  38952, 257536, 365062, 385081.    

    Ex:    Arrange the following numbers in descending order :  19710, 887151, 453212, 925473
    Sol.     The greatest number is 925473. Other numbers smaller than 925473 in order are 887151, 453212 and 19710.
    ∴   The numbers in descending order are 925473, 887151, 453212, 19710.]

ROMAN NUMERALS
One of the earliest systems of writing numerals is the Roman Numeral system. This system is still in use in many places. For example, some faces of clocks show hours in Roman numerals; we use Roman numerals to write numbered list; etc.
Unlike the Hindu-Arabic numeral system, Roman numeral system uses seven basic symbols to represent different numbers. The symbols are as follows :

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  for 12,... till XX for 20. Some more Roman numerals are :
 I     V     X      L      C      D        M
 1    5     10    50   100   500    1000
 The rules for the system are :
 (a)    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.

 (b)    A symbol is not repeated more than three times. But the symbols V, L and D are never repeated.

 (c)    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

(d)    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

(e)  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.
     Following these rules we get,
     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

Ex :    Write in Roman Numerals 
         (i) 69        (ii) 98.
Sol:  (i) 69 = 60 + 9  = (50 + 10) + 9 = LX + IX  = LX IX 
         (ii) 98 = 90 + 8     = (100 – 10) + 8 = XC + VIII = XCVIII

USING BRACKETS
Meera bought 6 notebooks from the market and the cost was Rs 10 per notebook. Her sister Seema also bought 7 notebooks of the same type. Find the total money they paid.
Seema calculated the 
6 × 10 + 7 × 10 = 60 + 70 = 130 
Meera calculated the amount like this amount like this
6 + 7 =13  and 13 × 10 = 130
 Ans. Rs 130.

We can see that Seema’s and Meera’s ways to get the answer are a bit different. 
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.
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.

FACE VALUE
Face value of a digit in a numeral is the value of the digit itself at whatever place it may be. 

PLACE VALUE
Place value of a digit in a given number is the value of the digit because of the place or the position of the digit in the number.
Place value of a digit = Face value of the digit  × value of the place  

Place value and Face value :
Every digit has two values – the place value and the face value. The face value of a digit does not change while its place value changes according to its position and number. 

Expanded form of a Number :
If we express a given number as the sum of its place value, it is called its expanded form. 

Ex:    Express 
        (i) 3,64,029    (ii) 2,75,00,386 in expanded form. 
Sol.  Place value of 3 = 3 x 100000 
        Place value of 6 = 6 x 10000
        Place value of 4 = 4 x 1000
        Place value of 0 = 0 x 100
        Place value of 2 = 2 x 10 
        Place value of 9 = 9 x 1
 ∴    The expanded form of 3,64,029 is  
        3 x 100000 + 6 x 10000 + 4 x 1000 + 0 x 100 + 2 x 10 + 9 x 1.    

Large Numbers in Practice
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).

We have done a lot of problems that have addition, subtraction, multiplication and division. 
We will try solving some more here.

Ex :    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?
Sol.    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.    

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

Comparing Numbers

Take two numbers. The number with greater number of digits is greater. However, if the two numbers have the same number of digits, then the number which has a larger left most digits is larger. If this digit also happens to be the same, then we proceed to next digit and use the same criterion and so on.

    Ex:    Which is greater of 270346 and 48356?
    Sol.     270346 has 6 digits
              48356 has 5 digits    
             6 digits are more than 5 digits   
         ∴   270346 is greater than 48356
        or    270346 > 48356
        Greater number has more number of digits.    

    Ex:    Find the greatest and the smallest numbers from the following group of numbers :
        23787,6895, 24569, 24659
    Sol.    Greatest number : 24659
              Smallest number : 6895.    

(i)    Making number without repetition of digits : In case of non-repetition of digits, it is better if we start making the number from left.

    Ex:    Write the greatest and the smallest 5-digit numbers by using each of the digits 8, 4, 7, 0, 2 only once.
    Sol.    For the greatest number, we write the greatest digit 8 in the T-thousands column. Next smaller digit in the thousands column and so on.
   ∴    The greatest number = 87420.

For the smallest number, we write the smallest digit in the T-thousands column. But here 0 is the smallest digit. 0 is not written on the extreme left of a number. So, we write 2 in the T-thousands column and 0 in the thousands column. Next digit greater than 2 is written in the hundreds column and so on.

∴   The smallest number = 20478

(ii)    Making number with repetition of digits :  In case of repetition of digit, it is better if we start making number from right.

    Ex:    Write the greatest and smallest numbers of 4 digits using all the digits 8,0, 5.
    Sol     For greatest number, select the smallest digit 0 and write in the ones column. Next greater digit is written in the tens column. Next greater digits 8 is written in the hundreds column. Since no digit greater than 8 given, so we repeat 8 in the thousands column.

∴    The greatest number = 8850
    For smallest number, select the greatest digit 8 and write in the ones column. Next smaller digit in tens column and so on. Repeat the smallest digit in the end. But here 5 is smaller than 8 and then comes 0 which cannot be repeated in the end.
So, we write 5 in the end and repeat 0 in the tens place.

∴   The smallest number = 5008.

ESTIMATION IN NUMBER OPERATION
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.

(i)    Estimating (Rounding) to the Nearest Ten : To round off a number to the nearest ten consider the ones digit. If the ones digit is 5 or greater than 5, then change the ten’s digit to the next higher digit and ones digit to zero. If the ones digit is less than 5, then leave the tens digit unchanged but change the ones digit to zero.
(ii)    Estimating (Rounding) to the Nearest Hundred : To round off a number to the nearest hundred, consider the tens digit. If the ten’s digit is 5 or greater than 5, then change the hundreds digit to the next higher digit and tens, ones digits to zeros. If the tens digit is less than 5, then leave the hundreds digit unchanged but change the tens and ones digits to zeros.
(iii)    Estimating (Rounding) to the nearest Thousand : To round off a number to the nearest thousand, consider the hundreds digit. If this digit is 5 or greater than 5, then change the thousands digit to the next higher digit and change all the other digits before that to zeros. If the hundreds digit is less than 5, then leave the thousands digit unchanged but change all the other digits before that to zeros.

Ex :     Estimate: 5,290 + 17,986.
Sol.:    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
           Estimated sum         =         23,000.

Ex :      Estimate: 5,673 – 436.
Sol.:     5,673 rounds off to              5,700
           – 436 rounds off to               – 400
           Estimated difference     =     5,300.    

Ex :    Estimate the following products :
           (i) 87 × 313    (ii) 9 × 795     (iii) 898 × 785
Sol.    (i)     87 is rounded off to 90
                  313 is rounded off to 300
∴   Estimated product = 90 × 300 = 27000    

        (ii)     9 is not rounded off [  ∴ it is a one-digit no.
                795 is rounded off to 800
 ∴     Estimated product = 9 × 800 = 7200

        (iii)    898 is rounded  off to 900
                785 is rounded off to 800
 ∴   Estimated product = 900 × 800 = 720000.

(iv)    Estimation in Quotients : In the process of estimation in quotients, we round off the dividend and the divisor before the process of division.

    Ex:    Estimate the following quotients :
                (a) 81  ÷ 17    (b) 7525 ÷ 365
    Sol.     (a)  81 is rounded to 80
                     17 is rounded to 20
            To get the estimated quotient think of dividing 80  by 20 or 8 by 2.
       

        (b)    7525 is rounded to 8000
                365 is rounded to 400
        To get the estimated quotient think of dividing 80 by 4.
        ∴     Estimated quotient = 20.