Our Numbers

1.0Introduction to Numbers

Numbers play an important role in our life. We use numbers in our day to day life to count things.

While counting we use numbers to represent any quantity, to measure any distance or length.

The counting numbers starting from 1,2,3,4,5 are termed as natural numbers.

The set of counting numbers and zero are known as whole numbers. Whole numbers are 0,1,2,3,4,5,6,7,
and so, on

The symbols used by different civilizations to represent numbers are as below:

Symbols One Two Three Four Five Six Seven Eight Nine Ten

Indo-
1 2 3 4 5 6 7 8 9 10
Arabic

Dev Nagri १ २ ३ ४ ५ ६ ७ ८ ९ १०

Arabs ١ ٢ ٣ ٤ ٥ ٦ ٧ ٨ ٩ ١٠

Roman I II III IV V VI VII VIII IX X

Even Natural Numbers: Numbers which are divisible by 2 are called even numbers, e.g. 2,4,6,8,10, ....

Odd Natural Numbers: Numbers which are not divisible by 2 are called odd numbers, e.g. 1,3,5,7,9,….

If at unit place, we have 0,2,4,6,8 then number is even otherwise it is odd.

2.0Comparing and building numbers

To put large numbers in order, you must check the number of digits in them first. If the number of digits
vary in each number, the smallest number is the one which is having the least number of digits and the
greatest number is the one which is having the maximum number of digits.

Comparing numbers with the same number of digits

Comparison of the numbers with the same number of digits starts from the left-hand side. You must
compare the face values of the digits having the same place value in the numbers until you come across
unequal digits.

 0 is the smallest whole number and largest whole number cannot be define because whole
number goes to infinite. Building numbers Now, you will learn to build numbers, under different
conditions.

 Greatest 4-digit number is 9999 and greatest 5-digit number is 99999 .

3.0Introducing 6-digit, 7-digit and 8-digit numbers

Till now you have learnt up to 5 digit numbers and you know that the greatest 5 digit number is 99,999 .
On adding 1 to it, we get the smallest 6-digit number. 99999+1=1,00,000, read as one lakh. The largest 6-
digit number is 9,99,999. On adding 1 to it, we get the smallest 7 -digit number. 9,99,999+1=10,00,000,
read as ten lakh. The largest 7 -digit number is 99,99,999. On adding 1 to it, we get the smallest 8 -digit
number. 99,99,999+1=1,00,00,000, read as one crore.

Ascending Order

When the numbers are arranged from the smallest to the largest number, those numbers are said to be
in an ascending order. The numbers are arranged from left to right in increasing order.

Descending Order

When the numbers are arranged from the largest to the smallest number, those numbers are said to be
in descending order. The numbers are arranged from left to right in decreasing order. Ascending order is
represented by < (less than) symbol, whereas descending order is represented by > (greater than)
symbol.

Shifting digits

Changing the position of digits in a number, changes magnitude of the number. Example: Take a number
257. The condition here is to exchange its hundreds and unit digit and form the new number. That is,
exchange 2 to 7 and 7 to 2 .

Here comes a question. Which is greater and which is least among the numbers? To find that express the
numbers formed in both ascending and descending order. The number before shifting is 257 .
Exchanging the hundreds and the unit digits, the number after shifting is 752 . That is, if we exchange the
hundreds and unit digit, the resultant number becomes greater.

4.0Place 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. The face value of a digit in a numeral
is its own value, at whatever place it may be. Place value or local value of a digit in a given number is the
value of the digit because of the place or position of the digit in the number.

Expanded form of a Number

If we express a given number as the sum of its place values, it is called its expanded form.

5.0Indian and International system of numeration

Suppose a newspaper report state 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 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, you need to understand both the systems and their relationship.

Indian system of numeration

The Indian system of numeration or Hindu-Arabic numeral system is a positional decimal numeral system
developed between the 1st and 5th centuries by Indian mathematicians, adopted by Persian and Arabian
mathematicians and spread to the western world by the High Middle Ages. It uses ten basic symbols
0,1,2,3,4,5, 6,7,8,9 (called digits) and the idea of place value. For a given numeral, we start from the
extreme right as : Ones, Tens, Hundreds, Thousands, Ten Thousands, Lakhs, Ten Lakhs, etc. Each place
represents ten times the one which is immediately to its right. Indian system of numbers

Crores Lakhs Thousands Ones

Ten One One

Ten Crore One Crore Ten Lakh One Lakh Ten One
Thousand Thousand Hundred

₹10,00,00,000 ₹1,00,00,000 ₹10,00,000 ₹1,00,000 ₹10,000 5 ₹1,000 4 ₹100 3 ₹10 2 ₹1 1

9 Digits 8 Digits 7 Digits 6 Digits Digits Digits Digits Digits Digit

Indian place-value chart

Crores Lakhs Thousands Ones

TC C TL L TTh Th H T 0

1 0 0 0 0 0 0 0

1C=1 crore =1,00,00,000

International system of numeration

International system of numeration is adopted by all the countries throughout the world. International
system of numbers

Millions Thousands Ones

Hundred Hundred Ten One One

Ten Million One Million Ten One
Million Thousand Thousand Thousand Hundred

100,000,000 10,000,000 1,000,000 100,000 10,000 1,000 100 10 1

9 Digits 8 Digits 7 Digits 6 Digits 5 Digits 4 Digits 3 Digits 2 Digits 1 Digit

International place-value chart

Millions Thousands Ones

HM TM M HTh TTh Th H T 0

1 0 0 0 0 0 0 0

1TM=10 million =10,000,000

6.0Use of commas

Commas help us in reading and writing large numbers. In our Indian system of numeration, commas are
used to mark thousands, lakhs and crores. The first comma comes after hundreds place and marks
thousands. The second comma comes after ten thousands place and marks lakh. The third comma
comes after ten lakh place and marks crore.

In International system of numeration, commas are used to mark thousands and millions. It comes after
every three digits from the right.

Express 643871 in both the systems of numeration. Explanation Indian: 6,43,871 Six lakh forty three
thousand eight hundred and seventy one

 Roman numbers were invented for the purpose of counting and performing other day-today
transactions.

7.0Roman Numerals

The Roman numerals is the numeral system of ancient Rome. It uses combinations of letters from the
Latin alphabet to signify values. The numbers 1 to 10 can be expressed in Roman numerals as follows:

I, II, III, IV, V, VI, VII, VIII, IX, and X.

This 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 Roman numeral system is decimal but not directly positional and does not include a zero.

Rules to form Roman numerals

We can form different roman numerals using the symbols and the following rules.

Rule 1: If a symbol is repeated one after the other, its value is added as many times as it occurs. For
example, III=1+1+1=3 XX=10+10=20

Rule 2 : The symbols I, X, C and M can be repeated up to a maximum of three times. For example,
I=1,II=2,III=3 X=10,XX=20,XXX=30 C=100,CC=200,CCC=300 M=1000,MM=2000,MMM=3000 Rule 3: The
symbols V, L and D (i.e., 5, 50, and 500, respectively) can never be repeated in a roman numeral. Rule 4:
If a symbol with a smaller value is written on the right of a symbol with a greater value, then its value is
added to the value of the greater symbol. For example, XII=10+2=12,LX=50+10=60, DCCCX
=500+300+10=810 Rule 5 : If a symbol with a smaller value is written on the left of a symbol with a larger
value, then its value is subtracted from the value of the greater symbol. For example,
IV=5−1=4,IX=10−1=9,CD=500−100=400 VI=5+1=6,XI=10+1=11,DC=500+100=600 Note: 'I' can be
subtracted from V and X once only. X can be subtracted from L and C once only. C can be subtracted from
D and M once only. Thus, I or V is never written to the left of L or C.L is never written to the left of C.

 Zero is the only number that can't be represented in Roman numberals.

 VC is not possible because V,L & D are never subtracted.

8.0Use of brackets
Raju brought 6 pencils from the market, each at Rs. 2. His brother Ramu also bought 8 pencils of the
same type. Raju and Ramu both calculated the total cost but in their own ways. Raju found that they
both spent Rs. 28 and he used the following method: (6×2)+(8×2) =(12+16) =28 Here number of
operations are two times multiplication and one time addition But Ramu found an easier way. He did
6+8=14 and then (2×14)=28. The use of brackets makes this sum easy. It can be done as follows : Rs.
2×(6+8) = Rs. (2×14) = Rs. 28 Here first solve the operation inside the bracket and then multiply it by the
number outside.

Now number of operations are one addition and one multiplication. So, second method takes less time.

9.0BODMAS Explanation

B→ Brackets first (parentheses) 0→Of DM → Division and Multiplication (start from left to right) AS →
Addition and Subtraction (start from left to right)

Note:

(i) Start Divide/Multiply from left side to right side since they perform equally. (ii) Start Add/Subtract
from left side to right side since they perform equally.

10.0Rounding Numbers

Rounding involves replacing one number with another number that's easier to work with. Rounded
numbers can be easier to use. Suppose you want to find 18×43, but had lost the calculator. You could
find an answer close to 18×43 by rounding to the nearest ten. "Rounding to the nearest ten" means
replacing a number with the nearest multiple of 10 . Replacing a number with a higher number is called
rounding up. Replacing a number with a lower number is called rounding down.

Procedure of round to different place values

You can round numbers to place values other than tens. Write the number. Underline the digit in the
position you want to round to.

 If the digit to the right of the underlined digit is 5 or more, round up.

 If the digit to the right of the underlined digit is 4 or less, round down.

Note: When we round a number to nearest place, all other digits to the right of the place becomes zero.
Ex round 24912 to nearest hundred, we will get 24900 . Digits to the right of 9 become zero.

 While converting 27381 into nearest thousands, we will focus on the digit which is on hundred
place.

Using rounded numbers

Now, you will learn more about using rounded numbers. You'll think about how much certain numbers
should be rounded. You'll also see how rounded numbers are useful for checking your work. People
round numbers to different place values depending on what the numbers are being used for.

The amount of rounding affects the accuracy

If you use rounding to estimate a sum, be careful how much you round. Rounding to higher place values
usually gives an estimate farther from the actual answer than rounding to lower place values.

Rounded numbers can be used to check work

Many times you'll want to check your work without doing the calculation all over again. Rounding is a
way to see if your answer is reasonable. Note: Using rounded numbers to check your answer won't ever
tell you that your answer is definitely right, only whether it is reasonable. Your answer might be close to
the real answer but could still be wrong.

11.0Estimation

Estimation means "making a good guess." We can use it if we don't need to know an exact answer, or if a
question has no exact right answer. You can estimate when there's no exact answer Sometimes in math
there is no exact right answer. You can use the information you do have to make an estimate.

Using estimation

Estimation is really useful in a lot of real-life situations, where you might not be able, or don't need, to
do an exact calculation. There are other times when it's better to figure out the exact answer.

Estimates aren't always a good idea There are some situations where you definitely shouldn't use an
estimate.

12.0Use of numbers in everyday life

Numbers are used immensely in our everyday life, such as measuring the length of a small object as
pencil, the distance between two given places, the weight of an orange, the weight of a ship, the amount
of juice in a glass and the amount of water in a lake. Small lengths are measured in millimeter ( mm ) and

the standard unit of length and we define it as : 1 m=100 cm=1000 mm ∴1 cm=10 mm

centimeter ( cm ) while bigger lengths are measured in meter ( m ) and kilometer ( km ). Meter ( m ) is

∴100 cm=100×10=1000 mm 1 km=1000 m Also, 1 km=(1000×1000)mm=1000000 mm Similarly, the units

of weight are as under: 1 g=1000mg 1 kg=1000gm 1 kg=(1000×1000)mg=1000000mg For capacity or
volume, 1ℓ=1000 mL and 1kℓ=1000ℓ 1kℓ=1000×1000 mℓ=1000000 mℓ

For Distance

km hm dam m dm cm mm

For Weight

kg hg dag g dg cg mg

For Liquids

kl hl dal l dl cl ml

Units of measurement

13.0Numerical Ability

How many odd numbers are there between 151 and 168 ?
  Explanation The odd numbers between 151 and 168 are - 153,155,157,159,161,163,165,167 The
total number is 8 .

(i) Find the smallest natural number. (ii) Find the number of four-digit natural numbers.

 Solution (i) The smallest natural number is 1 . (ii) The number of four-digit natural numbers is
9000 .

Compare 45967 and 45861. Explanation As number of digits are same so starting from the left hand side,

both the numbers we find that 9 in 45967 is greater than 8 in 45861. ∴45967>45861
we notice that 2 digits are the same. 45967 and 45861 On comparing the digits at the hundred places in

Make the greatest and the smallest four-digit numbers by using different digits such that digit 6 is always
in the tens place.

 Explanation We know that the digits written in the descending order are 9,8,7,6,5,4,3,2,1,0.
Keeping 6 in the tens place, we have Greatest number =9867 Smallest number =1062

Make the smallest and the greatest 5 -digit numbers using the digits 4, 6, 3, 1 and 0 only once.

 Solution: Smallest number: 10,346 Greatest number: 64,310

Arrange the numbers in ascending order.

 Explanation

Arrange the numbers in descending order.

 Explanation

Express the following in expanded form. (i) 3,54,039 (ii) 3,85,00,386

value of 5=5×10000 Place value of 0=0×100 Place value of 9=9×1 ∴ The expanded form of
 Explanation (i) Place value of 3=3×100000 Place value of 4=4×1000 Place value of 3=3×10 Place

3,54,029 is 3×100000+5×10000+4×1000+0×100+3×10+9×1. (ii) Likewise, the expanded form of

3,85,00,386 is 3×10000000+8×1000000+5×100000+0×10000+0×1000+3×100+8×10+6×1.

C TL L T-Th Th H T 0

10,000,000 1,000,000 100,000 10,000 1,000 100 10 1

(i) 3 5 4 0 3 9

(ii) 3 8 5 0 0 3 8 6
The population of Rajasthan is 5,64,73,122, and of Goa is 13,43,998 and of Karnataka is 5,27,33,958.
What is the combined population of the three states?

Karnataka is 5,27,33,958 ∴ Total population of three states

 Solution Population of Rajasthan is 5,64,73,122 Population of Goa is 13,43,998 Population of

=5,64,73,122+5,27,33,958+13,43,998=11,05,51,078 i.e., Eleven crore five lakh fifty-one thousand

seventy-eight.

What must be added to 34,52,629 to make it equal to 6 crores?

 Solution 6 crores =6,00,00,000 ∴ Required number =6,00,00,000−34,52,629

6,00,00,000−34,52,6295,65,47,371

=5,65,47,371

There are 785 students on roll in a residential public school. If the annual fee per student is Rs. 62,606.
What is the total fee collected annually by the school.

 Solution Annual fee of one student = Rs. 62,606 62606 Number of students =785 Total Annual
collection of fee = Rs. 62,606×785 = Rs. 4,91,45,710

Find the number of pages in a book which has on an average 207 words on a page and contains 201411
words altogether.

 Solution Number of pages =201411÷207 Thus, the number of pages in the book =973.

Write the numeral for each of the following numbers: (i) Ninety-eight crore two lakh seventy five. (ii) Six
million, four hundred and twelve thousand, two hundred and twenty.

 Solution: (i) Ninety-eight crore two lakh seventy-five is 98,02,00,075. (ii) Six million, four hundred
and twelve thousand, two hundred and twenty is 6,412,220.

Write the following in Roman numerals: (i) 52 (ii) 44 (iii) 85 (iv) 49 (v) 99

 Explanation (i) 52=50+2=L+II=LII (ii) 44=40+4=XL+IV=XLIV (iii) 85=80+5= LXXX +V= LXXXV (iv)
49=40+9= XL + IX = XLIX (v) 99=90+9=XC+IX=XCIX

Write the following in Hindu-Arabic numerals: (i) XLV (ii) LXIII (iii) LXXVI (iv) XCII (v) XXXVIII

 Solution (i) XLV=XL+V=(50−10)+5=40+5=45 (ii) LXIII =L+X+III=50+10+3=63 (iii) LXXVI

=L+XX+VI=50+(2×10)+6=76 (iv) XCII = XC + II =(100−10)+2=90+2=92 (v) XXXVIII = XXX + VIII
=(3×10)+8=30+8=38

78− [5 + 3 of (25-2×10)]

 Explanation: 78−[5+3 of (25−2×10)] =78−[5+3 of (25−20)] (Simplifying 'multiplication' 2×10=20 )

=78−[5+3 of 5] (Simplifying 'subtraction' 25-20=5) =78−[5+3×5] (Simplifying 'of) =78−[5+15]
(Simplifying 'multiplication' 3×5=15 ) =78−20 (Simplifying 'addition' 5+15=20 ) =58 (Simplifying
'subtraction' 78-20 =58 )

Simplify: 25-[ 22 - {17-(5-2) } ]

  Solution 25−[22−{17−(5−2)}] =25−[22−{17−3}] =25−[22−14] =25−8=17

Round 18×43 to the nearest ten.

 Explanation You need to decide whether to round up or down. Look at the digit in the ones
place: If the ones digit is 5 or more, round up. If the ones digit is 4 or less, round down. Start
with 18 : The digit in the ones place is 8 and 8 is more than 5 , so round up. 18 rounded up to the
nearest ten is 20 . Next, 43: The digit in the ones place is 3 and 3 is less than 5 , so round down.
43 rounded down to the nearest ten is 40 . By rounding, you can replace 18×43 with 20×40. This
is much easier to solve: 20×40=800 800 is fairly close to the real answer: 18×43=774

Round 25,281 to the nearest hundred.

 Explanation Write the number, and underline the hundreds digit: 25, 281 You're rounding to the
nearest hundred, so that's going to be either 25,200 or 25,300. The digit to the right of the
underline is 8 . That's greater than 5 , so round up. So, 25,281 rounds up to 25,300 , to the
nearest hundred. Rounding a number to the nearest 1000 To round off a number to the nearest
thousand, we get the nearest multiples of 1000 for that number. Rule : Look at the digit in the
hundreds place. If it is 5 or more, then round up, i.e., replace the digits at ones, tens and
hundreds place by 0 and add 1 to the digit at thousands place. If it is less than 5 , then round
down, i.e., replace the digits at ones, tens and hundreds place by zeros and leave the digit at
thousands place unchanged.

Round 8691 to the nearest thousand.

 Explanation

Round 4392 to the nearest thousand.

 Explanation

Round off the following numbers to the nearest tens, hundreds, thousands. (i) 7848 (ii) 5164

 Explanation (i) 7848 Rounded off to nearest 1078507848 Rounded off to nearest 1007800
7848 Rounded off to nearest 10008000 (ii) 5164 Rounded off to nearest 10
51605164 Rounded off to nearest 1005200 5164 Rounded off to nearest 10005000

Lucas wants to add 3439 and 5482. He doesn't need an exact answer, so he decides to use rounding.
Look at Lucas's work below. How could he have found a more accurate answer?

Actual:

+3439+54828921

Rounded to the nearest thousand:

+3000+50008000
Rounded to the nearest hundred:

+3400+55008900

 Explanation Lucas rounded to the nearest thousand, so he got an estimate of 8000 . If he had
rounded to the nearest hundred, he would have got 8900 , which is much closer to the actual
8921.

Calculate 2343 +5077. Then check your work by rounding to the nearest hundred.

 Solution Actual sum: 2343 Rounded sum: 2300

+507774207400+5100

The answer to the rounded sum is close to the answer to the actualsum, so the answer to the rounded
sum is reasonable.

Martin is trying to solve 29.6×9.8. He gets the answer 192.08. Check Martin's answer by rounding to the
nearest ten.

 Solution Rounded product : 30×10=300 Martin's answer is a long way from the rounded answer,
so it looks like his answer of 192.08 might be wrong. In fact, 29.6×9.8=290.08 This is much closer
to the rounded estimate.

Carla has a tall bookshelf and a short bookshelf. When full, the tall bookshelf can hold about 60 books.
Estimate from the picture how many books the small bookshelf will hold.

 Explanation There is no exact number of books you can fit on a bookshelf, because not all books
are the same size. To estimate the answer, compare the bookshelves. The tall one has 3 shelves,
and the small one only 2 . All the shelves has the same size, so the small bookshelf will hold
around two-thirds the number of books. So, you can estimate that the small bookshelf will hold
about 40 books. You can estimate if you don't need an exact answer You don't always need to
use an exact figure. Sometimes an estimate is enough.

A auto started its journey and reached different places with a speed of 20 km/hour. The journey is shown
below.

(i) Find the total distance covered by the auto from A to D. (ii) Find the total distance covered by the auto
from D to G. (iii) Find the total distance covered by the auto, if it starts from A and returns back to A. (iv)
Find the difference of distances from C to D and D to E ? (v) Find out the time taken by the auto to reach
(a) A to B (b) C to D (c) E to G (d) Total journey

 Explanation (i) Total distance covered by the auto from A to D =4180+3650+2360 =10,190 km (ii)
Total distance covered by the auto from D to G =8250+4930+2610 =15790 km (iii) Total distance
=4180+3650+2360+8250+4930+2610+1120 =27100 km (iv) Difference of distances from C to D
and D to E =8250−2360=5890 km (v) (a) 204180=209hrs (b) 202360=118hrs (c) 204930+2610
=377hrs (d) speed Totaldistance =2027100=1355hrs
14.0Memory map