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Number System
Introduction
In this chapter, we will study Hindu-Arabic system of numeration and place value and face value
of each digit in a number. We will also learn how to form numbers using the given digits and some
divisibility tests.
Natural Numbers (N)
The counting numbers 1, 2, 3, 4, … are called natural numbers.
We use N to denote the collection of all natural numbers.
So, N = {1, 2, 3, …}.
Maths Info
We denote a set/collection by
braces { · }
Points to remember
Maths Info
• 1 is the smallest natural number.
1 is the only natural number
• On adding 1 to a natural number, we get its successor.
which has a successor but no
Similarly, on subtracting 1 from the given natural number,
predecessor.
we get its predecessor.
• There are infinite natural numbers. Thus, there is no largest natural number.
Whole Numbers (W)
All natural numbers together with zero form the collection of whole numbers. We use W to denote
the collection of whole numbers. So, W = {0, 1, 2, 3, …}.
Points to remember
• 0 is the smallest whole number.
• On adding 1 to a whole number, we get its successor.
Similarly, on subtracting 1 from the given whole number,
we get its predecessor.
• There are infinite whole numbers. Thus, there is no largest
whole number.
Maths Info
0 is the only whole number
which has a successor but no
predecessor.
Integers (Z or I)
The collection of negatives of natural numbers, zero and natural
numbers forms the set of integers. We use Z or I to denote the
collection of integers. So, Z = {…, –3, –2, –1, 0, 1, 2, 3, …}.
Points to remember
• On adding 1 to an integer, we get its successor. Similarly, on
subtracting 1 from the given integer, we get its predecessor.
• There are infinite integers. Thus, there is no smallest or
largest integer.
Maths Info
Negatives of natural numbers
1, 2, 3, … are –1, –2, –3, … .
Maths Info
Every integer has a successor
and a predecessor.
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Real Numbers (R)
The set of all the numbers, i.e., positive, negative, zero and also
the fractional part between any two numbers is known as the set of
real numbers. We use R to denote the set of real numbers. There are
infinitely many real numbers. Thus, there is no smallest or largest
real number.
Maths Info
0 is neither a positive nor a
negative number. It is known
as a neutral number.
Even Natural Numbers (E)
The collection of natural numbers, divisible by 2, forms the set of even natural numbers. We use E to
denote the set of even natural numbers. So, E = {2, 4, 6, 8, …}.
Points to remember
• The smallest even natural number is 2.
• There are infinitely many even natural numbers. Thus, there is no largest even natural number.
Odd Natural Numbers (O)
The collection of natural numbers, not divisible by 2, forms the set of odd natural numbers. We use
O to denote the set of odd natural numbers. So, O = {1, 3, 5, 7, …}.
Points to remember
• The smallest odd natural number is 1.
• There are infinitely many odd natural numbers. Thus, there is no largest odd natural number.
Prime Numbers (P)
A natural number greater than 1 which is divisible only by 1 and
itself is known as a prime number. We use P to denote the set of
prime numbers. So, P = {2, 3, 5, 7, …}.
Maths Info
2 is the only prime number
which is even.
Points to remember
• 2 is the smallest prime number.
• There are infinitely many prime numbers. Thus, there is no largest prime number.
• 0 and 1 are not prime numbers.
Unit
A unit is something that denotes a single entity. For example, a crayon, a girl, a pencil, etc.
Number
A number denotes how many times a unit is taken. For example,
(a) Tina has four crayons. (4 crayons contains unit crayon, 4 times)
(b) There are twenty girls in the class. (20 girls contains unit girl, 20 times)
(c) Rahul has two pencils. (2 pencils contains unit pencil, 2 times)
Numeral
A numeral is a single digit or a group of digits used to represent a
number. For example, 2 and 421 are numerals.
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Maths Info
The ten symbols 0, 1, 2, 3, 4, 5,
6, 7, 8 and 9 are called digits or
figures.
Numeration
Expression of a number written in figures, into words is known as numeration. For example,
(a) The number 5 is written in words as five.
(b) The number 21 is written in words as twenty-one.
Exercise 1.1
1. Write True or False.
(a) 0 is the smallest whole number.
(b) There are finite natural numbers.
(c) All real numbers are integers.
(d) 0 is the smallest even natural number.
(e) The successor of a single-digit number is never a two-digit number.
(f) The predecessor of a two-digit number is always a two-digit number.
2. Which is the only even prime number? Is it bigger or smaller than the neutral number?
3. Consider the numbers 9, 8, 13, 17 and 19.
(a) Which of these are even?
(b) Which of these are odd?
(c) Which of these are prime?
4. Fill in the blanks.
(a) 20 cm contains unit __________, 20 __________.
(b) 10 ft contains unit __________, __________ times.
(c) 100 pens contains unit __________, 100 __________.
(d) 7 days contains unit __________, __________ times.
Hindu-Arabic System of Numeration
The decimal system which we use in Mathematics is the Hindu-Arabic system of numeration. It is
named so because it was originated by the Hindu mathematicians, then it was further developed by
the Arabs and finally it was introduced to western Europe. This system uses 10 digits (from 0 to 9)
and is based on the concept of place value.
Maths Info
In this system,
Hindu-Arabic number system
Ten ‘ones’ make one ‘ten’ i.e., 10 × 1 = 10
is also known as denary system
and ten is the base of this system.
Ten ‘tens’ make one ‘hundred’ i.e., 10 × 10 = 100
Ten ‘hundreds’ make one ‘thousand’ i.e., 10 × 100 = 1000
Face Value
The face value of a digit, at any place in a numeral is the value of the digit itself. Face value is also
known as true value. For example, the face value of 9 in 900 and in 56097 is 9.
Place Value
The place value of a digit in a numeral is the product of the face value and the value of the place of
the digit in the given numeral. Place value is also known as local value. For example, consider the
numeral 986.
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• The digit 6 is at one’s place. So, its place value is 6 × 1 = 6
• The digit 8 is at ten’s place. So, its place value is 8 × 10 = 80
• The digit 9 is at hundred’s place. So, its place value is 9 × 100 = 900
Point to remember
The face value of a digit always remains the same in any
numeral but its place value changes. For example, in 709,
the place value of 7 is 700 and in 672 the place value of 7 is
70, but in both the numerals, the face value of 7 is 7 itself.
Maths Info
0 is the only digit whose face
value and place value are same.
Indian Place Value Chart
Although the Hindu-Arabic system is used all over the world, the expression of a number in words
is different in India from the rest of the world. In Indian system of numeration, commas are used to
mark thousands, lakhs and crores.
Thousand
Hundred
Ten
One
10,00,00,000 1,00,00,000 10,00,000 1,00,000
Ones
Ten
Thousand
Thousands
Lakh
Ten Lakh
Lakhs
Crore
Ten Crore
Crores
10,000
1,000
100
10
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Example 1: Insert commas and write the number names of the following numbers according to the
Indian place value chart.
(a) 981765 (b) 75698027
Solution: Number
(a) 9,81,765
(b) 7,56,98,027
Number name
Nine lakh eighty-one thousand seven hundred sixty-five
Seven crore fifty-six lakh ninety-eight thousand twenty-seven
International Place Value Chart
Maths Info
In International system of numeration, commas are used to mark
thousands, millions and billions.
Ten
Billion
Billion
Hundred
Million
Ten
Million
Million
Hundred
Thousand
Ten
Thousand
Thousand
Hundred
Ten
One
1,000,000,000
100,000,000
10,000,000
1,000,000
100,000
10,000
1,000
100
10
1
Ones
10,000,000,000
Thousands
Hundred
Billion
Millions
100,000,000,000
Billions
We do not use commas while
writing number names.
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Example 2: Insert commas and write the number names of the following numbers according to the
International place value chart.
(a) 981765 (b) 75698027
Solution: Number
Number name
(a)
981,765
Nine hundred eighty-one thousand seven hundred sixty-five
(b)
75,698,027 Seventy-five million six hundred ninety-eight thousand twenty-seven
Example 3: Write the place value of the two 4’s in 34,048.
Solution: In 34,048, one 4 is at the ten’s place. So, its place value is 4 × 10 = 40
The other 4 is at the thousand’s place. So, its place value is 4 × 1,000 = 4,000
Example 4: Write the place value of the two 7’s in the number 7,12,657 and find their sum.
Solution: In 7,12,657, one 7 is at the one’s place. So, its place value is 7 × 1 = 7
The other 7 is at the lakh’s place. So, its place value is 7 × 1,00,000 = 7,00,000
The sum of the two place values is 7 + 7,00,000 = 7,00,007
Expanded Form of a Number
On the basis of the place values of the digits, we write the expanded form of a number. For example,
(a) 4,75,320 = 4 × 1,00,000 + 7 × 10,000 + 5 × 1,000 + 3 × 100 + 2 × 10 + 0 × 1
(b) 5,67,53,412 = 5 × 1,00,00,000 = 6 × 10,00,000 + 7 × 1,00,000 + 5 × 10,000 + 3 × 1,000 + 4 × 100 +
1 × 10 + 2 × 1
Comparison of Numbers
Out of any two given numbers, one is always bigger than the other. To check which number is bigger,
follow the given rules.
(a) The number with more digits is always bigger.
(b) If both the numbers have the same number of digits, then start comparing the digits from the
leftmost side. The number with a larger digit at the corresponding place is bigger.
Example 5: Which is greater, 76,83,45,200 or 76,38,45,200?
Solution: The number of digits in both the numbers is nine.
7 6 8 3 4 5 2 0 0
7 6 3 8 4 5 2 0 0
= = 8 > 3
\ 76,83,45,200 > 76,38,45,200
Example 6: Arrange 31,22,567, 34,21,657, 3,43,56,210, 3,44,59,610 and 4,44,02,057 in ascending
order.
Solution: 31,22,567 and 34,21,657 are 7-digit numbers. Let’s compare them.
3 1 2 2 5 6 7
3 4 2 1 6 5 7
= 4 > 1
\ 31,22,567 < 34,21,657
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Now, let’s compare 3,43,56,210, 3,44,59,610 and 4,44,02,057.
They all are 8-digit numbers.
As the leftmost digits are 3, 3 and 4 respectively, 4,44,02,057 is the greatest number.
Now, 3 4 3 5 6 2 1 0
3 4 4 5 9 6 1 0
= = 4 > 3
\ 3,43,56,210 < 3,44,59,610
Hence, the numbers in ascending order are:
31,22,567 < 34,21,656 < 3,43,56,210 < 3,44,59,610 < 4,44,02,057
Exercise 1.2
1. Consider the number 51078.
(a) The local value of 1 = __________
(c) The local value of 0 = __________
(e) The local value of 7 = __________
(b) The true value of 1 = __________
(d) The true value of 5 = __________
(f) The local value of 5 = __________
2. Write the place value of two 3’s in 73231 and find their difference.
3. Which digit in the number 4578 has the same face value and place value?
4. Find the sum of the face value and the place value of the digit 2 in 720.
5. Insert commas and write the number names according to the Indian system of numeration.
(a)87651055
(b)39754081
(c)467481405
6. Insert commas and write the number names according to the International system of numeration.
(a)767835410
(b)67234189
7. Which number is smaller?
(a) 1,25,678 or 1,26,578
(c) 5,67,30,001 or 4,98,71,002
(c)24892348
(b) 4,50,98,145 or 4,51,87,546
(d) 3,49,88,067 or 3,49,89,167
8. Arrange the following numbers in descending order.
78,13,018 70,99,954 11,48,05,654 1,95,45,678 2,95,00,141 2,95,10,200
Forming Numbers
To form numbers using the given digits, always remember to check whether the digits can be repeated
or not. If the given digits are to be used without repetition, then each of the given digit can be used
only once. If the given digits are to be used with repetition, then we can use digits any number of
times accordingly.
Example 7: Using the digits 2 and 6, form all the possible 2-digit numbers such that no digit is
repeated.
Solution: The 2-digit numbers are 26 and 62.
Note: The numbers 22 and 66 are not considered as no digit is to be repeated.
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Example 8: Using the digits 1, 5 and 9, form all the 2-digit and 3-digit numbers such that no digit
is repeated.
Solution: The 2-digit numbers are 15, 19, 51, 59, 91 and 95.
The 3-digit numbers are 159, 195, 915, 951, 519 and 591.
Note: Numbers like 11, 55, 99, 111, 191, 515, etc. are not considered as no digit is to be repeated.
Forming the smallest number using the given digits
To form the smallest number with the given digits, place the smallest
digit at the extreme left position, and then place the other digits in
ascending order of their values.
Maths Info
0 cannot be used at the extreme
left position to form any number
as a number can never begin
with zero.
Example 9: Form the smallest 3-digit number using the digits 4, 9 and 0 without repetition.
Solution: Place the smallest digit (except 0) i.e., 4 at the extreme left. Then, place 0 and then 9.
So, the required smallest 3-digit number is 409.
Note: If 0 is used at the extreme left position, then the number formed will be 049, which is a
2-digit number.
Forming the greatest number using the given digits
To form the greatest number with the given digits, place the greatest
digit at the extreme left position, and then place the other digits in
descending order of their values.
Maths Info
0 is always placed at the last
position to form the greatest
number with the given digits.
Example 10:Form the greatest 3-digit number using the digits 8, 0 and 7 without repetition.
Solution: Place the greatest digit i.e., 8 at the extreme left. Then, place 7 and then 0. So, the
required greatest 3-digit number is 870.
Example 11: Form the smallest and the greatest 6-digit number using the digits 8, 9, 1, 0, 4, 6 without
repetition.
Solution: To form the smallest 6-digit number, place the smallest digit (except 0) i.e., 1 at the
extreme left position. Then, place the remaining digits (8, 9, 0, 4, 6) in ascending order
i.e., 0, 4, 6, 8 and 9. So, the required number is 1,04,689.
To form the greatest 6-digit number, place the greatest digit i.e., 9 at the extreme left
position. Then, place the remaining digits (8, 1, 0, 4, 6) in descending order i.e., 8, 6,
4, 1 and 0. So, the required number is 9,86,410.
Example 12:Form the smallest 6-digit number.
Solution: The smallest digit except 0 is 1. So, place 1 at the extreme left and then five 0’s on its
right. So, the required number is 1,00,000.
Example 13:Form the greatest 5-digit number.
Solution: The greatest digit is 9. So, place 9 at all the five places to get the required number as 99,999.
Example 14:How many 4-digit numbers are there between 999 and 4,000?
Solution: The required 4-digit numbers are from 1,000 to 3,999.
Numbers from 1,000 to 3,999= 1,000 + (1,001 to 2,000) + (2,001 to 3,000) + (3,001 to 3,999)
Number of these numbers = 1 + 1,000 + 1,000 + 999 = 3,000
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Exercise 1.3
1. Form the smallest 3-digit numbers using the given digits without repetition.
(a) 4, 9, 2
(b) 3, 9, 0
(c) 0, 2, 1
(d) 9, 8, 1
2. Form the greatest 5-digit number using the digits 9, 1, 0, 6 and 2 without repetition.
3. Form the smallest and the greatest 4-digit numbers using the digits 5, 6, 9 and 0.
4. Form the greatest and the smallest 4-digit numbers.
5. Form the smallest 6-digit number with 4 at the ten’s place.
6. Fill in the blanks.
(a) The smallest 5-digit number is __________.
(b) The sum of the largest 4-digit number and the smallest 5-digit number is __________.
(c) In the greatest 6-digit number, the digit at the hundred’s place is __________.
(d) In the smallest 6-digit number, the digit at the ten’s place is __________.
7. How many 3-digit numbers are there between 99 and 3,000?
Divisibility Tests
(a) Divisibility by 2: A number is divisible by 2, if the digit at its one’s place is zero or an even
number.
For example, 8, 14, 64, 120, etc. are all divisible by 2.
(b) Divisibility by 3: A number is divisible by 3, if the sum of all its digits is divisible by 3.
For example, 33, 12, 69, 3,321, etc. are divisible by 3 as,
Sum of digits of 33 = 3 + 3 = 6, which is divisible by 3.
Sum of digits of 12 = 1 + 2 = 3, which is divisible by 3.
Sum of digits of 69 = 6 + 9 = 15, which is divisible by 3.
Sum of digits of 3,321 = 3 + 3 + 2 + 1 = 9, which is divisible by 3.
(c) Divisibility by 4: A number is divisible by 4, if its last two digits are divisible by 4.
For example, 24, 140, 80,512, 64,120, etc. are all divisible by 4 as their last two digits are
divisible by 4.
(d) Divisibility by 5: A number is divisible by 5, if its digit at the one’s place is 0 or 5.
For example, 50, 65, 120, 1,145, etc. are divisible by 5.
(e) Divisibility by 6: A number is divisible by 6, if the number is divisible by both 2 and 3.
For example, 21,390 is divisible by 2, as the digit at its one’s place is zero. Also, 21,390 is
divisible by 3 as, 2 + 1 + 3 + 9 + 0 = 15, which is divisible by 3. Therefore, 21,390 is divisible
by 6.
(f) Divisibility by 8: A number is divisible by 8, if its last three digits are divisible by 8.
For example, 45,640 is divisible by 8 as 640 is divisible by 8.
(g) Divisibility by 9: A number is divisible by 9, if the sum of all its digits is divisible by 9.
For example, 4,509 is divisible by 9 as, 4,509 = 4 + 5 + 0 + 9 = 18, which is divisible by 9.
(h) Divisibility by 10: A number is divisible by 10, if the digit at its one’s place is 0.
For example, 20, 40, 190, 5,330, etc. are divisible by 10.
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(i) Divisibility by 11: A number is divisible by 11, if the difference between the sum of the digits
at its odd places (starting from right) and the sum of the digits at its even places is either zero
or a multiple of 11.
For example, 50,875 is divisible by 11 as,
Sum of the digits at odd places = 5 + 8 + 5 = 18
Sum of the digits at even places = 7 + 0 = 7
Difference between the sums = 18 – 7 = 11
Example 15:Consider the numbers 29, 30, 65, 2,198 and 16,791.
(a) Which of these are divisible by 2?
(b) Which of these are divisible by 3?
(c) Which of these are divisible by 5?
(d) Which of these are divisible by 10?
Solution: (a) In 30 and 2,198, the digits at the one’s place are 0 and 8 respectively. Thus, 30 and
2,198 are divisible by 2.
(b) Sum of digits of 29 = 2 + 9 = 11, which is not divisible by 3.
Sum of digits of 30 = 3 + 0 = 3, which is divisible by 3.
Sum of digits of 65 = 6 + 5 = 11, which is not divisible by 3.
Sum of digits of 2,198 = 2 + 1 + 9 + 8 = 20, which is not divisible by 3.
Sum of digits of 16,791 = 1 + 6 + 7 + 9 + 1 = 24, which is divisible by 3.
Thus, 30 and 16,791 are divisible by 3.
(c) In 30 and 65, the digits at the one’s place are 0 and 5 respectively. Thus, 30 and 65
are divisible by 5.
(d)The only number having 0 at the one’s place is 30. So, 30 is the only number
divisible by 10.
SUMMARY
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The counting numbers 1, 2, 3, 4, … are natural numbers.
All natural numbers together with 0 form the set of whole numbers.
The collection of natural numbers, negatives of natural numbers and 0 forms the set of integers.
The real number system consists of all the numbers.
Natural numbers divisible by 2 are known as even natural numbers.
Natural numbers not divisible by 2 are known as odd natural numbers.
Natural numbers greater than 1, which are divisible only by 1 and itself are prime numbers.
Face value of a digit in a number is the value of the digit itself.
Place value of a digit = Face value × Value of the place of the digit.
If the digit at the one’s place is 0 or an even number, then the number is divisible by 2.
If the sum of all the digits of a number is divisible by 3, then the number is divisible by 3.
If the last two digits of a number are divisible by 4, then the number is divisible by 4.
If the digit at the one’s place is 0 or 5, then the number is divisible by 5.
If a number is divisible by both 2 and 3, then the number is divisible by 6.
If the last three digits of a number are divisible by 8, then the number is divisible by 8.
If the sum of all the digits of a number is divisible by 9, then the number is divisible by 9.
If the digit at the one’s place is 0, then the number is divisible by 10.
If the difference between the sum of the digits at the odd places and the sum of the digits at
the even places is either zero or a multiple of 11, then the number is divisible by 11.
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REVIEW EXERCISE
Mental Maths
1. The sum of the smallest natural number and the smallest whole number is ____________.
2. The face value of 6 in 1,260 is ____________.
3. In 9,870, the face value of 9 is ____________ and its place value is ____________.
4. Fill in the blanks.
(a) 1 thousand = __________ hundreds
(b) 1 lakh = __________ thousands
(c) 1 crore = __________ lakhs
(d) 1 million = __________ lakhs
(e) 1 million = __________ hundred thousands
(f) 1 crore = __________ lakhs
5. Fill in the boxes with ‘<’ or ‘>’.
(a)14,34,952 (c)7,81,49,512 14,33,899
6,99,45,678
(b)22,02,222 (d)1,23,45,670 22,20,222
1,23,45,677
6. Write True or False.
(a) Every whole number is a natural number. (b) Every integer is a whole number.
(c) Zero is the smallest real number.
(d) 1 is a prime number.
Solve and Answer
1. Insert commas and write the number names of the following numbers according to the Indian
system of numeration.
(a)7828790
(b)8123456
(c)12456789
(d)34685811
2. Insert commas and write the number names of the following numbers according to the International
system of numeration.
(a)1427323
(b)44556710
(c)29175434370
(d)2170034001
3. Write the numerals for the following number names according to the Indian system of numeration.
(a) Seventy-five lakh fifty-five thousand eight hundred four.
(b) Nine crore fourty-nine lakh six hundred.
(c) Five crore eighty-four lakh two thousand five hundred nine.
4. Write the place value of the two 8’s in 80,681.
5. Find the difference between the face value and the place value of 2 in 1,912.
6. Form the smallest 6-digit number using the digits 9, 5, 4, 0, 2 and 1.
7. Write the largest 6-digit number with 8 at the ten’s place.
8. Write the smallest 5-digit number with 9 at the hundred’s place.
9. Consider the numbers 9, 12, 50, 121, 323 and 990.
(a) Which of these are divisible by 6?
(b) Which of these are divisible by 4?
(c) Which of these are divisible by 5?
(d) Which of these are divisible by 10?
10. Write the greatest 2-digit number divisible by 3.
11. Write the smallest 4-digit number divisible by 5.
12. From the numbers 5,064, 3,501, 3,344 and 9,522, find the numbers divisible by 8, 9 and 11.
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