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Number System
Introduction
In this chapter, we will study about the number system and number line. We will also learn about the
four fundamental operations on whole numbers and their properties.
Natural Numbers (N)
The counting numbers 1, 2, 3, 4, … are called natural numbers. The set of natural numbers is given
by N = {1, 2, 3, 4 …}.
Even natural numbers (E)
The collection of natural numbers, divisible by 2, forms the set of even natural numbers. It is given
by E = {2, 4, 6, 8, …}. An even natural number can be represented by 2n where n ∈ N.
Odd natural numbers (O)
The collection of natural numbers, not divisible by 2, forms the set of odd natural numbers. It is given
by O = {1, 3, 5, 7, …}. An odd natural number can be represented by 2n – 1, where n ∈ N.
Whole Numbers (W)
All natural numbers together with zero form the collection of whole numbers. The set of whole numbers
is given by W = {0, 1, 2, 3, …}.
Integers (Z)
The collection of negatives of natural numbers, zero and natural numbers form the set of integers. It
is denoted by Z or I. So, Z = {…, –3, –2, –1, 0, 1, 2, 3, …}. The set of positive integers is given by
Z+ = {1, 2, 3, 4, …} and the set of negative integers is given by Z – = {–1, –2, –3, –4, …}.
Prime Numbers (P)
A natural number greater than 1 which is divisible only by 1 and itself is known as a prime number.
The set of prime numbers is given by P = {2, 3, 5, 7, …}. Natural numbers which are not prime
numbers are known as composite numbers. For example, 4, 6, 8, 9, … etc.
Face Value
The face value or true value of a digit at any place in a numeral is the value of the digit itself. For
example, the face value of 5 in 659 is 5.
Place Value
The place value or local 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. For example, the place value of 7 in 37,580 is 7,000.
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Number Line
A number line is a straight line on which points are marked to divide it into equal parts. The middle
point of this line is marked as zero. To the left of zero, equally spaced points are marked as negative
numbers and to the right of zero, they are marked as positive numbers. Thus, the number line is
obtained as given below.
–4
–3
–2
–1
0
1
2
3
4
For any two numbers on the number line, the number which is on the right is greater than the number
on the left.
Example 1: How many 2-digit whole numbers are there?
Solution:
10, 11, 12, …, 99 are 2-digit whole numbers.
\ Number of 2-digit numbers = 99 – 10 + 1 = 90
Maths Info
Largest number of the series –
Smallest number of the series + 1 =
Total numbers in the series
Example 2: How many 4-digit whole numbers are there?
Solution:
1,000, 1,001, 1,002, …, 9,999 are 4-digit whole numbers.
\ Number of 4-digit numbers = 9,999 – 1,000 + 1 = 9,000
Example 3:
Solution:
Write all the 2-digit numbers formed using the digits 3, 6 and 4 when:
(a) repetition of digits is allowed
(b) repetition of digits is not allowed
(a) The 2-digit numbers are 36, 63, 64, 46, 34, 43, 33, 66 and 44.
(b) The 2-digit numbers are 36, 63, 64, 46, 34 and 43.
Example 4: Write all the 3-digit numbers formed using the digits 9, 7 and 5 when repetition of digits
is not allowed.
Solution:
The 3-digit numbers are 975, 957, 579, 597, 759 and 795.
Example 5:
Solution:
What is the smallest 6-digit number formed when:
(a) repetition of digits is allowed
(b) repetition of digits is not allowed
(a) The 6-digit number is 1,00,000.
(b) The 6-digit number is 1,02,345.
Example 6: Find the difference between the two place values of 8 in the number 5,86,890.
Solution:
In 5,86,890, one 8 is at hundred’s place. So, its place value is 800.
The other 8 is at ten thousand’s place. So, its place value is 80,000.
Difference between the two place values = 80,000 – 800 = 79,200
Exercise 1.1
1. Write True or False.
(a) All natural numbers are whole numbers.
(b) Zero is the only number whose face value and place value is same.
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(c) There are infinite even prime numbers.
(d) The set of negative integers is finite.
(e)If x > y, then –x > –y.
2. Using a number line, fill in the boxes with ‘<’ or ‘>’.
(a)0 (d)–19 –16
(b) –24 9
(e) –5 24
–9
(c) 4 (f) –13 –5
–16
3. From the numbers 16, 19, 21, 23, 31, 37, 39 and 51, write which are:
(a)prime
(b)composite
4. Write all the 2-digit numbers formed using the digits 2, 0 and 7 when:
(a) repetition of digits is allowed (b) repetition of digits is not allowed
5. Write all the 3-digit numbers formed using the digits 4, 9, 7 and 5 when repetition of digits is
not allowed.
6. How many 3-digit whole numbers are there?
7. How many 5-digit whole numbers are there?
8. Find the difference between the smallest 4-digit natural number and the smallest 4-digit whole
number formed when repetition of digits is allowed.
9. Write the greatest 5-digit number using distinct natural numbers.
Fundamental Operations
We have already studied the operations of addition, subtraction, multiplication and division on whole
numbers. Now, let’s study some properties of these fundamental operations.
Properties of addition
Closure
The sum of whole numbers is always a whole number. Mathematically, if a and b are whole numbers,
then a + b is also a whole number. For example,
(a) 32 + 0 = 32
(b) 206 + 19 = 225
Commutative
The sum of two whole numbers remains the same irrespective of the order in which they are added.
Mathematically, a + b = b + a, where a and b are whole numbers. For example,
(a) 25 + 9 = 34 = 9 + 25
(b) 130 + 21 = 151 = 21 + 130
Associative
The sum of any three whole numbers remains the same even if their grouping is changed. Mathematically,
(a + b) + c = a + (b + c) = a + b + c, where a, b and c are whole numbers. For example,
(a) (34 + 61) + 75 = 95 + 75 = 170
Also, 34 + (61 + 75) = 34 + 136 = 170
\ (34 + 61) + 75 = 34 + (61 + 75) = 170
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(b) (104 + 88) + 112 = 192 + 112 = 304
Also, 104 + (88 + 112) = 104 + 200 = 304
\ (104 + 88) + 112 = 104 + (88 + 112) = 304
Additive identity
The sum of any whole number and 0 is always the number itself. So, 0 is called the additive identity
for whole numbers. Mathematically, a + 0 = 0 + a = a, where a is any whole number. For example,
(a) 39 + 0 = 0 + 39 = 39
(b) 210 + 0 = 0 + 210 = 210
Cancellation law
Cancellation law states that if a, b and c are whole numbers, then
a+ b =c+ b ⇒ a = c
For example, a + 14 = 5 + 14 ⇒ a = 5
Properties of subtraction
Closure
Whole numbers are not closed under subtraction as the difference of two whole numbers need not be
a whole number. For example, 5 – 10 = –5, which is not a whole number.
Commutative
Whole numbers do not obey commutative law under subtraction as a – b need not be equal to b – a,
where a and b are whole numbers. For example, 12 – 6 ≠ 6 – 12.
Associative
Whole numbers do not obey associative law under subtraction as (a – b) – c need not be equal to
a – (b – c), where a, b and c are whole numbers. For example, (5 – 2) – 1 ≠ 5 – (2 –1) as
(5 – 2) – 1 = 2 and 5 – (2 – 1) = 4
Cancellation law
Cancellation law states that if a, b and c are whole numbers, then a – b = c – b ⇒ a = c.
For example, a – 19 = 26 – 19 ⇒ a = 26
Properties of multiplication
Closure
The product of whole numbers is always a whole number. Mathematically, if a and b are whole
numbers, then a × b is also a whole number. For example,
(a) 12 × 9 = 108
(b) 125 × 8 = 1,000
Commutative
The product of two whole numbers remains the same irrespective of the order in which they are
multiplied. Mathematically, a × b = b × a, where a and b are whole numbers. For example,
(a) 95 × 6 = 570 = 6 × 95
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(b) 107 × 7 = 749 = 7 × 107
Associative
The product of any three numbers remains the same even if their grouping is changed. Mathematically,
(a × b) × c = a × (b × c) = a × b × c where a, b and c are whole numbers. For example,
(a) (5 × 9) × 3 = 45 × 3 = 135
Also, 5 × (9 × 3) = 5 × 27 = 135
\ (5 × 9) × 3 = 5 × (9 × 3)
(b) (3 × 7) × 6 = 21 × 6 = 126
Also, 3 × (7 × 6) = 3 × 42 = 126
\ (3 × 7) × 6 = 3 × (7 × 6)
Property of zero
The product of any whole number and 0 is always 0. For example,
(a) 4 × 0 = 0
(b) 947 × 0 = 0
Multiplicative identity
The product of any whole number and 1 is the number itself. So, number 1 is called the multiplicative
identity for whole numbers. Mathematically, a × 1 = 1 × a = a, where a is any whole number. For
example,
(a) 34 × 1 = 1 × 34 = 34
(b) 104 × 1 = 1 × 104 = 104
Cancellation law
Cancellation law states that if a, b and c are whole numbers, then a × b = c × b ⇒ a = c.
For example, 19 × 6 = a × 6 ⇒ a = 19
Distributive
According to distributive property of multiplication over addition, if a, b and c are whole numbers, then
a × (b + c) = (a × b) + (a × c)
Similarly, by distributive property of multiplication over subtraction, we have
a × (b – c) = (a × b) – (a × c)
For example,
(a) 65 × (5 + 3) = (65 × 5) + (65 × 3) = 325 + 195 = 520
Also, 65 × (5 + 3) = 65 × 8 = 520
\ 65 × (5 + 3) = (65 × 5) + (65 × 3)
(b) 3 × (45 – 8) = (3 × 45) – (3 × 8) = 135 – 24 = 111
Also, 3 × (45 – 8) = 3 × 37 = 111
\ 3 × (45 – 8) = (3 × 45) – (3 × 8)
Example 7: Find the following products using distributive property.
(a) 105 × 33 (b) 999 × 55
Solution:
(a) 105 × 33 = (100 + 5) × 33
5
= (100 × 33) + (5 × 33)
= 3,300 + 165 = 3,465
(b) 999 × 55 = (1,000 – 1) × 55
= (1,000 × 55) – (1 × 55)
= 55,000 – 55
= 54,945
[Using a × (b + c) = (a × b) + (a × c)]
[Using a × (b – c) = (a × b) – (a × c)]
Properties of division
Closure
The quotient obtained on dividing two whole numbers need not be a whole number. For example,
10 5
10 ÷ 4 =
= , which is not a whole number. So, whole numbers do not obey the closure property
4
2
under division.
Commutative
Whole numbers do not obey commutative law under division as a ÷ b need not be equal to b ÷ a,
1
where a and b are whole numbers. For example, 25 ÷ 5 ≠ 5 ÷ 25 as 25 ÷ 5 = 5 and 5 ÷ 25 =
5
Associative
Whole numbers do not obey associative law under division as (a ÷ b) ÷ c need not be equal to
a ÷ (b ÷ c), where a, b and c are whole numbers. For example, (45 ÷ 9) ÷ 3 ≠ 45 ÷ (9 ÷ 3) as
45
5
=9
(45 ÷ 9) ÷ 3 = and 45 ÷ (9 ÷ 3) =
3
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Property of zero
If 0 is divided by any non-zero whole number, the quotient is always 0. For example,
(a) 0 ÷ 9 = 0
(b) 0 ÷ 124 = 0
Identity
If a is any whole number then a ÷ 1 = 1. For example,
(a) 21 ÷ 1 = 21
(b) 105 ÷ 1 = 105
Division algorithm
If a and b are whole numbers and a > b (b ≠ 0), then there exist two other whole numbers q and r such
that a = bq + r, where r = 0 or r < b. This relation is known as division algorithm or rule of division.
We can also say that,
Maths Info
Dividend = Divisor × Quotient + Remainder
Division by 0 is not defined.
Example 8: Divide 1,509 by 27 and verify the division algorithm.
Solution:
Dividend = 1,509, divisor = 27, quotient = 55, remainder = 24
Divisor × Quotient + Remainder= 27 × 55 + 24
= 1,485 + 24 = 1,509
= Dividend
\ Division algorithm is verified.
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55
27 1509
–135
159
–135
24
Example 9: Find the number which when divided by 28 gives the quotient 13 and remainder 4.
Solution:
Divisor = 28, quotient = 13 and remainder = 4
Required number = Dividend = Divisor × Quotient + Remainder
⇒ Dividend = 28 × 13 + 4
= 364 + 4 = 368
Example 10:Find the number which on dividing 2,558 gives 150 as quotient and 8 as remainder.
Solution:
Dividend = 2,558, quotient = 150 and remainder = 8
Dividend = Divisor × Quotient + Remainder
⇒
2,558 = Divisor × 150 + 8
⇒
2,558 – 8 = Divisor × 150
2,550
⇒
= Divisor
150
⇒
Divisor = 17
\ Required number = 17
41
Example 11: Find the smallest 4-digit number which is exactly divisible by 24.
Solution:
The smallest 4-digit number is 1,000.
1,000 divided by 24 gives 16 as remainder.
So, if we add 24 – 16 = 8 in 1,000, then the sum will be exactly
divisible by 24.
Hence, the required number is 1,008.
24 1000
–96
40
–24
16
Note: If we subtract 16 from 1,000, then also the result will be exactly divisible by 24, but in that
case the number will be a 3-digit number.
Example 12:Find the largest 5-digit number which is exactly divisible by 52.
Solution:
The greatest 5-digit number is 99,999.
99,999 divided by 52 gives 3 as remainder.
So, the required number is 99,999 – 3 = 99,996
1923
52 99999
–52
479
–468
119
–104
159
–156
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Exercise 1.2
1. Fill in the blanks.
(a) 657 + _______ = 757 + 657
(b) 35 + (49 + _______) = 35 + 49 + 97
(c) 74 × (_______ – 201) = 74 × 225 – 74 × 201
(d)
_______ × (109 + 35) = 24 × 109 + 24 × 35
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2. Find the sum of the following numbers using the most convenient grouping.
(a) 3,526, 516, 474
(b) 1,486, 285, 1,014, 215
(c) 2,547, 108, 242, 1,953
(d) 500, 516, 358, 484, 442
3. Find the following products using distributive property.
(a) 456 × 102
(b) 999 × 54
(c) 13 × 955
(d) 1,004 × 36
4. Find the following products using the most convenient grouping.
(a) 25 × 595 × 4 (b) 5 × 123 × 20 (c) 16 × 5,456 × 125
5. Divide and verify the division algorithm.
(a) 2,542 ÷ 44 (b) 8,924 ÷ 58 (c) 17,654 ÷ 251
6. Find the number which when divided by 37 gives the quotient 16 and remainder 8.
7. Find the largest 6-digit number which is exactly divisible by 41.
8. Find the smallest 6-digit number which is exactly divisible by 180.
9. On dividing 3,487 by 112, the remainder is found to be 15. Find the quotient.
10. Find the number which on dividing 5,498 gives 43 as quotient and 123 as remainder.
SUMMARY
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The counting numbers 1, 2, 3, 4, … are called natural numbers.
The collection of natural numbers, divisible by 2, forms the set of even natural numbers.
The collection of natural numbers, not divisible by 2, forms the set of odd natural numbers.
All natural numbers together with zero form the collection of whole numbers.
The collection of negatives of natural numbers, zero and natural numbers forms the set of integers.
A natural number greater than 1 which is divisible only by 1 and itself is known as a prime
number.
The face value of a digit at any place in a numeral is the value of the digit itself.
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.
A number line is a straight line on which points are marked to divide it into equal parts.
The set of whole numbers is closed, commutative and associative under addition and cancellation
law also holds true.
0 is the additive identity of whole numbers.
The set of whole numbers is not closed, not commutative and not associative under subtraction
but cancellation law holds true.
The set of whole numbers is closed, commutative and associative under multiplication and
cancellation law also holds true.
1 is the multiplicative identity of whole numbers.
For any three whole numbers a, b and c, a × (b + c) = (a × b) + (a × c) and a × (b – c) =
(a × b) – (a × c)
The set of whole numbers is not closed, not commutative and not associative under division.
Division algorithm states that if a and b are whole numbers (a > b, b ≠ 0), then there exist two
other whole numbers q and r such that a = bq + r, where r = 0 or r < b.
REVIEW EXERCISE
Mental Maths
1. Write True or False.
(a) 2 is the only even prime number.
(b) Whole numbers are closed under addition and subtraction.
(c) Whole numbers are associative under multiplication.
(d) 1 is the additive identity of whole numbers.
(e) For whole numbers a, b and c, a × (b + c) = (a × b) + (b × c).
(f)If a is a whole number which is divisible by b (b ≠ 0) and a = bq + r, then r = 0.
2. Fill in the blanks.
(a) {–1, –2, –3, –4, …} is the set of ___________ integers.
(b) 86 × (_______ + _____) = (86 × 63) + (86 × 95)
(c)
_______ is the multiplicative identity of whole numbers.
(d) Dividend = Divisor × ___________ + ___________
Solve and Answer
1. What is the difference in the place values of two odd digits in each of the following?
(a)2,756
(b)56,289
(c)24,305
(d)48,770
2. Write all the 4-digit numbers using the digits 1, 0, 9 and 7 without repetition.
3. Write all possible 2-digit numbers using the digits 5, 4 and 8 if repetition of digits is not allowed.
4. How many natural numbers are there between 9 and 29, both inclusive?
5. How many natural numbers are there between 80 and 125, both inclusive?
6. What is the product of the smallest 4-digit number and the greatest 3-digit number?
7. Find the sum of the smallest and the largest 4-digit number formed by using the digits 0, 3, 5 and
7 without repetition.
8. Find the sum of the following numbers and verify the commutative law of addition.
(a) 549, 6,134
(b) 954, 24,055
(c) 44,096, 55,013
9. Find the following products using distributive property or the most convenient grouping.
(a) 106 × 643
(b) 25 × 84 × 8
(c) 474 × 96
(d) 744 × 102
(e) 50 × 8 × 45
(f) 40 × 552 × 50
10. Find the greatest 5-digit number which is exactly divisible by 225.
11. Find the smallest 4-digit number which is exactly divisible by 23.
12. Find the number which on dividing 66,495 gives 554 as quotient and 15 as remainder.
13. Find the remainder when 54,978 is divided by 113 and the quotient is 486.
14. Find the least number which is to be added to 1,193 so that the sum is exactly divisible by 214.
15. Find the least number which is to be subtracted from 21,325 so that the difference is exactly
divisible by 126.
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