Download Chapter 3: Numbers

ICSE - Class 8 - Mathematics
Chapter 3: Numbers (Lecture Notes)
Natural Numbers: The counting numbers are called natural numbers.
N = {1, 2, 3, 4, 5, 6, 7 …}
1
2
3
4
5
6
7
Whole Numbers: All natural numbers together along with 0 (zero) form the set W of all whole
numbers.
W = {0, 1, 2, 3, 4, 5, 6, 7 …}
0
1
2
3
4
5
6
Integers: All natural numbers, negative natural numbers and 0 (zero) together form a set Z of all
Integers.
Z = {…, -4, -3, -2, -1, 0, 1, 2, 3, 4 …}
-4
-3
-2
-1
0
1
2
3
Properties of Addition of Integers
1. The sum of two integers is always an integer
2. If a and b are two integers, then:
a+b=b+a
3. Associative Law: If a, b, and c are integers, then:
(a + b) + c = a + (b + c)
4. Existence of Additive Identity: 0 is the additive identity for integers. Therefore for any
integer a we will have:
a+0=0+a=a
5. Existence of additive inverse: For each integer a, there exists another integer (-a) such
that a + (-a) = 0
1
For more information please go to: https://icsemath.com/
Note: If a be any integer, then (a+1) will be called the successor of a and (a + 1) is call
the predecessor of a is (a-1)
Properties of Subtraction
1. If a and b are two integers, then (a – b) is also an integer.
2. For any integer a, we have (a – 0) = a. But (0 – a) ≠ a
3. If a, b, c are integers and a > b, then
(a –c) > (b – c)
Properties of Multiplication of Integers
1. Closure Property: The product of two integer is always an integer
2. Commutative Law: For any two integers a and b we have:
a×b=b×a
3. Associative Law: For any three integers a, b, and c we have
(a × b) × c = (a × (b × c)
4. Distributive Law of Multiplication over Addition: For any there integers a, b, and c we
have
a × (b + c) = a × b + a × c
5. Existence of Multiplicative Identity: The integer 1 is a multiplicative identity for
Integers. So, for any integer a we have:
a×1=1×a=a
6. Property of Zero: For any integer we have:
a×0=0×a=0
Properties of Multiplication of Integers
𝑎
1. If a and b are integers, then ( ) is not necessarily an integer.
𝑏
𝑎
2. If a is an integer and a ≠ 0, then ( ) = 1
𝑏
𝑎
3. If a is an integer, then ( ) = 1
1
0
𝑎
4. If a is a non-zero integer, then ( ) = 0 but ( ) is not defined.
𝑎
0
𝑎
𝑏
𝑏
𝑐
5. If a, b, c are integers, then ( ) ÷ c ≠ a ÷ ( ), unless c = 1
6. If a, b, c, are integers and a > b then
a. (a ÷ c) > (b ÷ c), if c is positive
b. (a ÷ c) < (b ÷ c), if c is negative
2
For more information please go to: https://icsemath.com/
Rational Numbers
𝑝
The numbers that can be expressed in the form , where p and q are integers and q ≠ 0 are
𝑞
called rational numbers. Therefore, the set Q of all rational numbers is given by
𝑝
Q = { : p. q Z and q ≠ 0}
𝑞
Examples:
2
5
,
1
19
,
−1
5
,
−117
−11
,
0
5
,
23
15
Note:
 Every integer is a rational number, since every integer a can be written as
 Every fraction is a rational number
 The square root of every perfect square is a rational number
3
E.g., √9 , √8
 Every terminating decimal is a rational number
E.g., 0.9 =
𝑎
1
9
10
 Every recurring decimal is a rational number
E.g., 0.222222 =
2
9
Positive Rational Numbers: A rational number is said to be positive the numerator or
2
−2
5
−5
denominator are either both positive and both negative. e.g., ,
Negative Rational Numbers: A rational number is said to be negative is any one of the
numerator or denominator is negative (they are of opposite sign) e.g.,
Equivalent Rational Numbers
𝑝
If is a rational number and m is a non-zero integer, then we have
𝑞
numbers
𝑝
𝑞
and
𝑝𝑚
𝑞𝑚
𝑝
𝑞
−1
5
=
,
1
−5
𝑝𝑚
𝑞𝑚
. We call these
as equivalent rational numbers.
Standard form of a Rational Numbers
𝑝
A rational number is said to be a standard rational number form if p and q are integers having
𝑞
no common divisors other than 1 and q is positive.
3
For more information please go to: https://icsemath.com/
Note: Every rational number can also be represented on a number line.
Addition of Rational Numbers
1. In order to add rational numbers, we must first convert them into rational numbers
with positive denominator.
2. When the rational numbers have the same denominator, then these numbers can be
added as follows:
𝑝 𝑚
𝑝+𝑚
+
=
𝑞
𝑞
𝑞
3. If the denominator is not same (or unequal), we find the LCM of the denominator and
express each one of the rational numbers having the LCM as its denominator. You
could use the following approach and reduce the number once you add them.
𝑝 𝑚
𝑝 ×𝑛+𝑚 ×𝑞
+
=
𝑞
𝑛
𝑞 ×𝑛
Properties of Addition of Rational Numbers
1. Closure Property: The sum of two rational numbers is also a rational number. That is,
𝑝
𝑚
𝑝
𝑚
if 𝑎𝑛𝑑 are rational numbers then + is also a rational number.
𝑞
𝑛
𝑞
2. Commutative Law: If
3. Associative Law: If
𝑝
𝑞
𝑝
𝑎𝑛𝑑
𝑞
𝑚 𝑥
,
𝑚
𝑛
𝑛
𝑝
are rational numbers, then +
𝑝
𝑚
𝑞
𝑛
𝑚
, are rational numbers, then ( + ) +
𝑛 𝑦
𝑞
𝑛
=
𝑥
𝑦
𝑚
𝑛
=
+
𝑝
𝑞
𝑝
𝑞
+(
𝑚
𝑛
𝑥
+ )
𝑦
4. Existence of Additive Identity: The rational number 0 is the additive identity for any
𝑝
𝑝
𝑝
rational number. Therefore ( + 0) = (0 + ) =
𝑞
𝑞
𝑞
Subtraction of Rational Numbers
𝑝
𝑚
1. If 𝑎𝑛𝑑 are two rational numbers with the same denominator, then
𝑞
𝑞
𝑝 𝑚
𝑝− 𝑚
−
=
𝑞
𝑞
𝑞 ×𝑛
2. If the denominator is not same (or unequal), we find the LCM of the denominator and
express each one of the rational numbers having the LCM as its denominator. You
could use the following approach and reduce the number once you subtract them.
𝑝 𝑚
𝑝 ×𝑛−𝑚 ×𝑞
−
=
𝑞
𝑛
𝑞 ×𝑛
4
For more information please go to: https://icsemath.com/
Multiplication of Rational Numbers
The product of two rational numbers
𝑝
and
𝑞
𝑚
𝑛
𝑝
𝑚
𝑞
𝑛
is given by ×
=
𝑝 ×𝑚
𝑞 ×𝑛
Properties of Multiplication
1. Closure Property: The product of two rational numbers is also a rational number.
𝑎
𝑐
𝑎×𝑐
𝑏
𝑑
𝑏×𝑑
Therefore if and are two rational numbers, then
2. Commutative Law: If
3. Associative Law: If
𝑝
𝑞
𝑝
𝑎𝑛𝑑
𝑞
,
𝑚
𝑛
𝑚
𝑛
𝑎𝑛𝑑
is also a rational number.
𝑝
are rational numbers, then ×
𝑚
𝑞
𝑥
𝑛
𝑝
𝑚
=
𝑛
×
𝑚
𝑝
𝑞
𝑥
𝑝
are rational numbers, then ( × ) × =
𝑦
𝑞
𝑛
𝑦
𝑞
×(
𝑚
𝑛
𝑥
× )
𝑦
4. Existence of Multiplicative Identity: The rational number 1 is a multiplicative identity for
𝑎
𝑎
𝑎
𝑎
rational numbers. For any rational number we have × 1 = 1 × =
𝑏
𝑏
𝑏
𝑏
5. Existence of Multiplicative Inverse: For every non zero rational number, there exists a
𝑎
𝑏
𝑏
𝑎
multiplicative inverse. For a rational number the multiplicative inverse would be such
that
𝑎
𝑏
×
𝑏
𝑎
=1
6. Distributive Law of Multiplication over Addition: For any three
𝑝
𝑚
𝑥
𝑝
×( + )=( ×
𝑞
𝑛
𝑦
𝑞
𝑚
𝑛
)+(
𝑝
𝑞
𝑝
𝑞
,
𝑚
𝑛
𝑥
𝑎𝑛𝑑 , we have
𝑦
𝑥
× )
𝑦
𝑎
7. Multiplicative Property of Zero: For any rational number that we have
𝑏
𝑎
𝑎
×0=0 × =0
𝑏
𝑏
Division of Rational Numbers
If
𝑎
𝑏
𝑐
𝑐
and are two rational numbers such that ≠ 0. Then, we define
𝑎
𝑑
𝑐
𝑎
𝑑
𝑏
𝑑
When is divided by , then is called the dividend,
𝑏
𝑐
𝑑
𝑎
𝑏
÷
𝑐
𝑑
=
𝑎
𝑏
×
𝑑
𝑐
is called the divisor and the result of the
division is called the quotient.
Properties of Division of Rational Numbers
1. Closure Property: The division of two rational numbers is also a rational number.
𝑎
𝑐
𝑎
𝑐
Therefore if and are two rational numbers, then ÷ is also a rational number
𝑐
𝑏
𝑑
𝑏
𝑑
given ≠ 0.
𝑑
𝑎
𝑎
𝑏
𝑏
2. For any rational number , we have
÷1 =
𝑎
𝑏
5
For more information please go to: https://icsemath.com/
3. For any non-zero rational number
𝑎
𝑏
, we have
𝑎
𝑏
÷
𝑎
𝑏
=1
How to find a Rational Number between Two Given Rational Numbers?
𝑎
𝑐
Let there be two rational numbers and . Then the rational number in between the two would
𝑏
1 𝑎
𝑐
2 𝑏
𝑑
𝑑
be ( + ) .
If you want to find a large number of rational numbers between two given rational numbers,
then first make the denominator equal by taking LCM. After that you can just insert values
between the two numerators.
Irrational Numbers
A number that can neither be expressed as a terminating nor as a repeating decimals, is called
an irrational number. These are non-terminating and non-repeating numbers. Examples
 Non-terminating and non-repeating numbers such as
 0.82485863738485738…. or 1.2456345789560….
 Square root of positive integers that are not perfect squares
 √2 , √5 , √7 …
 Cube root of numbers that are not perfect cubes
3
3
3
 √2 , √3 , √4 …
 Number π is irrational. It has a value that is non-terminating and non-repeating. We
only approximate it to
22
7
.
Properties of Irrational Numbers
1. Sum of two irrational numbers need not be an irrational number
a. Take (5 + √2 ) and (5 − √2 ) as two irrational numbers. The sum is 10 which
is a rational number.
2. Difference of two irrational numbers need not be an irrational number
a. Take (5 + √2 ) and (7 + √2 ) as two irrational numbers. The difference is 12
which is a rational number
3. Product of two irrational numbers need not be an irrational number
a. Take (5 + √2 ) and (5 − √2 ) as two irrational numbers. The product is 23
which is a rational number.
4. Quotient of two irrational numbers need not be an irrational number
a. Take (9√2 ) and (3√2 ) as two irrational numbers. The quotient is 3 which is a
rational number.
6
For more information please go to: https://icsemath.com/
5. Commutative Law:
𝑝
𝑚
𝑝
𝑚
a. If 𝑎𝑛𝑑 are irrational numbers, then × =
b. If
𝑞
𝑝
𝑞
𝑎𝑛𝑑
𝑛
𝑚
𝑛
𝑞
𝑛
𝑝
𝑚
𝑞
𝑛
are rational numbers, then +
𝑚
=
𝑛
𝑚
𝑛
𝑝
×
+
𝑞
𝑝
𝑞
6. Associative Law
a. If
b. If
𝑝
𝑞
𝑝
𝑞
,
,
𝑚
𝑛
𝑚
𝑛
𝑎𝑛𝑑
𝑎𝑛𝑑
𝑥
𝑝
𝑥
𝑦
𝑞
𝑚
𝑥
𝑥
𝑝
𝑥
𝑞
𝑝
𝑝
𝑚
are irrational numbers, then ( + ) + =
𝑞
𝑛
𝑦
7. Distributive Law
a. For any three irrational numbers
𝑝
𝑚
are irrational numbers, then ( × ) × =
𝑦
𝑞
𝑛
𝑦
𝑝
×( + )=( ×
𝑛
𝑦
𝑞
𝑚
𝑛
)+(
𝑝
𝑞
𝑝
𝑞
,
𝑚
𝑛
𝑞
×(
𝑚
𝑛
𝑚
𝑥
× )
𝑦
𝑥
+( + )
𝑛
𝑦
𝑥
𝑎𝑛𝑑 , we have
𝑦
𝑥
× )
𝑦
8. The sum or difference of a rational and an irrational numbers is always irrational.
a. Example 2 + √5 is irrational
b. Example 2 - √5 is irrational
9. The product or quotient of a rational and an irrational numbers is always irrational
a. Example 2 × √5 is irrational
b. Example 2 ÷ √5 is irrational
Real Numbers
All rational and irrational numbers forms the set of all real numbers.
Real number Line
 The real number line represents all real numbers.
 On a real number line, each point corresponds to a point and conversely, every point
on the real number line corresponds to a real number.
 Between any two real numbers, there exist infinite real numbers.
Properties of Real Numbers
1. Closure property
 The sum of two real numbers is always a real number.
 The product of two real numbers is always a real number.
2. Commutative law
 𝑎+𝑏 =𝑏+𝑎
 𝑎 ×𝑏 =𝑏 ×𝑎
3. Associative law
7
For more information please go to: https://icsemath.com/
 (𝑎 + 𝑏 ) + 𝑐 = 𝑎 + ( 𝑏 + 𝑐)
 (𝑎 × 𝑏) × 𝑐 = 𝑎 × ( 𝑏 × 𝑐)
4. Distributive Laws of multiplication over addition
 𝑎 × (𝑏 + 𝑐) = 𝑎 × 𝑏 + 𝑎 × 𝑐
5. If a ≠ 0, then
1
𝑎
is called the reciprocal of a and 𝑎 ×
1
𝑎
=1
Comparison of two irrational numbers
If a and b are two irrational numbers such that a < b, then √𝑎 < √𝑏
8
For more information please go to: https://icsemath.com/