1.1
Introduction
to Numbers:
1.1.1Natural Numbers
Let us look at some objects and the numbers
associated with them, We have 1 head, 2
eyes. There are 4 vedas (Rugveda, Yajurveda, samaveda and Atharvaveda)
Hand has 5
fingers. Indian music has 7 Svaras ( Sa Ri Ga Ma Pa Da Ni Sa), 10
fingers . . .
These type of numbers
which we come across in our daily life are called natural Numbers. Few more
examples are 3, 6, 8,9 …100…1000 and so on
The set (group) of natural numbers is
conventionally denoted by N.
N = {1, 2,
3…..}
Note : 0 is not a natural number(why? : Can we associate/identify an object with 0?
Properties
of Natural Numbers
Observe the result of addition and multiplication
of 2 natural numbers: 3+2 = 5, 3*4 =12
We notice that addition and multiplication of two
natural numbers is again a natural number. Then, we say that set of natural
numbers satisfies ‘closure property’ with
respect to addition and multiplication.
Note that 32 = 1 which is a natural number, but
34 = 1 is not a natural number.
Since the subtraction of two natural numbers does not always result in a natural number, subtraction does not satisfy closure property.
Also note that 4/2 =2 which is natural number but 2/4 =
1/2 is not a natural number
Since the division of two natural numbers does not always result in a natural number, division does not satisfy closure property.
Have you observed the following?
2+3 = 3+2 , 4+5 = 5+4
Thus for any two numbers a and b a+b = b+a
This property is true for all natural numbers and
this property is called ‘commutative property’ with
respect to addition
Notice 32 _{} 23 and 54 _{} 45
Thus
subtraction does not satisfy commutative
property.
2*3 =3*2, 6*5 =5*6
Thus for any two numbers a and b a*b = b*a_{}
This property is true for all natural numbers and
this property is called ‘commutative property’
with respect to multiplication.
4÷2 _{}2÷4 , 3÷2 _{} 2÷3
Thus
division does not satisfy commutative property.
(2+3)+6 = 2+(3+6) , (4+5)+8 =4+(5+8)
Thus for any three numbers a, b and c (a+b)+c = a+(b+c)
This property is true for all natural numbers and
this property is called ‘associative property’
with respect to addition
(43)2 = 3
4(32) = 1
_{} (43)2 _{}4(32)
Thus
subtraction does not satisfy associative
property
(2*3)*6 = 2*(3*6) , (4*5)*8 =4*(5*8)
Thus for any three numbers a, b and c (a*b)*c = a*(b*c)
This property is true for all natural numbers and
this property is called ‘associative property’
with respect to multiplication.
(8÷2) ÷2
= 4÷2 =2
8÷ (2÷2) = 8÷1 =8
Since both results are different,
division does not satisfy associative property
Hence for any 3 numbers a, b and c: (a÷b) ÷c _{}a÷ (b÷c)
1.1.2 Whole Numbers:
We have observed that 0 is not a natural number,
but what is special about this number?
1+0 = 1, 20
=2, 2*0 =0 and 3÷0 is undefined.
You must have heard that Zero is
The 0 and group of natural numbers together is
called set(group) of whole numbers and is denoted by
W = {0, 1, 2, 3, 4…..}
Whatever the properties natural numbers have, same
properties hold good for whole numbers also.
Properties that do not hold good for natural
numbers will not hold good for whole numbers also.
Note :
0 is the ‘identity
element’ of addition because;
addition of 0 to any number does not change the value of the number
1 is the ‘identity
element’ of multiplication because;
multiplication of any number by 1 does not change the value of the number
1.1.3 Integers:
We read in news papers that a city has recorded a
temperature of 5^{0}C. This means that the temperature is 5^{0}
C below 0^{0}.
How do we understand such numbers which we
encounter in our daily life, better?
The integers {1, 2, 3, 4 ….} are called set of positive
integers.
The integers {4,3, 2, 1 ….} are called set of negative
integers.
By convention 0 is neither a positive number nor a negative number.
The group Z = {…..4, 3, 2, 1, 0, 1,
2, 3, 4….} is called set of integers.
Note that this set consists of positive integers,
negative integers and zero.
The integers can be represented on a number line as
shown below.
Note that all positive integers are represented to
the right of 0 and all negative integers are represented to the left of 0.
If numbers are represented on the number line as
given below
What is the distance between 2 and 0 and distance
between 2 and 0?
In both the cases the distance is same and it is 2
units.
Definition: ‘Absolute
value’ of any number is the same number with + sign. The absolute
value of positive number is the number itself. Absolute value of negative
number is the same number with positive sign. If x is a number its absolute
value is denoted by x
Ex. Absolute value of +5 is +5 or 5 and absolute value of 5 is +5 or 5
Absolute value of positive number = Absolute value
of negative number
We can symbolically say:
1.1.3.1
Addition of integers:
1)
Addition of two positive integers: (+1) + (+4 ) We
observe from the diagram that (+1) + (+4) = +5 To
add two positive integers, we add the two and give the sum a positive sign. 

2)
Addition of two negative integers: (1)
+ (4) We
observe from the diagram that (1) + (4) = 5 = To
add two negative integers, we add their absolute values and give the sum a negative sign.  {1 + +4} =  (1+4) = 5 

3) Addition of positive and negative integers: 

3.1) (+5) +
(3 ) We
observe from the diagram that (+5) + (3) = +2 To
add one positive integer and another negative integer find the difference
between their absolute values and give the sign of the integer having greater
absolute value, to the difference. 

3.2) (4) + (+2 ) We
observe from the diagram that (4) + (+2) = 2 To
add one positive integer and another negative integer find the difference
between their absolute values and give the sign of the integer having greater
absolute value, to the difference. 

Explanation:
3.1) Let us take the case of (+5) + (3 ). Their absolute values are 5 and 3. The
difference between these two is 2(=53). The integer having greater absolute
value is +5 whose sign is +
Therefore the result will have positive
sign.
(+5) + (3 ) = Sign of +5[ Absolute value of (+5) – Absolute value of
(3)] = +[ 5 – 3]= +2 : +[+5  3] = + [ 53] = 2
3.2) Let us take the case
of (4) + (+2 ). Their
absolute values are 4 and 2. The difference between these two is 2(=42). The
integer having greater absolute value is 4 whose sign is 
Therefore the result will have negative
sign
(4) + (+2 ) = Sign of 4[Absolute value of (4) – Absolute value of
(+2)] =  [ 4 – 2] =  2 : [4  +2] =  [42] = 2
Observations:
1. The sum of two positive integers
is positive.
2. The sum of two negative integers
is negative.
3. The sum of a positive integer and
a negative integer is positive if the +ve
integer is greater than the absolute value of the –ve
integer.
4. The sum of a positive integer and
a negative integer is negative if the absolute value of
the ve
integer is greater than the +ve integer.
1.1.3.2
Subtraction of integers:
Note the following:
Statement 
Conversion 
Simplification 
Result 
Subtract 3 from 5

5
3 
5
+ (3) 
 8 
Subtract 3 from
5 
5
3 
5 + (3) 
2 
Subtract
3 from 5 
5
–(3) 
5
+ (+3) 
2 
Subtract 3 from
5 
5
– (3) 
5 + (+3) 
8 
Subtraction of integers is done in two steps as follows:
1. Change the sign of subtrahend
2. Add by following the rules of
addition of integers.
1.1.3.3
Multiplication of integers:
We know that multiplication
is a repetitive process of addition. Let us see the result of product of
numbers when multiplier/multiplicand is negative.
1.1.3.3.1
When negative number is multiplied by positive number:
When 
Multiplier 
Multiplicand 
Product 
Result 

Multiplier is Reduced by 1 
3 
2 
6 
Product reduces by 2 

2 
2 
4 

1 
2 
2 

0 
2 
0 

1 
2 
2 

2 
2 
4 

3 
2 
6 
Product is negative.
1.1.3.3.2
When positive number is
multiplied by negative number:
When 
Multiplier 
Multiplicand 
Product 
Result 

Multiplicand is Reduced by 1 
2 
3 
6 
Product reduces by 2 

2 
2 
4 

2 
1 
2 

2 
0 
0 

2 
1 
2 

2 
2 
4 

2 
3 
6 
Product is negative.
1.1.3.3.3
When negative number is
multiplied by negative number:
When 
Multiplier 
Multiplicand 
Product 
Result 

Multiplier is Reduced by 1 
3 
2 
6 
Product increases by 2. 

2 
2 
4 

1 
2 
2 

0 
2 
0 

1 
2 
+2 

2 
2 
+4 

3 
2 
+6 
Product
is positive.
We know Quotient = Dividend/Divisor
_{} Quotient*Divisor =
Dividend
The rules of multiplication applicable to negative
numbers, apply for divisions when Dividend or Divisor is negative.
1.1.3.4
Multiplication/Division of integers:
Operand 
Operation 
Operand 
Result 
+ 
Multiplication Or Division 
+ 
+ 
+ 
 
 

 
+ 
 

 
 
+ 
The above table can be expanded to be read as:
1 
If a positive integer 
is multiplied by 
another positive integer 
then the product is positive 
2 
If a positive integer 
another negative integer 
then the product is negative 

3 
If a negative integer 
another positive integer 
then the product is negative 

4 
If a negative integer 
another negative integer 
then the product is positive 
1 
If a positive integer 
is divided by 
another positive integer 
then the quotient is positive 
2 
If a positive integer 
another negative integer 
then the quotient is negative 

3 
If a negative integer 
another positive integer 
then the quotient is negative 

4 
If a negative integer 
another negative integer 
then the quotient is positive 
Example
: Simplify {(10)*(8)*(4)} ÷{(5)*(+2)}
(10)*(8) = +80 ( _{} product of 2 –negative
integers is positive)
(10)*(8)*(+4) = (+80)*(4) =  320( _{} product of positive and negative integers is negative)
(5)*(+2) = 10( _{} product of negative and positive integers is negative)
_{}{(10)*(8)*(4)} ÷{(5)*(+2)}
= (320) ÷ (10) = +32 ( _{} quotient of 2 –negative integers is positive)
1.1.3.5
Properties of Integers:
We notice:
4+ 3 = 7 : 4+(4)= 0: 4*3
= 12 : (4)*(+3) = 12
Also 34 = 1 is
an integer: 33 =0 is an integer
We notice that addition, subtraction and
multiplication of two integers is again an integer.
Thus, set of integers satisfy closure property with respect to addition,
subtraction and multiplication
4÷2 =2 but 2÷4 is not an integer
Since result of division of two integers is not
always an integer division does not
satisfy closure property.
We also notice the following:
For any two integers a and b a+b
= b+a
For any two integers a and b a*b = b*a
For any three integers a, b and c, (a+b)+c = a+(b+c)
For any three integers a, b and c, (a*b)*c =
a*(b*c)
Thus, set of integers satisfy
commutative and associative properties with respect to addition and
multiplication.
Like in the case of whole numbers, 0 is the
identity element for addition and 1 is the identity element for multiplication.
Can we find a corresponding integer for every
integer such that their sum is zero?
(3,3), (4, 4) are such pairs of integers such
that their sum is zero. An integer is called the
‘additive inverse’ if the sum of that integer and its additive
inverse is zero.
(Integer + Its additive inverse = 0)
3 is the additive inverse of 3, 3 is additive
inverse of 3.
1.1.4 Rational Numbers:
We have come across fractions such as 1/2, 1/4 ,1/2,3/4…….
They are all got by dividing an integer by another non zero integer.
Definition:
Rational numbers are set of numbers represented by the form _{} in the simplest form, where p and q are integers with q_{}0. The set is denoted by Q.
Q = { 1, 3/4 2/3 1/2, , 0, 1/4 ,1/2, 3/4,2,4…..} or symbolically
Q = { _{}: p, q _{}Z and q_{}0}
Like integers rational numbers can also be represented
on a number line as shown below:
Operations
on Rational numbers:
Before any operation, convert mixed number into
improper fraction if the number given
is a mixed number.
General
procedure for Addition/Subtraction:
In order to add/subtract two rationals
we need to have same denominators for both the rationals
Step1 : Find LCM of
denominators
Step2 : Convert the given rationals to rationals of same denominator
Step3 : Add/Subtract
numerators and then divide the result
by the common denominator.
Step4 : Simplify if necessary.
1.1.4 Problem 1: Add 7/5 and
2/3
Solution:
LCM of 5 and 3 is 15.
7/5 =
7/5*3/3 = 21/15
2/3 = 2/3*5/5 = 10/15
_{} 7/5 + (2/3) = 21/15
10/15 = (2110)/15 = 11/15
1.1.4 Problem 2: Subtract 3/2
from 8/5
Solution:
LCM of 2 and 5 is 10
8/5 = 8/5*2/2 = 16/10(note 8*10 = 5*16)
3/2 = 3/2*5/5 = 15/10 (note 3*10 = 2*15)
_{}8/5 (3/2) = 16/10+15/10 =
(16+15)/10 = 1/10
Note: We observe, if a/b and c/d
are two rational numbers then
1. They are equal if and only if ad=bc
2. a/b >
c/d implies(means) ad>bc( Ex 1/2 > 3/7)
3. a/b <
c/d implies ad<bc(Ex 1/2 < 5/9)
4. Every integer is a rational number
(If n is a integer then n =n/1)
5. Between two rational numbers there
exists many (infinite) rational numbers.
(If a and b are
rational numbers then (a+b)/2, (a+b)/3,
(a+b)/4….. (a+b)/n
are all rational numbers.)
General
procedure for multiplication:
Step 1: Find
the product of numerators and denominators
Step 2: Divide the product of numerators by the
product of denominators and simplify the result if necessary
1.1.4 Problem 3: Multiply 2/7 by 8/11.
Solution:
Product of numerators = 2*(8) = 16
Product of denominators = 7*11 = 77
_{}2/7 *8/11 = 16/77
General
procedure for Division:
Step 1 : Find out the
multiplicative inverse of divisor.
Step 2 : Multiply dividend
by the number arrived in step1
1.1.4 Problem 4: Divide 2(1/2) by
8/15
Solution:
1. Convert mixed dividend to improper fraction
2(1/2) = 5/2
2. The multiplicative inverse of divisor is 15/8
_{} 2(1/2) ÷
(8/15) =(5/2) *(15/8) = 75/16
Grouping of numbers:
Let us group the different types of numbers studied
so far. Let few numbers be
1 3/4 7/8  5/6 6/5
5 7 1/3 8
0 100 3 5
1) First we have the group of natural numbers: N =
{ 1 5 7 100 }
Let us
include 0 in to the above group.
2) Then we have the group
of whole numbers: W = {0 1 5 7 100 }
Let us include negative integers in to the above
group.
3) Then we have the group of Integers: Z = { 8 5
3 0 1 5 7 100 }
Let us include rational numbers in to the above
group.
4) Then we have the group of rational
numbers: Q = { 8 5
3 5/6 0 1/3 3/4 7/8 1 6/5 5 7 100 }
The adjacent figure represents the relationship
between set of natural numbers (N), set of whole numbers (W), set of integers
(Z) and set of rational numbers (Q)
We notice that N is a sub group of W which itself
is a sub group of Z which it self is a sub group of Q
Thus symbolically N_{}W_{} Z_{}Q.
Observations:
1. All the rules that apply for operations on set
of natural numbers hold good for set of whole numbers.
2. All the rules that apply for operations on set
of whole numbers hold good for set of integers.
3. All the rules that apply for operations on set
of integers hold good for set of rational numbers.
The operations which apply are
addition/subtraction/multiplication/divisions.
The rules which apply are closure, commutative and
associative properties.
Note: Set of rational numbers is not closed under division because (1/2) ÷0
is undefined.
Like in the case of natural numbers, set of
rational numbers is not commutative with
respect to subtraction and division.
For example 1/2 –1/4 = 1/4 and 1/4 1/2 =  1/4
Since 1/4 _{} 1/4
, 1/2 –1/4 _{} 1/4 1/2
For example
(1/2) ÷ (1/3) = 3/2 and (1/3) ÷ (1/2) = 2/3
Since 3/2_{} 2/3, (1/2) ÷ (1/3) _{}(1/3) ÷ (1/2)
Similarly associative property does not hold good for subtraction and division of
rational numbers.
Like in the case of integers
1. 0 is the identity element for addition of
rational numbers.
2. 1 is the identity element for multiplication of
rational numbers.
3. Every rational number has an additive inverse
(for every p/q, we have p/q as its additive inverse)
4 Every non zero rational number has a
multiplicative inverse (reciprocal.)
For example
(2/3)* (3/2) =1
Thus, (3/2) is the multiplicative inverse
(reciprocal) of (2/3)
1.1 Summary of learning
No 
Points
studied 
1 
Properties of numbers, whole
numbers, integers, rational numbers ( associative, commutative) 
2 
Representation of numbers on the
number line. 
Additional Points:
Distance
between two numbers on the number line:
If x and y are two numbers on the number line then
either x>y or y>x. Let us assume y>x then:
No 
Possible values of x and y 
Figure 
Distance between x and y 
1 
x>
0 and y>0 

yx 
2 
x<0
and y <0 

PQ = OPOQ = x(y) =x+y = yx 
3 
x<0
and y>0 

PQ = OP+OQ =x+y =yx 
The distance between two numbers x and y on the
number line = xy = yx
Let us consider the statement x _{} a
Since x = +x or –x
We have one of the following:
x _{} a or –x _{} a (i.e. x or –x is to the right of ‘a’ on number line)
Let us take the case when –x _{}a
We will show that this means x _{}a
Let us assume to the contrary that x > a (Note:
x can be in between –a and a or x can be greater than a)
(i) If x is in between –a
and a then –x also will be in between –a and a and hence –x < a which is not
what we started with
(ii) If x is > a then –x < a which is not
what we started with
Thus our assumption of x > a is not correct which means that x _{} a
Similarly x _{} a
implies that x _{} a.
1.1.4 Problem 4: Solve 8x5 _{} 3
Solution:
Case 1:
(8x5) _{} 3
i.e. 8x 5 +5 _{} 3+5 (Add 5 to both
sides)
i.e. 8x _{} 8
i.e. x _{} 1
Case 2:
(8x5) _{} 3
i.e. 8x +5 5 _{} 35 (Subtract 5 from
both sides)
i.e. 8x _{} 2
i.e. x _{} 2/(8) (Divide both sides by 8)
= (1/4)
i.e. x _{} 1/4