The real numbers consist of rational numbers like integers, fractions, and repeating decimals as well as irrational numbers like square roots and pi. Real numbers have the properties that between any two real numbers there are an infinite number of real numbers, and real numbers are closed under addition, subtraction, multiplication, and division except by zero. Rational numbers can be written as fractions of integers, while irrational numbers cannot be written as fractions.

1. WWELCOMEWELCOME

2. REAL NUMBERSREAL NUMBERS
S b itt d bSubmitted by
Sh YShareena .Y
M th tiMathematics
R N 18014350013Reg.No. 18014350013

3. Real NumbersReal Numbers
The set of Real
Numbers consists of
all rational and
irrational numbers.

4. E l f R lExamples of Real
NumbersNumbers
1 11 1
2,2, , , 2, 3, 5,.............
2 3

2 3

5. Properties of Real NumbersProperties of Real Numbers
The sets N, Z,Q and Q are subsets of RThe sets N, Z,Q and Q are subsets of R
N-Natural Number, Z – Integers
R Rational Numbers Q  Irrational NumbersR- Rational Numbers, Q - Irrational Numbers
R – Real Numbers
 l b iBetween any two real numbers we can write as
many real numbers as we alike.
The set of real numbers is closed under addition,
subtraction, multiplication, division (except division
b )by zero)

6. Real Numbers
Rational Numbers Irrational Numbers
Integers Non
Negative Positive
integers
Negative Positive
Negative
Integers
Zero
N t l
Negative Positive
Natural
numbers

7. Rational NumbersRational Numbers
A b hi h b
P
A number which can be
expressed in the form of p
q
expressed in the form of p,
P/q where p,qz and q  0P/q where p,qz and q  0
is called a rational number

8. Examples of Rational NumbersExamples of Rational Numbers
3 is a rational number since 633 is a rational number since,
where 6,2z and 2 0.
63
2

is a rational number since 3,5z
d 5 0
3
5
and 5  0.
is a rational number since ‐2,3z2 is a rational number since 2,3z
and 3  0
2
3


9. Equivalent Rational NumbersEquivalent Rational Numbers
The rational numbers do not have aThe rational numbers do not have a
unique representation in the formq p
where p and q are integers andP
q
q  0.
1 2 1 0 3 0 1 31 2 1 0 3 0 1 3:
2 4 2 0 6 0 2 6
e g    

10. Irrational NumbersIrrational Numbers
Numbers which are not rational butNumbers which are not rational but
which can be represented by points on
th b li ith th ti lthe number line with the rational
numbers are called irrational numbers.
In other words, a number ‘s’ is called
irrational, if it cannot be written in theirrational, if it cannot be written in the
form p/q, where p and q are integers
and q0and q0.

11. Examples of irrational p
numbers
1 12 2 2 3 51 12 ,2 , , , 2 , 3 , 5..........
2 3

are irrational numbers.