3. REVIEW OF REPRESENTATION OF NATURAL NUMBERS,
INTEGERS, RATIONAL NUMBERS ON THE NUMBER LINE.
REPRESENTATION OF TERMINATING/NON TERMINATING
RECCURING DECIMALS,ON THE NUMBER LINE THROUGH
SUCCESSIVE MAGNIFICATION.RATIONAL NUMBERS AS
RECCURING/TERMINATING DECIMALS.
EXAMPLES OF NONRECCURING/NON TERMINATING
DECIMALS SUCH AS √2, √3, √5 ETC. EXISTENCE OF NON-
RATIONAL NUMBERS (IRRATIONAL NUMBERS) SUCH √2 ,√3
AND THEIR REPRESENTATION ON THE NUMBER LINE AND
CONVERSELY, EVERY POINT ON THE NUMBER LINE
REPRESENTS A UNIQUE REAL NUMBER.

4.  Natural Numbers-Counting
numbers are known as
natural numbers.
Thus,1,2,3,4,5,6,7,…..,etc., are
all natural numbers.
 Whole Numbers-All
natural numbers together
with 0 form the collection
of all whole numbers.
Thus,1,2,3,4,5,6,7,….,etc., are
all whole numbers.
 Integers-All natural
numbers,0 and negatives of
natural numbers form the
collection of all integers.
Thus,..., -5, -4, -3, -2, -1, 0, 1, 2, 3,
4, 5,…..,etc., are all integers.

5. On the number line, each point corresponds to
a unique real number. And ,every real number
can be represented by a unique point on the
real line.
Between any two real numbers, there exist
infinitely many real numbers.

6. (i) CLOSURE PROPERTY- The sum of two real numbers
is always a real number.
(ii) ASSOCIATIVE LAW- (a+b)+c=a+(b+c) for all real
numbers a, b, c.
(iii) COMMUTATIVE LAW- a+b=b+a for all real numbers a
and b.
(iv) EXISTENCE OF ADDITIVE IDENTITY- Clearly 0 is a
real number such that 0+a=a+0=a for every real
number a.
0 is called the additive identity for real numbers.
(v) EXISTENCE OF ADDITIVE INVERSE-For each real
number a ,there exists a real number(-a)such that a+(-
a)=(-a)+a=0.
a and(-a) are called the additive inverse of each other.

7. (i) CLOSURE PROPERTY-The product of two real numbers
is always a real number.
(ii) ASSOCIATIVE LAW- (ab)c=a(bc) for all real numbers
a,b,c.
(iii)COMMUTATIVE LAW-ab=ba for all real numbers a and b.
(iv)EXISTENCE OF MULTIPLICATIVE IDENTITY-Clearly, 1 is a
real number such that 1 x a=a x 1=a for every real number a.
1 is called the multiplicative identity for real numbers.
(v) EXISTENCE OF MULTIPLICATIVE INVERSE –For each
nonzero real number a, there exists a real number
(1/a)such that a x 1/a=1/a x a=1. a and 1/a are called the
multiplicative inverse of each other.

8. For all positive real numbers a
and b, we have:
(i) √ab=√a x √b
(ii)√a/b=√a/√b

9. The numbers in the form p/q, where p and q are integers
and q≠0,are known as rational numbers.
REMARKS- (i) 0 is a rational number, since we can
write,0=0/1.
(ii) Every natural number is a rational
number, since we can write, 1=1/1, 2=2/1,
3=3/1, etc.
(iii) Every integer is a rational number ,since an
integer is a rational number ,since an
integer can be written as a/1.

10. If a/b and c/d are two rational numbers
then their sum and product are given by
a/b + c/d= a x d + b x c/b x d and a/b x c/d=
a x c/b x d.
a/b and c/d are said to be equal if ad=bc.

11. Rational numbers follow the commutative and associative law
of addition and multiplication .They also follow the
distributive law of multiplication over addition.
If a, b and c are three numbers, then
a+b=b+a (commutative law of addition)
a x b=b x a (commutative law of multiplication)
a+(b+c)=(a+b)+c (associative law of addition)
a x (b x c)=(a x b)x c (associative law of multiplication)
a x (b+c)=a x b + a x c (law of distribution)
(b +c)a=b x a + c x a (law of distribution)

12. A number which can neither be
expressed as a terminating decimal
nor as a repeating decimal, is called
an irrational number.
Thus, non terminating, non
repeating decimals are irrational
numbers.

13. (i)Irrational numbers satisfy the commutative,
associative and distributive laws for addition and
multiplication.
(ii)-(i)Sum of a rational and irrational is irrational.
(ii)Difference of a rational and an irrational is
irrational.
(iii)Product of a rational and an irrational is
irrational.
(iv)Quotient of a rational and an irrational is
irrational.