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 ,√3AND THEIR REPRESENTATION ON THE NUMBER LINE AND CONVERSELY, EVERY POINT ON THE NUMBER LINE REPRESENTS A UNIQUE REAL NUMBER.

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.

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.

(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.

(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.

For all positive real numbers a and b, we have:

(i) √ab=√a x √b

(ii)√a/b=√a/√b

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 aninteger is a rational number ,since an integer can be written as a/1.

If a/b and c/d are two rational numbers then their sum and product are given bya/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.

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)

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.

(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.