God gave us natural numbers, while other types of numbers were created by humans. The German mathematician Kronecker expressed that natural numbers play a significant role in human thought. The document then defines various types of numbers - natural numbers, whole numbers, integers, rational numbers, irrational numbers, and real numbers. It provides properties and examples for each number type. Rational numbers can be expressed as terminating or repeating decimals, while irrational numbers cannot be expressed as fractions. Together, rational and irrational numbers form the set of real numbers.

3. God gave us the natural number, all else is the work of man". It was
exclaimed by L. Kronecker (1823-1891), the reputed German
Mathematician. This statement reveals in a nut shell the significant
role of the universe of numbers played in the evolution of human
thought
.
N : The set of natural numbers
,
W : The set of whole numbers,
Z : The set of Integers, R
Q : The set of rationals,
R : The set of Real Numbers.
In our day to life we deal with different types of numbers which can

4. 1.The set of whole number is infinite (unlimited elements).
2.This set has the smallest members as '0'. i.e. '0' the
smallest whole number. i.e., set W contain '0' as a member.
3.The set of whole numbers has no largest member.
4.Every natural number is a whole number but every whol
number is not natural number.
5.Non-zero smallest whole number is '1'.
. Natural Numbers (N) : The counting numbers 1, 2, 3, ..... are known
as natural numbers. The collection of natural number is denoted by 'N'
N = {1, 2, 3, 4...
Whole numbers (W) : The number '0' together with the natural
numbers 1, 2, 3, .... are known as whole numbers. The collection
of whole number is denoted by 'W' W = {0, 1, 2, 3, 4...∞}

5. 1.This set Z is infinite.
2.It has neither the greatest nor the least element.
3.Every natural number is an integer.
4.Every whole number is an integer.
5.The set of non-negative integer = {0, 1, 2, 3, 4,....}
6.The set of non-positive integer = {......–4, – 3, – 2, –1, 0}
6.
3. Integers (I or Z) : All natural numbers, 0 and negative of natural
numbers are called integers. The collection of integers is denoted by
Z or I. Integers (I or Z) : I or Z = {–∞, ..... –3, –2, –1, 0, +1, +2, +3 ....
+∞} Positive integers : {1, 2, 3...}, Negative integers : {.... –4, –3, –2,
–1}
4. Rational numbers :– These are real numbers which can be expressed in
the form of pq , where p and q are integers and . Ex. (2over 3) ,(37over
15) , (-17over 19), –3, 0, 10, 4.33, 7.123123123.........
8.

6. 1.All natural numbers, whole numbers & integer are rational numbers.
2. Every terminating decimal is a rational number.
3.Every recurring decimal is a rational number.
4.A non- terminating repeating decimal is called a recurring decimal.
5. Between any two rational numbers there are an infinite number of
rational numbers.
6.This property is known as the density of rational numbers.
7.Every rational number can be represented either as a terminating
decimal or as a non-terminating repeating (recurring) decimals.
8. Types of rational numbers :– (a) Terminating decimal numbers and (b)
Non-terminating repeating (recurring) decimal numbers

7. 5
. Irrational numbers :– A number is called irrational
number, if it can not be written in the form p q , where p &
q are integers and . All Non-terminating & Non-repeating
decimal numbers are Irrational numbers.
6. Real numbers :– The totality of rational
numbers and irrational numbers is called the set of
real number i.e. rational numbers and irrational
numbers taken together are called real numbers.
Every real number is either a rational number or
an irrational number.

8. (A) 1st method : Find a rational number between x and y then, is a rational number lying
between x and y.
(B) 2nd method : Find n rational number between x and y (when x and y is non fraction number)
then we use formula.
(C) 3rd method : Find n rational number between x and y (when x and y is fraction Number)
then we use formula
then n rational number lying between x and y are (x + d), (x + 2d), (x + 3d) .....(x + nd)
Remark : x = First Rational Number, y = Second Rational Number, n = No. of Rational Number
FINDING RATIONAL NUMBERS BETWEEN
TWO NUMBERS

9. Every rational number can be expressed as terminating decimal or non-terminating decima
1. Terminating Decimal : The word "terminate" means "end". A decimal that ends is
terminating decimal.
O R
A terminating decial doesn't keep going. A terminating decimal will have a finite number
digits after the decimal point.
2. Non terminating & Repeating (Recurring decimal) :– A decimal in which a
digit or a set of finite number of digits repeats periodically is called Non-
terminating repeating (recurring) decimals.

11. NATURE OF THE DECIMAL EXPANSION OF RATIONAL NUMBERS
Theorem-1 : Let x be a rational number whose decimal
expansion terminates. Then we can express x in the
form (pover q), where p and q are co-primes, and the prime
factorisation of q is of the form 2m × 5n,
where m,n are non-negative integers.
Theorem-2 : Let x = (pover q) be a rational number, such that the
prime factorisation of q is of the form 2m × 5n, where m,n are
non-negative integers . Then, x has a decimal expansion which
terminates.

12. Theorem-3 : Let x = (pover q) be a rational number, such that
the prime factorisation of q is not of the form 2m× 5n, where m,n
are non-negative integers . Then, x has a decimal expansion
which is nonterminating repeating.

13. RATIONAL NUMBER IN ITS LOWEST
TERMS
 The numerator and denominator of such a rational
number in its lowest terms are co-primes. E.g
Express 45/60 in its lowest terms.
 Sol-H.C.F of 45 and 60=15
 45/60=3/4(dividing both terms by 15)

14. NEGATIVE RATIONAL NUMBERS
 If in a rational number,one out of its numerator and
the denominator is positive and the other is
negative, then it is called a negative rational
number. Eg—3/11,16/-17
 Zero rational number is 0=0/1=0/-1.

15. EQUIVALENT RATIONAL NUMBERS
 We know that the rational numbers do not have a
unique representation in the form p/q, where p & q
are integers and q is not equal to 0 such rational
numbers are called equivalent rationals(or
fractions).

16. EXERCISE TIME…………………
 Find six rational numbers between 3 and 4.
 Sol.
 we know that there are an infinite number of rational
numbers between 3 and 4 can be …
 Let x=3 and y=4 also n=6
 therefore, d=(y-x)/(n+1)=(4-3)/(6+1)=1/7
 since the six rational numbers between x and y are;
 (x+d),(x+2d),(x+3d),(x+4d),(x+5d),(x+6d).
 Therefore, the six rational numbers are 3 and 4 are;
 (3+1/7),(3+2/7),(3+3/7),(3+5/7) and (3+6/7)
 i.e. 22/7,23/7,24/7,25/7,26/7 and 27/7.

17.  Express o.99999 in the form of p/q.Are you
surprised by your answer?
Solution. Let x=0.99999…………………(1)
multiply both sides by 10,we have
10*x=10*(0.99999)
10x=9.9999…………………………………….(2)
Subtracting 1 from 2,we get
10x-x=(9.9999…)-(0.9999)
Or 9x=9
Or x=9/9=1
Thus,0.9999…=1
As 0.9999.. Goes on forever, there is nogap between 1 and
0.9999….
Hence both are equal.

18.  Simplify each of the following expressions;
Solution;
(i) (3+√3)(2+√2)
Solution. (3+√3)(2+√2)=2(3+√3)+√2(3+√3)
=( 2*3+2√3)+(3√2+√2*√3)
= 6+2√3+3√2+√6