# Number System

**Number is a mathematical object used to count, measure and label.****In Indian system, numbers are expressed by means of symbols 1, 2 ,3 ,4, 5, 6, 7, 8, 9 nad 0 called digits.**

**Types of Numbers**

**Numbers can be classified into sets the same number can be written in many different ways.**

**Natural numbers (N),****whole Numbers (W),****integers (Z),****Rational Numbers (Q),****Irrational numbers ( ),****Real numbers ( R ),****Complex Numbers (C)**

Natural numbers :- All Counting numbers are called Natural numbers, denoted by N

- Natural numbers are {1, 2, 3 . . .}
- The smallest natural number is 1 and the Natural numbers are infinite.
- No largest natural number exists.

**Whole Numbers :- **

**All counting numbers together with zero are called whole numbers.**

**whole numbers denoted by W****whole numbers W= {0 ,1, 2, 3, . . .}****0 is a whole number which is not a natural number.**

**Integers : **

**All counting numbers , 0 and negatives of counting numbers are called integers.****Negative integers are usually written a negative sign.****Integers are denoted by Z****Integers= z=****{…,- 3,- 2,- 1,0,1,2,3,…}****{1,2,3 …} is the set of positive integers****{-1,-2,-3,-4,…….} is the set of negative integers****0 is neither positive nor negative.****The set of all integers is represented by Z****Z comes from the German word “ Zahlen ” which means ‘ to count ’**

**Rational Number ∶-**

**A number which can be expressed as p/q where both p and q are **** integers and q ≠0 is called Rational Number.**

** When all the integers and fractions are put together ,then they are called **** rational number.**

**Rational Number are denoted by Q (Q means quotient) ,****Q ={p/q, p,q∈Z,q ≠0 }**

** For example∶(-5)/6, 8/9,(-7)/(11 ) etc.are rational numbers.**

**All fractions are rational numbers but all rational numbers need not necessarily be fractions.**

**Irrational Number**

**The decimal numbers which can not be expressed in the form of rational numbers is called Irrational Numbers.**

**The non terminating or non recurring decimals can not be expressed in the form of rational numbers.**

**Numbers which cannot be expressed in the form of p/q, where p, q are integers and q not equal to zero are called irational numbers.****The decimal form of an Irrational Number is a non terminating or non recurring decimal.****Square root of a number which is not perfect square, is an irrational number.**

__Real numbers:-__
The collection of both Rational and Irrational numbers together is called Real numbers.

- Real number s are denoted by R

Complex number:-

A complex number is a number that can be expressed in the form of a + ib

where a and b are realnumbers

and i is a solution of the equation i ^{2}=-1

i is called an imaginary number