Number System | Real Numbers | Definations

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.

  1. Natural numbers (N),
  2. whole Numbers (W),
  3. integers (Z),
  4. Rational Numbers (Q),
  5. Irrational numbers ( ),
  6. Real numbers ( R ),
  7. 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

  •  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 

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