Introduction

In Hindu Arabic system, we use ten symbols 0, 1, 2, 3, 4, 5, 6, 7, 8  and  9  called digits are used to represent any number. A group of  figures, denoting a number is called a numeral ( or symbol ).

For a given numeral, we start from extreme right as, unit’s place, ten’s place, hundred’s place and so on.

Ex – 1  :   We represent a number,  ‘968054745’  as shown below  :

968054752  =   9 x 10⁸  +  6 x 10⁷   +  8 x 10⁶   +  0 x 10⁵   +  5 x 10⁴   +  4 x 10³   +  7 x 10²   +  4 x 10¹   +  5 x 10⁰ 

We read it as :

‘ Ninety-six crores, eighty lacs, fifty-four thousand, seven hundred and forty five.’

☼ NUMBER  SYSTEM

• Natural  Numbers  :
  • N = The set of  natural numbers  (  means,  positive integers )

     = { 1, 2, 3, 4, 5, . . .   }

• Whole  Numbers   :
  • W = The set of  whole numbers

     = Zero  +  positive integers  ( Set  N )

     = { 0, 1, 2, 3, 4, 5, . . .   }

• Integers  :
  • I  or  Z = The set of  integers

= Negative integers  +   zero  +  positive integers

= Negative integers  +   Set  W ( means,    zero  +  positive integers  )

= { . . .  ,  – 3, – 2, – 1, 0, 1, 2, 3,  . . .   }

• Rational  Numbers  :
  • Q = The set of  rational numbers

= Fractions  +   Integers ( Set  I )

= { . . .  ,  – 3, – 2, , – 1 , -1  , ,  0, 1, 2, , 3,  . . .   }

={ . . .  ,  – 3, – 2, – 1, 0, 1, 2, 3,  . . .   }

• Real  Numbers  :
  • R = The set of  real numbers

= Rational  numbers ( Set  Q )   +   Irrational numbers [ The numbers whose nth root is not a  rational number]

= Negative integers  +   Set  W ( means,    zero  +  positive integers  )

=

= { . . .  ,  – 3, – 2, – 1, 0, 1, 2, 3,  . . .   }

• Complex  Numbers  :
  • C = The set of  Complex numbers

= Real  numbers ( Set  R )   +   Imaginary  numbers[ e.g.  2 + 3i, 5 – 7i, – 2 – i,   etc. Where,  i  = ]

= Negative integers  +   Set  W ( means,    zero  +  positive integers  )

=

= { . . .  ,  – 3, – 2, – 1, 0, 1, 2, 3,  . . .   }

Hence,    N    W   I    Q    R    C

☼ Various  Types  of   Numbers

• Positive Integers  :   The set  N  =    =   { 1, 2, 3, 4, 5, . . .   } is the set of  positive integers. Clearly, positive integers and natural numbers are synonyms.
• Negative Integers  :   The set     =   { – 1, – 2, – 3, – 4, – 5, . . .   } is the set of  negative integers.
• Non-negative  Integers  :   The set  W  =  { 0, 1, 2, 3, 4, 5, . . .   } is the set of  non-negative integers. Clearly, non-negative integers and whole numbers are synonyms.
• Rational  Numbers  :   The numbers of  the form  where  p  and  q  are integers and  q   0,  are known as rational numbers.  e.g. , ,   etc.
• = The set of  positive real numbers.
• =The set of  negative real numbers.
• =The set of  positive real numbers including zero.
• =The set of  negative real numbers including zero.
• = The set of  integers except zero.
• =  W
• Terminating  &  Repeating  Decimals  :   Every rational number has a particular characteristic that it when expressed in decimal form is expressible either in terminating decimals or in repeating decimals ( recurring numbers ).

e.g. ( terminating decimals )

( repeating decimals )

• Irrational  Numbers  :   The numbers which when expressed in decimal form are neither terminating decimals  nor  repeating decimals, are  known as irrational numbers.

e.g. 

• Even  and  Odd  Numbers  :   Integers divisible by 2 are known as even integers, while those which are not divisible by 2 are known as odd integers.

e.g. …, – 6, – 4, – 2, 0, 2, 4, 6, …  are  called even integers.And,  … , – 7, – 5, – 3, – 1, 1, 3, 5, 7, …  are called odd integers.

• Consecutive Integers  :   Consecutive integers are numbers differing by 1 in ascending order.

e.g. 19, 20, 21, 22, . . .  are consecutive integers.

• Modulus of a Number  :   For a real number x, its modulus is defined as  :

e.g.

• Factors  ( Divisors )  &  Multiples  :   An integer n is called a factor or a divisor of another integer m, if n divides m exactly.  (i.e., when n divides m, the remainder is zero). And we say that m is a multiple of n.

e.g. 5 and 7 are factors of  35.  35 is called multiple of  5 and 7 both.

If    ( where ,    are distinct  primes )

then   the  number of  factors of 

e.g.

The number of  factors  of   37800 = ( 3 + 1 ) · ( 3 + 1 ) · ( 2 + 1 ) · ( 1 + 1 )

= 4  ·  4  ·  3  ·  2

= 96

• Prime  and  Composite  Numbers  :   A positive integer greater than 1 is  called a prime number if it has exactly two factors, namely  1  and  itself.

If  a  number is not a prime number ( a positive integer greater than 1 ), then it is called a composite number. i.e. composite number has always at least  3 distinct factors.

e.g. 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, . . .   are prime numbers.

• • There are only 25 prime numbers between 1 and 100.
  • Test  of  a  prime number  :   Let  x  be a given number and  k  be an integer very near to , such that  < k

If  x  is not divisible by any prime number less than  k,  then x is a prime;  otherwise, it is not a prime.

e.g. (i) Is  247  a  prime  number  ?

            Yes,  it  is  a prime number.

       (ii) Is  391  a  prime  number  ?

No, it is not  a  prime number. i.e.  it is a composite number.

1  is neither prime nor a composite number.  ( 1 has exactly one factor = itself  )

  1  is neutral number.

Number  of  even  prime  is  1.      (    2  is  the  only  even  number  which  is  a  prime. )