Numbers and number systems

In this post let us clearly know about the classification of the number system.before classifying numbers let us know about numbers and number systems.

•  what is a number?
  =>A number can be defined as a group of digits which have some value.
• what is a number system?
  =>The systematic way in which numbers are organized
  e.g: decimal, binary, octal, hexadecimal..........
• what is base or radix?
  => base can be defined as a number of digits used in that number system.

Classification of numbers based on base or radix:

One can form number system of any radix but to make it simple, the most often used number system is decimal system(it have digits 0-9) and computers use only binary representation (digits 0,1)(As computers have only ON or OFF state)

Most commonly used systems are: binary(base 2),octal(base 8),decimal(base 10),hexadecimal(base 16).

System                         Digits

Binary                      0,1

Octal                        0,1,2,3,4,5,6,7

Decimal                   0,1,2,3,4,5,6,7,8,9

Hexadecimal            0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F              

conversion of any radix to a decimal number.

for example value of (1563).10 in decimal representation is 

1*10^3 + 5*10^2 + 6*10^1 + 3*10^0

similarly (10011)

1*2^4 + 0*2^3 + 0*2^2 + 1*2^1 + 1*2^0

conversion of decimal to another radix:

It is as same as dividing a number by radix multiple times and noting down the remainder.

Q)convert (467).10 to hexadecimal number.

                base   number         remainder

                   16| 467              3

                      ------------------------------

                  16 | 29              D(13)

                      ------------------------------

                          1

so (467).10 is 1D3 in hexadecimal representation

if it is decimal to octal, we need to divide with 8.

For simplicity, in conversion of the number from one radix to other, convert it to decimal and then to the other radix.

Decimal number system classification:

Decimal numbers can be classified into 2 categories 

1. Real numbers: Which exists really.
2. Imaginary numbers: which are assumed to exist.(eg: square root(-25) is 5i )(here i is imaginary or it can be defined as square root of -1)

Real numbers can be further classified into 2 categories:

1. Rational numbers: Any number which can be represented in form of p/q (where p,q are natural numbers)
   eg: 5.5=11/2
2. Irrational numbers: Numbers which can't be represented in form of p/q
   eg: pi which cannot be defined in p/q form(22/7 is just an approximate)
   

Rational numbers can be classified into 2 types:

1. Integral  (-ve ,0, +ve numbers)
2. Fraction