Arithmetics for Computer

Number System Base

What is a base for number system types?

Actually most of the numbering system will have a base. Each base maximum number just can represent up to single digit or number only.

For example:

-          Binary (Base 2): 0, 1

-          Octal (Base 8): 0, 1, 2, 3, 4, 5, 6, 7

-          Decimal (Base 10): 0, 1, 2, 3, 4, 5, 6, 7, 8, 9

-          Hexadecimal (base 16): 0, 1, 2, 3, 4, 5, 7, 8, 9, A, B, C, D, E, F

In general, we can represent numbers in the base,b, as follow :

a = a_{n .}bⁿ + a_n-1 . b^n-1 + a_n-2 . b^n-2 +…+ a₂ .b² + a₁ .b¹ + a₀ .b⁰

a = number ;     n = place of number, n^th ;                     b = base ;

Example:

345 change to base 8

345= (3*(8²))+(4*(8¹))+(5*(8⁰))

       =229₈

Decimal (Base 10 Integers)

-         An integer in decimal system is a sequence of digits.

-        Value assigned weight is composed by 10 digits starting. Each digit is one of the following:

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

-          They position weight structure is base on their  positive and negative values to determined .

       …10⁵  10⁴  10³  10²  10¹  10⁰        ( positive value )

      10²  10¹  10⁰  10^-1  10^-2  10^-3 …     ( negative value )

 

Binary ( Base 2 Integers )

-          A number in binary system is a sequence of bits, possibly followed by a binary point and then a sequence of bits.

 The contraction of the words ‘binary’ and ‘digit’ is in term of ‘bit’.

-          Number of Base 2 just consists two digit 0 and 1 only.

-          The actual value ( in Base 10 ) represented by the binary number is :

-          The binary number  of weight structure is:

2^n-1… 2³  2²  2¹  2⁰   .  2^-1  2^-2 … 2^-n

                      Binary | points

-          The size of binary number will able to determined they are Least Significant Bit (LSB) and Most Significant Bits (MSB).

   ( MSB )----100101110----( LSB )

Octal ( Base 8 Integers )

-          The possible number for base 8 is  0, 1, 2, 3, 4, 5, 6, 7

-          The actual value ( in Base 10 ) represented by the octal digit sequence is :

Hexadecimal ( Base 16 Integers )

-          The possible digits in a hexadecimal number are :

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

Where A, B, C, D, E, F stand for 10, 11, 12, 13, 14, 15 in decimal respectively.

-          Hexadecimal numbers have the relationship between the Base 2 and Base 16. Sixteen is a power of 2 (16 = 2⁴⁾ .Therefore, four digits in a binary number is equal to a single hexadecimal digit. So that, the conversion between binary and hexadecimal numbers is easy and more suitable to present it by divide them into group which contain 4-bit binary numbers.

-          Each digit in Base 16 position will represent in power of 16.

-          The actual value ( in Base 10 ) represented by the hexadecimal digit sequence:

Ang Kuan Kee B031210344