Binary representation of numbers

 Binary representation of numbers


In this we will learn how to represent a number in Binary format, Octa, Hex formats

1 - 0001            6 - 0110            11 - 1011 (B)

2 - 0010            7 - 0111            12 - 1100 (C)

3 - 0011            8 - 1000            13 - 1101 (D)

4 - 0100            9 - 1001            14 - 1110 (E)

5 - 0101           10 - 1010 (A)    15 - 1111 (F)

Ex 1: Represent 56 in binary format

       56 - (2^5) * 1 + (2^4) * 1 + (2^3) * 1 +  (2^2) * 0 + (2^1) * 0 + (2^0) * 0   

            - (111000)

Ex 2: Represent -56 in binary format

There are multiple ways to represent negative numbers in binary format

1) With sign magnitude, Generally 

    sign bit - 1 -> Negative numbers

    sign bit - 0 -> Positive numbers

    3 - 00000011

   -3 - 10000011

2) Using 2's complement

-47 -> 1) binary representation of Positive 47 - 101111

           2) pad the bits based on format -> 00101111

           3) Invert the bits -> 11010000

           4) Add 1 to LSB -> 11010001 


Ex3: Addition of two negative numbers 

-47  -> 11010001

-112 -> 10010000

         -----------------

           101100001 -> -159 -> Overflow


In this case we need to saturate the value because with 8 bits we can store from -128 to +127 beyond that we saturate the values 


IEEE 754 representation of numbers

IEEE -754 is technical standards used to represent floating point numbers. IEEE(Institute of Electrical & Electronic) engineers found hard to identify the floating point they used IEEE format. It is still exiting in most of intel, Mac PCs


It has 3 components 

 
        Sign               Exponent             Mantissa     Bias

           1                    8                           23                127        - Single Precision Floating point

           1                    11                         52                1023       - Double Precision Floating Point


Ex:1

85.125

85 = 1010101

0.125 = 001

85.125 = 1010101.001

       =1.010101001 x 2^6 

1. Single precision:

biased exponent 127+6=133

133 = 10000101

Normalised mantisa = 010101001  we will add 0's to complete the 23 bits

The IEEE 754 Single precision is:

= 0 10000101 01010100100000000000000

This can be written in hexadecimal form 42AA4000


2. Double precision:

biased exponent 1023+6=1029

1029 = 10000000101

Normalised mantisa = 010101001

we will add 0's to complete the 52 bits

The IEEE 754 Double precision is:

= 0 10000000101 0101010010000000000000000000000000000000000000000000

This can be written in hexadecimal form 4055480000000000 


    

    






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