Question

Hex value 0xC29C3480 represents an IEEE-754 single-precision (32 bit) floating-point number. Work out the equivalent decimal number. Show all workings (e.g. converting exponent and mantissa)

Answer 1

Answer:

floating point Decimal Number  = -78.1025390625

Explanation:

Hex Value =0xC29C3480

Convert this hex value into binary number, Use only C29C3480 as hexadecimal number. Each digit in hexadecimal will convert into 4 bits of binary. To convert this number into decimal we use hexadecimal to binary table. The conversion is as below:

1100 0010 1001 1100 0011 0100 1000 0000

Now this is a 32 bit number in binary, the bits are arranged as follows

1                        10000101                       00111000011010010000000

(Sign bit)            (Exponent bits)                      (Fraction bits)

In this 32 bit number, 32nd bit is sign bit,number is positive if the bit is 0 other wise number is negative. From 31st to 23rd bits 8 bits that are known as exponent bits, while from 22nd to 0 total 23 bits that are called fraction bit.

1. As the 32 bit is 1, so given number is negative.
2. Now convert the Exponent bits into decimal values as

10000101 = 1x2⁷+0x2⁶+0x2⁵+0x2⁴+0x2³+1x2²+0x2¹+1x2⁰

                = 1x2⁷ + 1x2² + 1x2⁰

                = 128 + 4 + 1 =  133

      3. Now we calculate the exponent bias, that will be calculated through, the formula.

       exponent bias = 2ⁿ⁻¹ - 1

there n is total number of bits in exponent value of 32 bit number, Now in this question n=8, because there are total 8 bits in exponent.

  so,       exponent bias = 2⁸⁻¹ - 1 = 2⁷-1= 128-1= 127

          e = decimal value of exponent - exponent bias

             =     133 - 127 = 6

       4. Now, we calculate the decimal value of fraction part of the number, which is called mantissa.

         Mantissa=0x2⁻¹+0x2⁻²+1x2⁻³+1x2⁻⁴+1x2⁻⁵+0x2⁻⁶+0x2⁻⁷+0x2⁻⁸+0x2⁻⁹+1x2⁻¹⁰+1x2⁻¹¹+0x2⁻¹²+1x2⁻¹³+0x2⁻¹⁴+0x2⁻¹⁵+1x2⁻¹⁶+0x2⁻¹⁷+0x2⁻¹⁸+0x2⁻¹⁹+0x2⁻²⁰+0x2⁻²¹+0x2⁻²²+0x2⁻²³

               = 1x2⁻³+1x2⁻⁴+1x2⁻⁵+1x2⁻¹⁰+1x2⁻¹¹+1x2⁻¹³+1x2⁻¹⁶

               =  0.125 + 0.0625 + 0.03125 + 0.0009765625 + 0.00048828125 +                     0.00012207031 + 0.00001525878

Mantissa = 0.22035217284

Now the formula to calculate the decimal number is given below:

decimal number = (-1)^s x (1+Mantissa) x 2^e

there s is the sign bit which is 1. and e =6.

so,

decimal number = (-1)^1 x (1+0.22035217284 ) x 2⁶

                            = -1 x 1.22035217284 x 64

                            = -78.1025390625