Question

Convert from IEEE 754 representation to decimal

- C210000

- 40D0000

- FE40000

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

Thank you so much!!

Answer 1

I'm assuming that these are in IEEE 754 Single precision format.

In the  IEEE 754 Single precision format, the binary number is of 32 bits and is divided into 3 components as follows:-

SIGN(1 bit)

EXPONENT(8 bits)

MANTISSA(23 bits)

The Bias of  IEEE 754 Single precision format is = 2^{(number of exponent bits) - 1} - 1 = 2^8-1 - 1 = 2⁷ - 1 = 128-1 = 127.

The equivalent decimal of the binary form in IEEE 754 Single precision format will be given by the formula = (-1)^SIGNX2^{(EXPONENT - BIAS)}X1.MANTISSA

a) C210000

0000 1100 0010 0001 0000 0000 0000 0000

SIGN = 0

EXPONENT = 00011000 = 24(in decimal)

MANTISSA = 01000010000000000000000

Now its equivalent decimlal = (-1)⁰x 2^{(24 - 127)}x1.01000010000000000000000

= 2^-103x(1 +2^-2 + 2^-7)

= 2^-103X1.2578125

b) 40D0000

0000 0100 0000 1101 0000 0000 0000 0000

SIGN = 0

EXPONENT = 00001000 = 8(in decimal)

MANTISSA = 00011010000000000000000

Now its equivalent decimlal = (-1)⁰x 2^{(8 - 127)}x1.00011010000000000000000

= 2^-119X(1 + 2^-4 + 2^-5 + 2^-7)

=2^-119X1.1015625

c) FE40000

0000 1111 1110 0100 0000 0000 0000 0000

SIGN = 0

EXPONENT = 00011111 = 31(in decimal)

MANTISSA = 11001000000000000000000

Now its equivalent decimlal = (-1)⁰x 2^{(31 - 127)}x1.11001000000000000000000

= 2^-96X(1 + 2^-1 + 2^-2 + 2^-5)

= 2^-96X 1.78125