An 9-bit floating-point number representation system has the following specifica

Question

An 9-bit floating-point number representation system has the following specification:

Among the 8 bits(B8,B7,B6,B5,B4,B3,B2,B1,B0 ), bit B8is the sign bit, bits B7,B6,B5,B4 is the exponent with a bias value (0100). Bits B3,B2,B1,B0  consist of the mantissa. The binary number that is represented is (-1)B8 * (0.,B3,B2,B1,B0) * 10B7B6B5B4 - 0100.

A)What are the largest positive and negative numbers this system can present?

B)Find a postive number x whic is representable by this system such that 1 + x = 1.

C)Find the largest positive number y which is representable by this system such that 1 + y = 1

Explanation / Answer

Answer:

A)

The general formula for the representation is:

(-1)B8 * (0. ,B3,B2,B1,B0) * 10B7B6B5B4 - 0100.

The largest positive and negative numbers are given are:

The bias value is 0100 which mean the decimal equivalent is 4.

The bias exponent (E), exponent(e) and the total max value in 4 bits is 15.

Therefore, the relation is given as,

            E = e+15

            4 = e+15

            e=4-15 = -11.

Thus the exponent value is -11.

The binary equivalent of -11 is = two’s complement of 11

                                                   = 2’s complement(1011)

                                                   = 0100+1

                                                   = 0101

Thus the bits in the exponent are 0101.

The positive integer is (-1)0*(0.B3, B2, B1, B0)* 100101-0100

(-1)0*(0.B3, B2, B1, B0)* 100001

The negative integer is (-1)1*(0.B3, B2, B1, B0)* 100001

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