numerical methods floating point problem A machine stores fl

numerical methods floating point problem

A machine stores floating point number in a 9-bit word. The first bit is for the sign of the number, the second bit is for the sign of the exponent, the next 3 bits are for the exponent, and the final 4 bits are for the mantissa. The number that (001111001)₂ represents in the above given 9-bit format is...?

Solution

Given

sign of mantissa

                          The first bit is 0, so the number is positive

                          The second bit is 0, so the exponent is positive

                           The third bit is 111, are the exponents so

                                 e(111)₂= (1*2²+1*2¹+1*2⁰)₉=(4+2+1)₉=(7)₉       (here 2⁰ =1)

                            The next four bits 1001, are the mantissa so,

                              m(1.1001)₂ = (1*2⁰+1*2^-1+0*2^-2 +0*2^-3+1*2^-4)₉  = (1.25)₉

                                                 The number in binary format then is

                                        =(1.1001)₂*2^-(111)₂     

                                    The number in 9-bit format is

                                       = 1.25*2^-6

                                                          = 0.01953125