Question

Determine the single precision 32-bit representation of the following decimal numbers: (50 points) 1. e. 42.424242 f. 76.234567 x 105 g. 1.4345678

Answer 1

1. 42.424242

0(sign bit which is +) 10000100(exponent which is 132) 01010011011001001101100(mantissa which is 2732652).

we can again calculate the decimal number from its binary:

$(-1)^{b_{31}}*2^{(b_{30}b_{29}.....b_{23})_2-127}*(1.b_{22}b_{21}....b_{0})_2$

where  $b_{i}$ is the bit at position i.

so, $(-1)^0*2^{132-127}*(1.01010011011001001101100)_2$

which will be: $1*2^{5}*1.3257575035095215$

which is 42.4242401

2. $76.234567*10^{-15}$

0 (sign +) 01010011(exponent which is 83) 01010111010101000111100 (mantissa which is 2861628)

so, $\\(-1)^0*2^{83-127}*(1.01010111010101000111100)_2 \\1*2^{-44}*1.341132640838623 \\5.68434189*10^{-14}*1.341132640838623 \\76.234528*10^{-15}$

3. 1.4345678

0 (sign +) 01111111 (exponent 127) 01101111001111111101011 (mantissa)

$\\(-1)^0*2^{127-127}*(1.01101111001111111101011)_2 \\1*2^{0}*1.4345678091049194 \\1*1.4345678091049194 \\1.4345678091049194$