Single-precision floating-point numbers has this formula:
\(\displaystyle (-1)^S \times 2^E \times M\)
where
\(\displaystyle S \rightarrow \text{sign digit}\)
\(\displaystyle E \rightarrow \text{biased exponent part} - 127\)
\(\displaystyle M \rightarrow \text{Mantissa} = 1 + \text{decimal part}\)
The main idea is how to get the values of these three variables from the \(\displaystyle \text{zeros}\) and \(\displaystyle \text{ones}\).
We have in \(\displaystyle \bold{(a)}\):
\(\displaystyle 0 \ 10000000 \ 11000000000000000000000\)
\(\displaystyle S\) is the first digit on the left. In this case \(\displaystyle S = 0\).
\(\displaystyle 10000000\) is the biased exponent part. In this case \(\displaystyle 10000000 = 128\), so \(\displaystyle E = 128 - 127 = 1\).
\(\displaystyle 11000000000000000000000\) is the decimal part. In this case \(\displaystyle 11000000000000000000000 = 2^{-1} + 2^{-2}\).
Then, \(\displaystyle M = 1 + \frac{1}{2} + \frac{1}{4} = 1.75\).
Goint back to the formula.
\(\displaystyle (-1)^S \times 2^E \times M\)
\(\displaystyle (-1)^S\) represents the sign of the number.
\(\displaystyle 2^E\) represents the biased exponent.
\(\displaystyle M\) represents \(\displaystyle \text{Mantissa}\).
Let us fill the formula with what we got.
\(\displaystyle (-1)^S \times 2^E \times M = (-1)^0 \times 2^1 \times 1.75 = 3.5 = +3.5\)
