I was studying the product with floating point and I saw this example. I made the translation, sorry if something is not grammatically correct.
![enter image description here][1]
I was studying the product with floating point and I saw this example. I made the translation, sorry if something is not grammatically correct.
![enter image description here][1]
Here is the complete calculation: $$ \begin{align*} &2.5 = (.101)_2 \times 2^2 \\ &12.125 = (.1100001)_2 \times 2^4 \\ &(.101)_2 \times 2^2 \cdot (.1100001)_2 \times 2^4 = (.0111100101)_2 \times 2^6 = (.111100101)_2 \times 2^5 \\ &(.111100101)_2 \times 2^5 = 30.3125 \end{align*} $$ Ignore what the book says, and just make sure that you understand why this calculation works out.
Also, actual floating point formats usually omit the leading 1 in the mantissa, which is usually interpreted as the first digit before the dot rather than the first after the dot. Using this convention, the exponents add without correction (under your convention, you have to subtract 1, essentially since 1/2 times 1/2 equals 1/4 rather than 1/2).