Converting base 2 to Base 10?

Question

How is $0.11 \cdot 2^{1}$ converted to $1.5$ in base 10? I am unable to understand the theory behind this, since I believe $0.11 * 2^{1}$ corresponds to $1.1$ but I dont see the conversion process?

Answer 1

$0.11\cdot 2^1$ does indeed correspond to $1.1$. In binary. Which is to say, it means $$ 1\cdot 2^0 + 1\cdot 2^{-1} $$ as per standard positional number system conventions. Writing this out in base ten does get you $1.5$.

Usually, when we write numbers in different bases close to one another, there is postential for such a misunderstanding. In those cases, it's common to see a subscript denoting which base we're using (this subscript uses base ten). So with that, the above paragraph would look like

$0.11_2\cdot 2^1$ does indeed correspond to $1.1_2$. In binary. Which is to say, it means $$ 1\cdot 2^0 + 1\cdot 2^{-1} $$ as per standard positional number system conventions. Writing this out in base ten does get you $1.5_{10}$.