How do I know that the number is halfway in between

Question

I was shown the following:

$$ 0.110 < 0.1101 < 0.111$$

and told that the middle number is halfway in between those two numbers. Is this correct? How can I see that?

Update:

If I add a zero to the first and the last number, I get:

$$ 0.1100 < 0.1101 < 0.1110$$

Now, let write out the last two digits and we can see the relationship:

$$ \underbrace{00 < 01 < 10}_{base-2} = \underbrace{0 < 1 < 2}_{base-10} $$

Answer 1

\begin{align} 0.1100 &= 2^{-1} + 2^{-2} \\ 0.1101 &= 2^{-1} + 2^{-2} + 2^{-4} \\ 0.1110 &= 2^{-1} + 2^{-2} + 2^{-3} = 2^{-1} + 2^{-2} + 2\cdot2^{-4} \end{align}

Therefore, the difference between $0.1100$ and $0.1101$ as well as between $0.1101$ and $0.1110$ is $2^{-4}$, and hence $0.1101$ does indeed lie halfway between the two other numbers.

Answer 2

Calculate the average of $0.110$ and $0.111$, by adding them and dividing the result by $2$ (keep in mind that it is binary), and see that you get $0.1101$.