Suppose I have an imaginary computer, with an infinite binary table, like the one below:
$$ \begin{array}{|c|c|c|c|c|c|c|c|c|} \cdots & 128s & 64s & 32s & 16s & 8s & 4s & 2s & 1s \end{array} $$
If you don't already know how to make integers with a binary table, this is how it's done. To make an integer place a $1$ under the desired columns of the table, then add the values of each column where there is a $1$ together, and you get your number. Here's an example:
$$ \begin{array}{|c|c|c|c|c|c|c|c|c|c|} 128s & 64s & 32s & 16s & 8s & 4s & 2s & 1s & \text{result}\\ 0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 6 \end{array} $$
My question is, how is it possible to make every [positive] [real] [whole] number with our imaginary infinite binary table? Will there be numbers that can't be made?
In order to represent non integer numbers, you must expand your table to the right, adding $1/2$, $1/4$, $1/8$ and so on. Just like decimal representation. For the sign, you must add an additional slot. Usually $0$ means positive and $1$ negative.