Unfortunately, there are problems with this simple design. Leaving out details of electrical engineering, we need N1 resistors to get N output. If we want 12 bits of resolution, 2^{12} = 4096, and that's a lot of resistors. Isn't there a way to use fewer resistors? It turns out that there's a method that uses only 2N resistors. It's called an R/2R ladder.

A 3level R/2R Network

Start at the far right. How does V_{3} compare to V_{2}? By analogy to the previous figure, V_{3} = V_{2}/2. At first, it looks like computing how V_{2} and V_{1} relate could be ugly. However, there are 2 routes to ground from V_{2}. Both of them have a total resistance of 2R. The effective resistance of two resistors in parallel relate as 1/R_{effective} = 1/R_{path 1} + 1/R_{path 2}. The effective resistance from V_{2} to ground via the two parallel paths is R, the same as the resistance of the individual resistors! This means that each numbered voltage is 1/2 the potential of the next lower numbered voltage. Suddenly, we have a way to get 1/2^{n} fractions of a reference potential for any n, just by using a large enough number of identical resistors. Actually, there's a limit; the precision of the resistors has to be such that the errors in each stage are smaller than the smallest fractional voltage. That means that, if we have a 12 bit ladder, all resistances must be precise to 1 part in 2^{12} = 1/4096. That's about 0.025%. That sounds difficult to achieve, but resistors can be trimmed to 0.1% fairly easily, 0.01% with effort, and even 0.001% if temperature is carefully controlled and adequate standards are available.
We thus have a way to look at a reference potential and 1/2, 1/4, 1/8, ... of that reference. How do we combine those potentials so we can get potentials that vary in small, equal steps from 0 to V_{in}? We need to combine the current in the R/2R network with one amplifier. Let's take a moment to examine the Inverting Operational Amplifier With Gain, one of the most common analog electronic circuits.