I happen to be studying the basics of quantum mechanics at the moment and have made acquaintance with the vector representation of quantum states, in particular the two states of electron spin. For this question let's just say the spin can be up or down. The state of the spin is represented by vector, and here's where my trouble in understanding shows up. I'll have to write the column vectors as row vectors, although the issues should still be clear. What is giving me trouble is that fact the the array/vector has two elements instead of just one. I would think a single digit, either "0" or "1" would be sufficient to describe a state as being either up or down. So my question is why are two element/components required to describe the two states of electron spin. There are only two states, so a single bit should be sufficient. Two bits allow for four possible states. That begs the question, what do the unused state vector definitions | 1 1 | and | 0 0 | then represent, and if they fail to represent anything meaningful why even allow for their expression, which is another way of asking my first question, why use two components in the vectors in the first place.