Why have additional symbols on a Turing machine?

What is the point of non-binary printed letters on a turing machine? I understand that these need to be omitted to get a computable number, but why are they used in the first place?

• You mean, why a TM could have an alphabet other than $\{0, 1\}$. Commented Jan 23, 2014 at 17:17
• Because TMs are a formalism to think about computation, and sometimes having more symbols than $0$ and $1$ makes thinking about problems easier. They are not a blueprint for hardware. Commented Jan 23, 2014 at 17:21
• @G.Bach Together with an example, this would make a fine answer, I think. Commented Jan 23, 2014 at 18:10
• Why even give a TM as an example? It seems sufficient to point out that the "point" of having more than two letters in the English alphabet is more or less the same as the "point" of allowing more than two letters in a TM's tape alphabet. The answer could range from "it's convenient" to "that's just how it ended up", with everything in between being equally as valid. Really, the question should be why computer hardware relies on binary representations, not vice versa. Another interesting question might discuss unary alphabets. The answer to this question, as asked, is "because!" Commented Jan 23, 2014 at 19:09
• By the way, for most interpretations of I understand that these need to be omitted to get a computable number I can imagine, your understanding is wrong. Commented Jan 23, 2014 at 19:11

There is a fine point regarding input encodings. Usually inputs are blank-separated or blank-terminated. If you have only two symbols and are using this convention, then everything has to be encoded in unary — $1^n0$ for the number $n$. This has a strong effect on input length and so on running times (making them appear much faster), so either you use an unconventional encoding (say encode "characters" in groups of two symbols), or you need to have at least three symbols (including the blank symbol).