A ditty is a short, simple song, like the ones I write. It comes from the Old French dite (composition), from the Latin dictatum (something dictated), from dictare (to dictate), a frequentative of dicere (to say, speak), which is related to dicare (to proclaim, dedicate), from the Proto-Indo-European root *deik- (to point out).
Some English words that come from the same root include dictate, diction, and digit, which came to be related to numbers as a result of counting on fingers. Other words that developed from this root include the Latin digitus (finger), the German zeigen, the Greek δίκη [díkê] (custom, right, judgement), and quite a few more.
The word teach also comes from the *deik-, via the Old English tæcan (to show, point out, declare, demonstrate; to give instruction, train, assign, direct; warn; persuade), from the Proto-Germanic *taikijan (to show).
Source: Online Etymology Dictionary, Oxford Dictionaries and the Indo-European Lexicon.
Being a programmer, “digit” immediately makes me think of “digital” computers, which have never used fingers… but, checking the Online Etymology Dictionary for other uses of *deik-, it’s also the root of “disk”.
I read somewhere that English ten and German zehn etc. are connected with German zehen (‘toes’) – which seems logical enough. If that is the case, then it seems plausible that there might be a connection between Latin decem, Welsh deg etc. and Latin digitus. Can anyone confirm this? Of course, any such connection would presumably go back long before the dawn of Latin.
Incidentally, http://www.etymonline.com/index.php?allowed_in_frame=0&search=finger suggests a possible connection between finger and P.I.E.penkwe- (‘five’ – cf. Lithuanian penki, German fünf).