CharacterRange oddity [closed]

Question

CharacterRange documentation says that it returns a list. However, if I define

letters[n_] := CharacterRange["\[Alpha]", "\[CurlyPhi]"][[;; n]]

and then ask for Part[letters, 3], I get Part::partd. If I ask for Last[letters] I get Last::normal. What gives?

Answer 1

While e.g., letters[5] (or any other integer between 1 and 22) is a list, letters is not. Colloquially, letters is a function, which is implementing in Mathematica as a symbol with DownValues.

You can think of DownValues as rules which are applied during the sequence of evaluation, so that letters[5] evaluates to {"α", "β", "γ", "δ", "ε"}.

In fact, you can query the DownValues of letters with fairly readable results:

DownValues[ letters ]

{ HoldPattern[letters[n_]] :> CharacterRange["α", "φ"][[1 ;; n]] }

_{(HoldPattern is just a way of making sure the left hand side of the rule doesn't evaluate before the rule can be applied.)}

Your problem was two-fold:

1. No subpart of letters[[3]] is of the form letters[n_], so the DownValue couldn't apply
2. More noticeably, Part immediately tries to evaluate, even if its first argument has insufficient length/depth, which results in your error.

If you'd like to use the syntax letters[[3]] to return "α", you can instead redefine letters:

ClearAll[ letters ]  (* Remove DownValues *)
letters = CharacterRange["α", "φ"];

letters

{"α", "β", "γ", "δ", "ε", "ζ", "η", "θ", "ι", "κ", "λ", "μ", "ν", "ξ", "ο", "π", "ρ", "ς", "σ", "τ", "υ", "φ"}

letters[[3]]

"γ"

Fun fact: while SetDelayed (:=) statements are stored as rules in DownValues, Set (=) statements are stored in OwnValues :

 OwnValues[ letters ]  (* after using `=` rather than `:=` *)

{ HoldPattern[letters] :> {"α", "β", "γ", "δ", "ε", "ζ", "η", "θ", "ι", "κ", "λ", "μ", "ν", "ξ", "ο", "π", "ρ", "ς", "σ", "τ", "υ", "φ"} }