# Mathematically express length of $n$-tuple

Wikipedia says "An $$n$$-tuple is a sequence (or ordered list) of $$n$$ elements, where $$n$$ is a non-negative integer."

so I have a $$n$$-tuple $$m$$, what of $$m$$ is $$n$$?

If I write $$f(m)=n$$, which function should $$f$$ be? Is there already a mathematical notation available?

• What about "dimension" ? – Peter Oct 13 '20 at 7:25
• As kind of implied in K.defaoite's answer, you wouldn't normally have a tuple out of the blue without declaring a name for its length at the same time. Like, at worst maybe something like "let $x_i\in\mathbb R^{n_i}$ for $1\le i\le k$". – Mark S. Oct 13 '20 at 10:36

An $$n$$ tuple comes from the $$n$$ fold Cartesian product of a set with itself. That is to say given $$x_1 \in S,...,x_n\in S$$, this is equivalent to $$\underline{x}=(x_1,...,x_n)\in \underbrace{S \times {}\dotsm{} \times S}_{n \text{ times}}$$ Which is often denoted $$\underline{x}\in S^n$$ It is technically incorrect to talk about the dimension of $$\underline{x}$$, i.e the expression $$\dim{\underline{x}}=n$$ it is however correct to say $$\dim(S^n)=n$$ but this is redundant because the $$n$$ superscript already implies the dimension of the set.
Following the title of your question, I would call it the length of the tuple, in analogy with the length of a word of a free monoid, since a word is an ordered sequence of letters. As for notation, the length of a word $$u$$ is usually denoted by $$|u|$$.