What is the decimal expansion of a real number?

Question

I'm curious as to what the decimal expansion of a real number is. Is it an expression? A series? A sequence? What sort of mathematical object is it?

For context, we may define an isomorphism between two Group objects as a Bijective Function mapping from one to the other that "preserves structure." In this sense, an isomorphism IS a function (albeit with some extra requirements).

Likewise, we may define a basis of a vector space as a maximal set of linearly independent vectors. In this sense, a basis IS a set.

Functions and sets clearly represent very fundamental concepts in mathematics. And I'm not asking for a good answer to 'what is a set?'

What I am asking, is what IS a decimal expansion, defined in a way similar to the above? Is a decimal expansion an expression? (Something that exists purely graphically) Is it a sequence of natural numbers? (This might make sense) Is it a series? (This would also make sense).

Answer 1

We all informally understand what a decimal expansion is, so the particular implementation might differ from situation to situation by what is convenient. One way I have seen is that a decimal expansion is a function $f: \mathbb Z \to \{0, 1, \ldots, 9\}$ such that there is some $n$ so that $f(m) = 0$ for all $m < n$, together with a number $s \in \{0, 1\}$. This then represents the number $$ (-1)^s\sum_{k \in \mathbb Z}f(k) \times 10^k, $$ and is, at least when $m$ is negative, usually written as $$ f(m)f(m+1)\ldots f(0).f(1)f(2)f(3)\ldots $$ when $s = 0$ or $$ -f(m)f(m+1)\ldots f(0).f(1)f(2)f(3)\ldots $$ when $s = 1$.

Answer 2

A decimal expansion is a series (which implies summation, as opposed to a sequence, which does not).

The series is $$\sum_{n\in\Bbb{N}} a_n 10^{i_0 - n}$$ with all $a_n \in \{ 0, 1, 2, \ldots , 9\}$ and $i_0 \in \Bbb{Z}$.

Answer 3

Is a decimal expansion an expression? (Something that exists purely graphically) Is it a sequence of natural numbers? (This might make sense) Is it a series? (This would also make sense).

When defined formally, it is usually defined as a sequence, but can also be formulated as a function. But a precise distinction between "expression", "sequence" and "series" is often not made, which makes such statements as " .999... = 1" problematic. Lay people often interpret this as meaning "If you write a decimal point and then a bunch of nines, that's equal to one if you have enough nines". This leads them to be skeptical of the claim, as they well should be: as an expression, the decimal expansion is never equal to one. You can never get to one by writing nines after the decimal point.

As a sequence, the decimal expansion isn't equal to one; we can say that it represents one, but there is a slight difference between a sequence representing a number and it being equal to it.

For " .999... = 1" to be seen as true, it must be interpreted as "The sequence of symbols decimal point nine nine nine ellipsis represents the number one". And that in turn is true within the framework of series and limits.