Notation of exactness

Question

I'm doubting my recent Facebook comment concerning the acceptance of 2 + 2 = 5:

The trick is that without a decimal dot, those numbers could be up to 0.5 bigger or smaller, depending on rounding system. http://en.wikipedia.org/wiki/Rounding

While googling, this is the closest supporting statement i could find, but it doesn't mention a final . to denote exactness:

A common convention in science and engineering is to express accuracy and/or precision implicitly by means of significant figures. Here, when not explicitly stated, the margin of error is understood to be one-half the value of the last significant place. For instance, a recording of 843.6 m, or 843.0 m, or 800.0 m would imply a margin of 0.05 m (the last significant place is the tenths place), while a recording of 8,436 m would imply a margin of error of 0.5 m (the last significant digits are the units).

I guess this proves the margin, but is it correct that "2." means "exactly two"?

Update: I found the notation here. It only means that "100." has three significant digits, for example, so there's still a margin of 0.5. Indeed this question is about chemistry, not pure math.

Answer 1

The difference here is between how science and math treat numbers. In math, we deal in a number's exact value, i.e. I can compute $\pi$ or $\sqrt{2}$ or $7/9$ with arbitrary precision. That is to say, I can perform calculations with these numbers with as little error as I like, in order to (typically) do anything I like with as little error as I like. For example, if I want to compute something using $e$, then I can compute it with arbitrary precision by computing the sum $2 + \sum_{n = 1}^{k} \frac{1}{n!}$ for some sufficiently large $k$, and have as good an approximation of $e$ as I want.

In the sciences, this is not so. We derive numbers from empirical observation. The experimental procedure puts limits on how well I can know a number. I have only finite information of this number, and of that information there's a question of its reliability. A favorite anecdote of mine is a scientist coming through a small town, and asking a local how old a rock is, to which the local replies, "7,000,005!" The scientist asks how he knows, to which the local says, "Well, a scientist came and said it was 7,000,000 years old, and that was 5 years ago."

To compensate for this, there are conventions of significant figures and so-called "scientific notation." These utilize dots to denote the information in a given experimental value. When these decimal points are absent, we assume the value to be exact (e.g. in math). So here, $2$ is an exact value, and so $2 + 2 = 4$. Further, even in scientific notation, assuming we're dealing with two numbers with two digits' information, say $2.3, 2.4$, then our sum would have two digits, so $2.3 + 2.4 = 4.7 \neq 5$. If, say, we had $1., 1.3, 2.4$, with $1.$ being only one digit, then we'd round our sum to one digit, so $1. + 1.3 + 2.4 = 5$.

Answer 2

That entirely depends on who you ask in what context, or what system you feed with that string.

See this conversation for example:

Eshell V5.9.1  (abort with ^G)
1> is_float(2).
false
2> is_float(2.).
* 1: syntax error before: ')'
2> is_float(2.0).
true