# Notation of exactness

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.

• Perhaps there are computer languages here $2.$ means exactly $2$. But I have not seen that convention in mathematics. – André Nicolas Nov 26 '14 at 16:38
• $\displaystyle\lim_{2\to\infty}(2+2)=5$. – Lucian Nov 26 '14 at 17:38
• @Lucian WolframAlpha says lim_(x->infinity) (2+2) = 4. – Cees Timmerman Nov 27 '14 at 8:47
• WA has lots of bugs. – Lucian Nov 27 '14 at 9:37

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 $1. + 1.3 + 2.4 = 5$.

• Okay, so in math "2 apples" are exactly two apples, but in other science i have to write "exactly" if it's not clear from the context? Then what about the mathematical proof that 2 + 2 = 5? – Cees Timmerman Nov 26 '14 at 15:47
• In science, you'll almost never encounter exact values, except in formulas. However, a number without dots is typically assumed to be exact, while a dot will denote the information in a number. For example, I might multiply by $\pi$ or $2$ in a formula, and those will be exact. But other constants, e.g. if I encounter $G$, will not be, and have information indicated by the dots. – AJY Nov 26 '14 at 15:51
• Almost never is not never. How is this correct when your answer is also correct? – Cees Timmerman Nov 26 '14 at 15:56
• Values denoted by letters may be mathematical constants (e.g. $\pi$) or experimental (e.g. $G$, the universal gravitation constant). A numerical number may be exact (e.g. $2, \frac{9}{7}$) or experimental (e.g. $9., 7.253$). For letters, it's typically known whether a letter is experimental or exact. For numericals, the dots or lack thereof denote exactitude or information. – AJY Nov 26 '14 at 16:08
• It appears that i misread that answer; it proves that the given proof is incorrect. – Cees Timmerman Nov 26 '14 at 16:11

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

• In Python, 2. is 2.0. As linked, 2 + 2 = 5 here. How is that possible if those values are exact? – Cees Timmerman Nov 26 '14 at 15:54
• I thought math as accepted here was universally true. – Cees Timmerman Nov 26 '14 at 16:00
• My bad, that answer proved the proof for 2 + 2 = 5 false. – Cees Timmerman Nov 26 '14 at 16:17
• We would need to dig deeper what a statement is and how to proof it wrong or false. Usually a proof means reducing the statement under consideration down to the basic truths, the axioms. Those axioms need to be assumed true or false. And then generate their own set of true, false and undecidable statements. So maybe the mathematical reasoning has a chance to be universal, the statements are not. See the axiom of parallels, the Banach Tarski pardox, Gödel and so on. – mvw Nov 26 '14 at 16:25
• So 2 + 2 = 4 is not universally true, because in apples, people may not believe that the second pair of apples exists? – Cees Timmerman Nov 26 '14 at 16:35