# Approximately not equal

What terms do you consider appropriate for the relations denoted by symbols like these:

$$\Large 1.≈\qquad 2.≉\qquad 3.⪅\qquad 4.⪉$$

1. The first one should be easy: “almost equal to” and “approximately equal to” are I think both clear and widely accepted. Personally I prefer “approximately (equal to)”, while Unicode calls this symbol “almost equal to”.
2. The second is harder already. Personally I'd call this “not approximately equal to”. I've heard others call it “approximately not equal” (in a different context). To me, “approximately” by itself means “almost but not exactly”, so it gets me wondering how something can be almost unequal. Is it just me, or would that term confuse others as well? If it's just me being confused, does that mean the term would be acceptable, or would it still sound strange or unprofessional, even though the meaning is clear? Unicode apparently calls this “not almost equal to”, but that might be for typographic reasons.
3. This one I'd call “less or approximately equal”. But could it also be called “approximately less or equal” without becoming ambiguous? Unicode says “less-than or approximate”, switching from almost equal to approximate.
4. This one is hard, I think. I could call it “not greater or approximately equal”, or I could call it “less than and not approximately equal”. The latter is more in line with the typographic rendering. Would something like “approximately less than” or “approximately strictly less than” make any sense to a common audience as well? By the way, Unicode uses “less-than and not approximate” for this symbol.

To clarify: I'm using the symbols to concisely describe the relations, but it's the relations themselves I want a term for, not the symbol I'm using.

As for the precise meaning, suppose that in a given context you have some small $\varepsilon$ defined. Then you could define the relations as

\begin{alignat*}{2} a≈b\;&:\Leftrightarrow&\;\lvert a-b\rvert&\leqq\varepsilon \\ a≉b\;&:\Leftrightarrow&\;\lvert a-b\rvert&\gt\varepsilon \\ a⪅b\;&:\Leftrightarrow&\;a-b&\leqq\varepsilon \\ a⪉b\;&:\Leftrightarrow&\;a-b&\lt-\varepsilon \end{alignat*}

• Could you clarify a little what you want? Are you looking for mostly unambiguous typographic descriptions, or the common terms applied to these symbols in actual math contexts? – GPhys Feb 19 '16 at 0:13
• I've never seen symbols 2 or 4 used before. I would decide what to call them based on what they were being used to mean. There isn't a standard meaning for them as far as I know, so it would depend on the context. – Nate Eldredge Feb 19 '16 at 0:20
• @GPhys: The latter. To me, these symbols convex fairly clear meaning (once you have e.g. an $\varepsilon$ defined so you know when to call something approximately equal). Now I'm looking for established / common terms for that meaning, irrespective of the symbol used for it. – MvG Feb 19 '16 at 1:06
• In Louis CK's show, there's a scene with parking signs reading, "2-hour parking 6AM-5AM, Mon thru Fri", "Parking of vehicles only authorized", "Parking permitted anytime [sic] after midnight", and a green disk with a horizontal white bar through it; Louie is trying to decide if it's legal to park. Now, I notice that the binary relation symbols in question come from Unicode, not from LaTeX. To me, the last three are ambiguous (non-associative parsing), and therefore "ill-advised". Symbols should clarify meaning, not merely abbreviate expression. :) – Andrew D. Hwang Jun 24 '16 at 12:26

$\require{cancel}$ Generally speaking, if we have two relations $R$ and $S$, then we define $$a\,\cancel {R\,}\,b\iff \lnot(a\,R\,b)\\ a\,{}^R_S\,b\iff (a\,R\, b) \vee(a\,S\,b).\tag{*}\label{moi}$$

There are some exeptions, which I personally try to avoid, like $A\subsetneqq B$.

Before I go on, I must point out a crucial inconsistency in the OP. In words the OP says: "approximately by itself means almost but not exactly". At the bottom however, it then reads $a\approx b\iff \vert a-b \vert \leqq \varepsilon$ and not $\underline {a\approx b \iff 0<\vert a-b\vert \leqq \varepsilon}$. For the record, I would like to agree on the underlined definition for "$\approx$", i.e. $a\approx a$ is false.

Now, the definitions of the symbols $\approx$, $\not\approx$, $\Large ^<_\approx$ and $\Large ^<_{\not \approx}$ is completely determined by \eqref{moi}.

Now for the relations themselves, I would just use words like close to eachother, far from eachother, just below/smaller, and a lot/way smaller.
N.B. I do think you need to be more precise about how $\vert a-b\vert$ and $a-b$ relate to zero.

• I hope this is what you wanted. If not, let me know. – gebruiker Jun 24 '16 at 11:25
• The last paragraph is what best answers my question, by providing suggested terms for the various relations. The rest shows me that even the symbols are less clear than I considered them to be. Personally I think that $⪉$ should fall into the same class as $⫋$, i.e. something where the definintion of $(*)$ would not apply since the connection would have to be $\wegde$ not $\vee$. I wasn't aware of the inconsistency, but you are right in diagnosing it. I guess I'd resolve it the other way, i.e. “approximately” as “almost but in general not exactly”, allowing for $a≈a$ as a special case. – MvG Jun 24 '16 at 12:22
• Exactly. The problem is that you probably don't want, $4\Large{{}^<_{\not\approx}}5$. But strictly speaking this would be true, according to (*). I thought that last paragraph was what you wanted, but I wasn't sure. Hence the extra stuff :) – gebruiker Jun 24 '16 at 12:47