# Reverse subtraction: has any culture had a symbol (call it $\oplus$) where $A \oplus B$ (read in the same direction as in the language) $:= B - A$?

The standard use of the minus sign is such that $$A-B$$ means you subtract B from A. Thus $$5-2 = 3.$$ Has any culture used a symbol (let's call it $$\oplus$$) where $$A \oplus B$$ means you subtract A from B? Thus you would get $$2\oplus5=3.$$ Note that in languages written from right to left, this question has to run in that direction. What I am interested in is the case when reading symbols written in the standard direction (whichever direction that is) you would say "A [symbol] B" and get the answer $$B-A$$.

That 3 is said to be equal to $$5 \;[\text{symbol for subtraction}] \;2$$ rather than $$2 \;[\text{symbol for subtraction}] \;5,$$ and more generally we write $$[\text{minuend}]\;[\text{symbol for subtraction}]\;[\text{subtrahend}],$$ is of course convention. I am interested in whether any cultures have used a symbol for "reverse subtraction"; and if so, how widespread its use has been, both in absolute terms and relative to the use of a minus symbol with the usual meaning. Perhaps there have been cases where the "reverse" symbol has been the standard one, or the only one; or perhaps two symbols have been in more or less equally widespread use, one for one meaning and one for the other?

• semi-related: we do have symbols for right- and left- matrix operations, at least in software. May 18, 2021 at 11:38
• And there's always the not-what-you-meant $-A + B$ notation :-) May 18, 2021 at 11:39

It may remain a mystery in some cases. Britannica states that the ancient Egyptians wrote right-to-left but also states that their algorithm for subtraction is not known.
Then again, Wichita says (sadly, without a picture),

In one example, from the Rhind Papyrus, addition and subtraction signs were represented through figures which resemble the legs of a person advancing for addition, and departing for subtraction.

They did have a horizontal "equals" glyph.

• Thanks for this. If the ancient Egyptian scripts said something like "2 (runs away from) 7 (to give) 5", that would be the kind of thing I'm looking for. But perhaps it was more like "7 (and then the following quantity runs away) 2 (to give) 5". Then there is Aymara culture. Their concept of the future is physically "behind", with the past conceived as physically "in front". That's the reverse of how relative position in time is considered to correspond to relative location in space in most cultures, so perhaps an Aymara notation for subtraction may put the subtrahend before the minuend.
– user13571
May 22, 2021 at 8:10
• Interesting how the conceptualisation that runs "this is an amount, and then I'm going to take this other amount away from it" is standard, perhaps even to the exclusion of the conceptualisation "this is an amount I want to take away, and here's a big heap I will take it away from". (There's division too. Nothing wrong with calling 5/2 "five over two" or "two under five" according to circumstances.)
– user13571
May 22, 2021 at 8:14

The closest thing I can think of are symbols that mean "(the absolute value of) the difference between", a sort of "commutative subtraction" operator:

$$4 \sim 5 = 5 \sim 4 = 1$$

I have access to a copy of Webster's New International Dictionary, Second Edition with a (publication?) date of 1949 on the title page, copyright date of 1934, and new words section copyright 1939 and 1945. The appendix "Arbitrary Signs and Symbols", section "Mathematics", sub-section "Relations and Operations" on page 3005 contains the following entries:

$$\sim$$ or $$\mathrel{-}:$$   Difference.

$$\sim$$ ~ or $$\bumpeq$$   The difference between ; used to indicate the difference between two quantities without designating which is the greater ; as, $$a \sim b$$ ; that is, the difference between $$a$$ and $$b$$.

$$\mathrel{-}:$$   The difference between; excess.   Rare.

In the Unicode Mathematical Operators block, these symbols are called

∼ U+223C TILDE OPERATOR ("= difference between")
≏ U+224F DIFFERENCE BETWEEN
∹ U+2239 EXCESS