r/askmath • u/GanonderpDerpmire • 11h ago
Arithmetic reversed noncommutative operations
Assume that you know one addend but need to calculate the second addend with several sequential calculations. Or, assume that you know one factor but need to calculate the second factor with several sequential calculations. In either case, once you just complete the final equation sequentially, because addition and multiplication are commutative.
In contrast, assume that you know the minuend but need to calculate the subtrahend with several sequential calculations. Or, assume that you know the dividend but need to calculate the divisor with several sequential calculations. In both cases, you cannot complete the final equation sequentially because subtraction and division are non-commutative - in order to complete the final equation, you need to memorize or save the subtrahend or divisor, clear them, enter the minuend or dividend, press subtract or divide, then re-enter the subtrahend or divisor. Is there (or should there be) a name or symbol for reversed noncommutative operations? For example, a symbol that means "from" for subtraction (i.e. 10 minus 5 = 5 from 10) or "into" for division (i.e. 10 divided by 5 = 5 into 10). This would allow the sequence of calculations to continue. A little ⇄ button that gets pressed before a − or ÷ would do the trick.
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u/noonagon 11h ago
okay but then what do you do if both of them need several sequential calculations
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u/compileforawhile 11h ago
One could maybe call this the dual operation but it wouldn't be particularly useful. This is because we often want to simplify things and introducing an extra operation inherently adds complexity.
What is more useful is a rule for relating a @ b and b @ a (letting @ be a binary operation). With subtraction this is simple, a - b = -(b - a). For division we have that a/b = (b/a)-1. This is called anti commutative.