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Wed, 15 Nov 2017 [ Credit where it is due: This was entirely Darius Bacon's idea. ] In connection with “Recognizing when two arithmetic expressions are essentially the same”, I had several conversations with people about ways to normalize numeric expressions. In that article I observed that while everyone knows the usual associative law for addition $$ (a + b) + c = a + (b + c)$$ nobody ever seems to mention the corresponding law for subtraction: $$ (a+b)-c = a + (b-c).$$ And while everyone “knows” that subtraction is not associative because $$(a - b) - c ≠ a - (b-c)$$ nobody ever seems to observe that there is an associative law for subtraction: $$\begin{align} (a - b) + c & = a - (b - c) \\ (a -b) -c & = a-(b+c).\end{align}$$ This asymmetry is kind of a nuisance, and suggests that a more symmetric notation might be better. Darius Bacon suggested a simple change that I think is an improvement:
The !!\star!! operation obeys the following elegant and simple laws: $$\begin{align} a\star\star & = a \\ (a+b)\star & = a\star + b\star \end{align} $$ Once we adopt !!\star!!, we get a huge payoff: We can eliminate subtraction:
The negation of !!a+b\star!! is $$(a+b\star)\star = a\star + b{\star\star} = a\star +b.$$ We no longer have the annoying notational asymmetry between !!a-b!! and !!-b + a!! where the plus sign appears from nowhere. Instead, one is !!a+b\star!! and the other is !!b\star+a!!, which is obviously just the usual commutativity of addition. The !!\star!! is of course nothing but a synonym for multiplication by !!-1!!. But it is a much less clumsy synonym. !!a\star!! means !!a\cdot(-1)!!, but with less inkjunk. In conventional notation the parentheses in !!a(-b)!! are essential and if you lose them the whole thing is ruined. But because !!\star!! is just a special case of multiplication, it associates with multiplication and division, so we don't have to worry about parentheses in !!(a\star)b = a(b\star) = (ab)\star!!. They are all equal to just !!ab\star!!. and you can drop the parentheses or include them or write the terms in any order, just as you like, just as you would with !!abc!!. The surprising associativity of subtraction is no longer surprising, because $$(a + b) - c = a + (b - c)$$ is now written as $$(a + b) + c\star = a + (b + c\star)$$ so it's just the usual associative law for addition; it is not even disguised. The same happens for the reverse associative laws for subtraction that nobody mentions; they become variations on $$ \begin{align} (a + b\star) + c\star & = a + (b\star + c\star) \\ & = a + (b+c)\star \end{align} $$ and such like. The !!\star!! is faster to read and faster to say. Instead of “minus one” or “negative one” or “times negative one”, you just say “star”. The !!\star!! is just a number, and it behaves like a number. Its role in an expression is the same as any other number's. It is just a special, one-off notation for a single, particularly important number. Open questions:
Curious footnote: While I was writing up the draft of this article, it had a reminder in it: “How did you and Darius come up with this?” I went back to our email to look, and I discovered the answer was:
I wish I could take more credit, but there it is. Hmm, maybe I will take credit for inspiring Darius! That should be worth at least fifty percent, perhaps more. [ This article had some perinatal problems. It escaped early from the laboratory, in a not-quite-finished state, so I apologize if you are seeing it twice. ] [Other articles in category /math] permanent link |