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What is the source, or "status", of the rule that multiplication is performed before addition?

Is it a definitive property of $\mathbb R$, a property that can be derived directly from the definition of $\mathbb R$, or simply a (universal but arbitrary) notational convention?

For example, in Spivak's Calculus (4E, p. 7) the distributive law ("P9") is stated as $$a \cdot (b+c) = a\cdot b + a\cdot c \text{ ,}$$ which is simply understood, without further justification, to mean $$a \cdot (b+c) = (a\cdot b) + (a\cdot c)\text{ .}$$ Should this interpretation have been derivable from the earlier-stated properties of $\mathbb R$, or is it simply being assumed as shared knowledge?

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Note that this is not a repeat of the about the pernicious memes regarding how to evaluate simple arithmetic problems, or about the "order of operation" rules used to to dismiss them; but rather about the origin and mathematical "status" of one of those rules. – raxacoricofallapatorius Jan 8 '12 at 19:19
It is merely a notational convention that allows one to omit parentheses in order to obtain more concise notation. – Bill Dubuque Jan 8 '12 at 19:22
@BillDubuque: That looks like an answer to me. – raxacoricofallapatorius Jan 8 '12 at 19:45
up vote 14 down vote accepted

Such operator precedence rules are merely syntactic conventions that are adopted for convenience. They improve the conciseness of commonly denoted expressions by allowing one to omit some parentheses, while preserving unique readability (parsing) of expressions. For example, such conventions enable the standard concise notation for ubiquitous polynomial expressions. For another example see my answer here. In other words, such syntactic conventions are simply optimizations of the language used to denote certain expressions. As conventions, they have little if any semantic significance.

[Comment migrated to an answer per request]

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Note that under a different convention precedence can be different. For example, under Łukasiewicz notation (Polish Notation or [R]PN) the order of operations is entirely different, in fact it is trivial.

I think that the questioner also wanted to know the source of such conventions. presents a quite thorough history of operator precedence and operation grouping. On point:

The convention that multiplication precedes addition and subtraction was in use in the earliest books employing symbolic algebra in the 16th century.

Although precedence seems to have existed in its current form for as long as the math has been used, there are some interesting exceptions. For example:

an early notation in which a multiplication would be replaced by a comma to indicate aggregation:

  n, n - 1 

would mean $n(n - 1)$


 n n - 1 

meant $n^2 - 1$

However - this is just the history, as @BillDubuque correctly states in his answer, these are all merely a notational convention.

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That multiplication precedes addition is a notational convention, not a mathematical proposition.

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Why repeat my comment that was posted a half-hour earlier, without adding anything new? – Bill Dubuque Jan 8 '12 at 20:42
Sorry, I missed the comments. – Michael Hardy Jan 8 '12 at 23:23

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