# Is there a way to explain or prove that there is not a single order of operations on the real numbers? [closed]

Given that addition and multiplication are both commutative and associative, Expressions containing only addition or multiplication can be evaluated in a any order. However, when the operation changes from addition to multiplication or from multiplication to division it is necessary to specify which operation to perform first because multiplication distributes over addition.

This seems somewhat unsatisfactory to me because it relies on the assumption that there is no actual preferred order of evaluation built into the real numbers. I was wondering if I might have some help fleshing this out in a way that explicitly addresses the putative assumption that there is no explicit order of operations on the real numbers.

Starting with the field axioms, can one prove that there is a single interpretation for unbracketed real-number expressions?

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• To correct for this, brackets are used to avoid ambiguity. – KennyB Aug 3 '17 at 23:04
• How is a abstract algebra not an appropriate tag? The question is about how to derive a statement from the axioms of a field. – Michael McGovern Aug 3 '17 at 23:07
• @Mr.Brooks so now I added the philosophy tag. – user451844 Aug 3 '17 at 23:10
• I think this question is not about properties of the reals or the integers so much as about properties of notations for computations. Consider these alternatives to parenthesized formulas for representing computations: evaluation trees and reverse Polish notation. Neither of these need "order of operations". – kimchi lover Aug 3 '17 at 23:15
• Order of operations is pure convention. For the most part it's arbitrary and a matter of convenience. You have to separate the notation from the actual mathematics. Fields don't care what order you apply their operations in, you end up with compositions of functions of two variables. If there's ambiguity, it's not well defined. – Matt Samuel Aug 3 '17 at 23:45

We could simply say as convention that all operations of more than two operands must absolutely have parenthesis around every pair and that if the parenthesis are missing they expression is meaningless.

So $a + b*c + d*e +f$ (as we know it) must be written as $(((a + (b*c)) + (d*e))+f)$ This is, of course, tedious.

We could have as a convention that we always go from left to right, always. So the expression $a + b*c + d*e + f$ would now mean $((a+b)*c + d)*e + f$ and to write the expression we meant we'd have to writh $a + (b*c) + (d*e) + f$.

This is perfectly acceptable.

However given that we have the distributive law that $a(b+c) = (ab) + (ac)$ there is an incentive to view $a*c + b*c$ so $(a*c) + (b*c) = (a+b)*c$ rather than as $(a*c + b)*c = a*c^2 + (b*c)$. Our convention is consistent, less capricious, and usefl and insightful.

• But my question was whether the convention we use is provably the only one constituent with the axioms of real numbers. – Michael McGovern Aug 4 '17 at 15:15
• They all are constituent with the axioms of the real numbers. But ours using precedence of multiplication has direct bearing to the distributive law. – fleablood Aug 4 '17 at 15:38

Suppose we restrict ourselves to just multiplication and addition. The distributive law usually reads: $$a(b+c) = ab+ac$$

Which we understand to mean: $a(b+c)=(ab)+(ac)$, not $a(b+a)c$ or $(a(b+c))a$. This is as close as I can find to an implied order of operations from the field axioms. Personally I'd argue this is the writers abusing a standard, not intended as an actual definition for order of operations on the reals. Nobody would object to field axioms in post script reading: $$\cdot (a, + (b, c)) = + (\cdot (a ,b), \cdot (a, c)).$$ (parenthesis added for notational prettiness, they aren't required for this to be unambiguous)