Original by Don Koks, 2022.

# What is 1 / 2 (3 + 4)?

This question has caused much fighting on the web.  Some people say it's 7/2.  Others say it's 1/14.  The "right answer" can only be set by some mathematical convention that everyone accepts; but no such convention is accepted by all.  And yet it turns out that physics has something to say on the subject.

The combatants cite all manner of rules to support their preferred answer to the question.  More on that later, but for now, we'll encapsulate the arguments in two rules:

• Presumably, those who say that the answer is 7/2 have applied the following rule (which we'll call rule A): "The operations +, −, ×, ÷ are implemented from left to right, with × and ÷ implemented before + and − are.  Juxtaposition is interpreted as the presence of ×.  Expressions in parentheses are evaluated independently of everything else".  Rule A says that 1 / 2 (3 + 4) = 1 / 2 × 7 = (1 / 2 ) × 7 = 7/2.

• Presumably, those who say that the answer is 1/14 have applied the following rule (which we'll call rule B): "The operations +, −, ×, ÷ are implemented from left to right, with × and ÷ implemented before + and − are.  Juxtaposition represents a multiplication that is implemented before anything else.  Expressions in parentheses are evaluated independently of everything else".  Rule B says that 1 / 2 (3 + 4) = 1 / (2 × 7) = 1/14.

Rule A is clearly simpler than rule B.  Irrespective of that, can we determine if one of the rules is more consistent than the other, by appealing to something that everyone agrees with?  It turns out that we can.  Consider that juxtaposing a number with a unit, such as "2 seconds", can be (and is) routinely treated mathematically as multiplication by that unit; namely, "2 × (one second)".  (For more on that, see the FAQ entry Can you take the logarithm of a dimensioned quantity?)  In the same way, "1/2 second" is routinely treated as "(1/2) × (one second)".

• Rule A interprets "2 seconds" to mean 2 seconds, and "1/2 second" to mean 1 / 2 × (one second), which is (1 / 2) × (one second), which is 0.5 seconds.

• Rule B interprets "2 seconds" to mean 2 seconds, and "1/2 second" to mean 1 / (2 × (one second)), which is 1 / (2 seconds), which is 0.5 hertz.

Independently of these rules, everyone agrees that "1/2 second" denotes 0.5 seconds, and not 0.5 hertz.  This is only consistent with applying rule A.  That is, everyone accepts rule A's interpretation of "1/2 second" and rejects rule B's interpretation of the same.  Because that's about the one thing that everyone does agree on, it makes sense to base a convention on that agreement, and thus accept rule A.  So, if you want consistency in your application of binary operators, follow rule A, and not rule B.  Consistency is a core principle of mathematics, and applying it forces us to conclude that 1 / 2 (3 + 4) equals 7/2.

Assorted variations on rule B exist that put all multiplications before divisions, or perhaps all divisions before multiplications.  Such rules probably owe their origin to a mis-reading of an acronym.  Rule A has traditionally been represented by some choice of simple acronym whose letters denote the operations in order of precedence.  "M" denotes multiplication, "D" division, and so on.  Both "MD" and "DM" that appear in such acronyms have traditionally been given a single meaning: "do the multiplications and divisions, with neither taking precedence over the other, apart from the usual left-to-right order".  (The same applies to "AS" and "SA".)  But such an acronym with, say, an "MD" is sometimes read completely literally, and it ends up being interpreted incorrectly as "do all multiplications first, and all divisions afterwards".  That was never the acronym's intent, but no simple acronym is capable of indicating that multiplication and division have rule A's equal order of precedence.  These acronyms should be understood for their original meaning, and should not be interpreted completely literally.  A better approach is not to use them at all.  It's really not difficult to learn an order of precedence without requiring an acronym.

Programming languages and modern calculators (but not some older ones) use rule A.  But rule B's practice of writing "a/bc" to mean a/(bc) appears widely in physics textbooks and some journals—it might even be sometimes introduced against authors' wishes or their knowledge by journal typesetters, because modern typesetters often have little maths knowledge.  Keep that in mind when reading journals.