Is 5/10/20/40 a valid mathematical expression?

Question

The Wikipedia article on Fractions says:

If, in a complex fraction, there is no unique way to tell which fraction lines takes precedence, then this expression is improperly formed, because of ambiguity. So 5/10/20/40 is not a valid mathematical expression, because of multiple possible interpretations [...]

The first sentence makes sense, but does the second sentence follow? WolframAlpha interprets that input without issue, as do popular programming languages.

Is the order of operations not accepted in formal math?

Answer 1

Yes, but that's because Wolfram Alpha does that by convention. When you type something like that in, you're probably "confusing" WA, and so it has to use its last resort, which is to apply the operations in the order in which they are typed. Even though WA can interpret it, it's still bad mathematics to write a complex fraction that way if you want anyone to know what you mean.

Answer 2

it's only because of the way it interprets it potentially. The reason the order of operations is needed is to stop ambiguous answers. in this case with parentheses added around the divisions you can make it equal 1, or ${5\over(10*20*40)}= {1\over1600} $,etc. some arithmetic without implied parentheses by order of operations would have 24 answers for just 4 operations.