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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?

Corey
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  • Regardless of whether it's valid or not, it can be considered ambiguous, and hence it is advised to include brackets. – Shuri2060 Aug 24 '17 at 23:50
  • Why is it not interpreted as $(5/10)/(20/40)=(1/2)/(1/2)=1$? – Sahiba Arora Aug 24 '17 at 23:50
  • I am not a math person. The reason I ask this question is because I edited the page to remove the inline expression and focus on the formal versions with varying line lengths below, which is what the first sentence refers to. My edit was not only reverted but accompanied by this note: "The un-parenthesized inline expression is invalid" – Corey Aug 24 '17 at 23:50
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    $1/2/3$ can be interpreted as $(1/2)/3=1/6$ or $1/(2/3)=3/2$. Do you see now why your example is ambiguous? – Batominovski Aug 24 '17 at 23:52
  • @Shuri2060 acknowledged, but the question is specifically about its validity. – Corey Aug 24 '17 at 23:52
  • @Batominovski my opinion about it's ambiguity is outside the scope of this question. – Corey Aug 24 '17 at 23:54
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    Note that computer languages will have an associativity defined for division. However that doesn't mean in manual usage there is a commonly defined associativity. So if you're writing for a computer, you're OK, but if you're writing for other people then explicitly define the order of operations - which can be done in several ways: $(a/b)/c$ or $\frac{\frac{a}b}{c}$ are a couple (it may be hard to tell but the horizontal line above $c$ is longer than the one above $b$). – Χpẘ Aug 24 '17 at 23:55
  • @Batominovski Sweet, thanks for the link. – Corey Aug 24 '17 at 23:57

2 Answers2

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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.

  • I missed a lot but just for demonstration ( done in PARI/GP):

    1+23-4/5=31/5; (1+2)3-4/5=41/5; (1+23)-4/5=31/5; (1+23-4)/5=3/5; 1+(23)-4/5=31/5; 1+(23-4)/5=7/5; 1+(23-4/5)=31/5; 1+2(3-4)/5=3/5; 1+2(3-4/5)=27/5; 1+23-(4/5)=31/5; 1+(23)-(4/5)=31/5; (1+2)3-(4/5)=41/5; (1+2)(3-4/5)=33/5; (1+2)(3-4)/5=-3/5; 1+2((3-4)/5)=3/5; 1+(2(3-4)/5)=3/5; (1+2*(3-4)/5)=3/5

     –  Aug 25 '17 at 00:26
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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.

Franklin Pezzuti Dyer
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