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In the UK, the abbreviation BIDMAS/BODMAS is used to remember the order of mathematical calculations:

Brackets

Indices / Others

Division

Multiplication

Addition

Subtraction

In the USA, the abbreviation PEMDAS is used instead:

Parenthesis

Exponents

Multiplication

Division

Addition

Subtraction

In the UK, division is first, followed by multiplication, but this order is reversed in the USA.

It seems to me that this can result in different answers for the same question, for example

What is $1 \div 2 \times 2$?

In the UK, with BIDMAS/BODMAS, division is performed first:

\begin{align} 1 \div 2 \times 2 &= 0.5 \times 2 \\ &= 1 \end{align}

Whereas in the USA, with PEMDAS, multiplication is performed first:

\begin{align} 1 \div 2 \times 2 &= 1 \div 4 \\ &= 0.25 \end{align}

So what is the actual value of $1 \div 2 \times 2$? Or am I misunderstanding something?

Aryan
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    Multiplication and division have equal priority, so evaluate from left to right (addition and subtraction also have equal priority) – J. W. Tanner Oct 27 '20 at 13:12
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    @J.W.Tanner so have I misunderstood the meaning of BIDMAS/BODMAS/PEMDAS? – Aryan Oct 27 '20 at 13:13
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    Any source that in the USA / in UK , this is the way how it is calculated ? – Peter Oct 27 '20 at 13:14
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    Yes. The letters have to go in some order. Ones next to each other (M,D) and (A,S) are of equal priority. We have many questions like this, but they are hard to find because the numbers change. – Ross Millikan Oct 27 '20 at 13:14
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    There is no reason why anyone should ever need to use a confusing expression like $1 \div 2 \times 2$. If someone writes this and causes confusion, it's the fault of whoever wrote it, not whoever is reading it. – Clive Newstead Oct 27 '20 at 13:15
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    To stress: no sensible person would write an expression like this unless the intent was to highlight the ambiguity. When in doubt, add parentheses to clarify. – lulu Oct 27 '20 at 13:15
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    Where I live, we are taught to avoid ambigous notation – Hagen von Eitzen Oct 27 '20 at 13:15
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    If you thought one says division takes precedence over multiplication and the other says the opposite, then you thought wrong – J. W. Tanner Oct 27 '20 at 13:15
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    @CliveNewstead: It is not at all confusing. See every single modern programming language. What you need is proper pedagogy teaching conventions, not knee-jerk reactions blaming established conventions. – user21820 Oct 28 '20 at 07:52
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    @lulu: You're wrong. See above comment. I am sure you also write expressions like "$3-2+1$" and nobody says that you are not a sensible person. Why do you have such a strange reaction to the use of "$÷,×$" in the same manner following the same left-to-right convention for equal-precedence infix binary operations?? – user21820 Oct 28 '20 at 07:54
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    @user21820 Because the latter, $3-2+1$, is not confusing and the other is. I don't think "confusing" is a subjective or ambiguous term. I have never seen anyone ask about an expression of the form $3-2+1$. By contrast, people are constantly arguing over $1\div 2\times 2$ and such. To me, that proves the notation is confusing, the fact that there is an official resolution to the ambiguity does not alter the fact that it causes confusion. I think the point of mathematical notation is to be clear and precise, so if simply adding parentheses removes a persistent confusion, I say do it. – lulu Oct 28 '20 at 11:11
  • @lulu: cf. this question and this question – J. W. Tanner Oct 28 '20 at 11:27
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    @J.W.Tanner Fair enough. "Less confusing", then. And it's important, perhaps, to remark that the ubiquity of, say, polynomial expressions, requires us to have a universally accepted convention for strings of additions and subtractions. If we had to constantly replace, say, $x^4-2x^3-x^2+x-1$ with, e.g., $((x^4-2x^3)+(-x^2+x))-1$ we'd go mad. There is no commonly used analog for strings of multiplies and divides. – lulu Oct 28 '20 at 11:37
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    @CliveNewstead As so often, I am reminded of an xkcd cartoon. – lulu Oct 28 '20 at 12:29
  • @CliveNewstead: It was your initial comment that was aggressive. I was just stating facts. Doesn't matter whether you want to believe facts or not. – user21820 Oct 28 '20 at 12:42
  • @lulu: I'm not saying don't put brackets. All I am saying is don't put brackets just as a knee-jerk reaction to the actual root problem of bad pedagogy. – user21820 Oct 28 '20 at 12:43
  • @CliveNewstead: To clarify, your initial comment gives the fallacious impression that if person X does not understand written expression E then it is always the fault of the author of E, and that is aggressive. Why do you not consider the possibility that X has a defective understanding due to those who did not teach them properly? – user21820 Oct 28 '20 at 13:01

2 Answers2

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In the USA, the M/D and A/S in PEMDAS are treated as single entities. They should be calculated from left to right, as they would be in the UK. PEMDAS is not a perfect method of understanding order of operations, and it can be confusing for those who are first learning it.

J. W. Tanner
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Sage Stark
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The "actual value" of that ambiguous expression depends on the conventions adopted by the evaluator - that might be a society, or a programming language. It's not a question with a mathematical answer.

Avoid writing those expressions. Use extra parentheses for clarity.

Ethan Bolker
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    Doesn’t everyone (at least those who read from left to right) have the convention to evaluate the left operation before the right one when the operations (e.g., multiplication and division) are at the same level in the hierarchy? – J. W. Tanner Oct 27 '20 at 13:21
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    @J.W.Tanner If everyone knew that the OP wouldn't have had to ask the question ... – Ethan Bolker Oct 27 '20 at 21:12
  • Have you ever seen a convention among those who read from left to right to evaluate the right operation before the left operation when the operations (e.g., multiplication and division) are at the same level in the hierarchy? – J. W. Tanner Oct 28 '20 at 02:41
  • @J.W.Tanner $a^{b^c}$ is properly read as $a^{(b^c)}$ for example. At least I think that is standard....I note that Wolfram Alpha agrees with me while Excel, surprisingly (at least to me), interprets, e.g., 2^3^4 as $8^4=4096$ – lulu Oct 28 '20 at 13:39
  • @lulu: yes, I had thought of that, and that's why I gave multiplication and division as the example – J. W. Tanner Oct 28 '20 at 13:45