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Educators developed an acronym to help students remember how to apply the order of operations. Unfortunately it has a flaw, it does not consider multiplication by juxtaposition. How this came to be is unknown, it could have been deliberate or simply an oversight.

When calculators came on the scene that did apply multiplication by juxtaposition then educators had a problem. In North America manufacturers were asked to drop multiplication by juxtaposition in calculators since it created a conflict with textbooks which applied PEMDAS strictly. A few manufacturers did comply (Casio comes to mind) which resolved the conflict in NA and assured of sales of calculators in that market. This then created another conflict in other nations where multiplication by juxtaposition was taught. Several manufacturers have subsequently switched back to applying multiplication by juxtaposition. And then we have online calculators which also disagree with each other.

This is a case of educators trying to make life easier for students and then breaking mathematics using arithmetic rules.

In my mind Arithmetic is a mechanical calculation to arrive at a single evaluated number, or the 'answer'. Mathematics goes way beyond Arithmetic and is more involved in solving problems through the application of logic.

Another reason North America caught the PEMDAS bug is that America loves multiple choice tests. It allows educators to grade students using automation and is focused on students getting the right 'answer'. I was always graded in school mostly on my method and I would only lose a small amount of my final grade if the answer I arrived at was "wrong' due to a calculation mistake. Showing your method was a requirement to get a passing grade. In the US that has largely been lost in grade schools. I believe this is why Americans that have been taught PEMDAS and have been graded on their ability to get the right answer argue for the strict application of PEMDAS rather than look at the problem mathematically. Adding parentheses to "encapsulate" multiplication by juxtaposition would make mathematical formulas harder to read and write (by hand) and certainly make solving mathematical problems burdensome. Mathematicians are less interested in the minutia of evaluation and focus on the problem at hand.

In my mind logic should prevail insead of being a slave to arithmetic rules that may have limitations.

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