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Why was the order of operations established in mathematics with multiplication taking precedence over addition, as dictated by the PEMDAS rule? What historical or practical factors influenced this choice? Additionally, what would be the implications for mathematical consistency and computation if this hierarchical order were changed?

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    $\begingroup$ This is probably better suited for History of Science and Mathematics Stack Exchange. I haven't bothered (at least at the present moment) to double-check this, but I'm pretty sure the addition/multiplication convention was in effect before superscripted numbers were used (or at least, before they were in common use) for repeated multiplication. Thus, expressions like $9xxx + 5yy$ were used around 1800 and earlier, I think. $\endgroup$ Nov 11, 2023 at 15:02
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    $\begingroup$ There is a small section here on the topic mathshistory.st-andrews.ac.uk/Miller/mathsym/operation $\endgroup$
    – Adam
    Nov 11, 2023 at 15:34
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    $\begingroup$ See hsm.stackexchange.com/questions/8417 $\endgroup$
    – user12357
    Nov 11, 2023 at 17:24

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I don't have historical proof but the following explanation feels so intuitive to me that I'd be shocked if there were a different reason.

When you use multiplication and addition for simple real-life applications, you're usually trying to find the total of "some number of this and some number of that."

For instance, if you buy two $3 \, \textrm{kg}$ bags of rice and four $5 \, \textrm{kg}$ bags of wheat, then how many kilograms of grain do you have in all?

$$(2 \times 3) + (4 \times 5),$$

not

$$2 \times (3 + 4) \times 5.$$

Because we do multiplication first by convention, we can just write

$$2 \times 3 + 4 \times 5$$

without parentheses and by default it will still represent the real-life situation that we are modeling.

But if we did addition first, then we would have to introduce explicit parentheses around the multiplications $(2 \times 3) + (4 \times 5)$ pretty much every time that we wanted to write down an expression for a real-life modeling situation, because $2 \times 3 + 4 \times 5$ by default would not represent what we are intending.

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    $\begingroup$ Yup, and when one sloppily composes a shopping list in the first place like "2 apple 3 banana" (where notation such as "apples: 2; bananas: 3" might be philosophically tidier), the implicit operations and intended groupings remain clear, coinciding with the usual order of operations. $\endgroup$ Nov 11, 2023 at 15:37
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    $\begingroup$ You're saying that in real life, more often do we add products, than multiply sums? Possible. $\endgroup$ Nov 21, 2023 at 21:10
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It might be related to the order (some of) these operations appear on the hyperoperation sequence

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