Q&A: Why is the Order of Operations the Way That It Is?

by Justin Skycak (@justinskycak) on

Cross-posted from here.

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Question

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?

Answer

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?

$$\begin{align*} (2 \times 3) + (4 \times 5), \end{align*}$$

not

$$\begin{align*} 2 \times (3 + 4) \times 5. \end{align*}$$


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

$$\begin{align*} 2 \times 3 + 4 \times 5 \end{align*}$$

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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