What is the logic behind PEMDAS?

Question

I know some places use BIDMAS as well. Overall though, to someone who knows little about mathematics, why is this the correct order? What’s the exact logic behind it

In: 6

Answer 1

Without having an order of operations, you would need to have parentheses everywhere, because all the basic arithmetic operations are only defined on two things at a time. Further, two different orders of doing the operations usually yields different results, so it does matter.

But why the particular order that we picked? The glib answer is “because people agreed.” A more insightful answer is that the more complicated operations happen sooner. Multiplication is repeated addition, so more complicated than addition, and happens sooner. Exponentiation is repeated multiplication, so more complicated than multiplication, and happens sooner. This makes the order of operations consistent, and does a good job of eliminating the need for many uses of parentheses.