### The PEMDAS Paradox - A resolution

I have spent 4 months arguing, (especially with Americans) over this matter, because they were taught differently, and in my opinion, wrongly.

At the crux of this matter are two principles that are ignored by many:
1. That multiplication is commutative, always.
2. That division is not commutative, and is only partly distributive.

Therefore, a rational person must ask themselves: when I find a math problem that involves both division and multiplication, how do I approach it, in order to retain the full power of the principles above?

Do I enter then shut the door, or do I shut the door then enter?

Do I reproduce then die, or do I die then reproduce?

That level of logic is what is missing.

Multiplication allows you (by commutation), to "walk in or out" as you wish. It allows you to "reproduce" early or late, for as long as you are alive. Once you divide, commutativity is lost (like death - reproduction is impossible afterwards).

Division, on the other hand, shuts the door; or is like the event of death. Once the door is shut, you cannot enter or leave. Once one is dead, they cannot reproduce!

How does this apply here?

Suppose we have 6 ÷ 2(2+1).
We cannot start with division, simply because it BLOCKS multiplication from being commutative.

You see, clearly, 2*(2+1) = (2+1)*2 and that MUST always remain valid and feasible. We cannot prefer or apply any method that denies this principle. This should be self explanatory.

Anyone who suggests anything that denies the commutativity of multiplication opposes math itself!

So, 6 ÷ 2(2+1) can only be resolved by safeguarding the commutativity of multiplication, before we "shut the door."

That is why 6 ÷ 2(3) = 6 ÷ 6 = 1, and not 9.

We cannot ever decide to die first, then start wondering whether reproduction is possible!