- Solve: 6 ÷ 3 ( 2 + 1 )
- How do you know that the answer you arrived at is correct?
So you’ve stumbled upon this blog, wondering why some nuanced anime critic who seems to think that talking about Philosophy is cool is now talking about Mathematics. Well, despite the fact that Philosophy critiques a multitude of disciplines, and goes hand-in-hand with just about any field, some of you are really stubborn about your worldviews in misunderstanding Philosophy, so I don’t bother trying to justify it.
You know what else some of you are really stubborn about? Order of Operations problems!
You don’t have to look very far for these. Somewhere on the Internet, you will find memes of Math problems, out to sarcastically expose how many people will get the wrong answer, as well as show how ineffectively the comment section will try to resolve these disputes!
But the thing that makes me want to punch my computer screen the most is the common justification for answers on these tricky memes: “PEMDAS.”
To be fair, those who know what they’re doing happen upon the answer correctly. However, I also see PEMDAS being used as the justification where the answer is flat-out wrong! So, what’s the deal with these divergences in evaluating problems, even if we agree on the rule? Let me explain.
What is the Order of Operations, and what does PEMDAS got to do with it?
Order of Operations is the Mathematical standard for solving, evaluating, or simplifying all Math problems that use a variety of operations. You will find these problems in elementary school, but the rule will carry on into your high school and university level courses that use Mathematics as well. Order of Operations standardize how to solve any calculation, so that we all arrive at the same answer to any specific problem without fail, and yet we still do, because of human fallibility.
The rules of the Order of Operations goes as follows:
- Grouped terms (e.g. terms inside parentheses, brackets) must be solved first. Grouping, by definition, are operating terms that are forced to be together.
- Exponential operations are solved second.
- Multiplication and Division operations are solved third.
- Addition and Subtraction operations are solved last.
So how do you memorize this order? A mnemonic device like PEMDAS will suffice. PEMDAS (or BODMAS) help students memorize the order based on which operation fits from beginning to end. The rules of PEMDAS are as follows:
- Parentheses (or Brackets)
- Exponentials (or Orders)
- Multiplication and Division, solved from left to right.
- Addition and Subtraction, solved from left to right.
And here is where the confusion lies. If you solve your Order of Operations using these rules above, you should be fine; and you can calculate all of your Math Problems in the correct order. But the problem lies not in these rules, but the misinterpretation of the mnemonic, because “PEMDAS” assumes that there are six orders of operation, not four! And the most common mistake that a people forget that Multiplication and Division, as well as Addition and Subtraction, are solved at the same time!
In other words, your PEMDAS rule could just as easily be PEMDSA, PEDMAS, or PEDMSA!
So which one of these mnemonics is the best one to use for all Order of Operations? None of them! Because unless you recall that there are only four orders of operation and not six, grounding yourself on a catchy letter sequence isn’t going to help matters.
So what grounds the Order of Operations?
Let’s be frank: rules in logic are important to follow. Otherwise, the logic won’t get us any closer to the truth or precision of problems that we face. But if you’re going to ground a rule in Mathematics, you want to make a system that uses the least amount of operations possible. That way, we have a foundation — a grounder — for all the rules that follow. And what might that be?
Imagine for a moment the most basic calculator in the world. So basic, the only thing it can do is add 1 to piles of more 1’s, but it can do it infinitely many times, and at lightning speed. What you get is the primitive recursive function.
Not to go into gruesome detail over this theory, but this one, basic function allows you to get three of the four operations we established before.
- For addition (A + B), you would take a pile of 1s, A, and add 1 to it B times. This is the first compiled function we can derive out of the primitive recursive function.
- For multiplication (A * B), you would use the addition function to add A from zero, B times. This is the second compiled function that we can derive.
- Finally for exponents (A^B), you would use the multiplication function on a base value A, B times. This is the third compiled function that we can derive.
- The grouping or “parentheses” rule is the only one that cannot be derived, as it is a forced rule that supersedes the other three.
These rules, as grounded by the primitive recursion function, establish the logic of the Order of Operations. Whenever you solve a Math problem, you are effectively working out the primitive recursive function backwards, from most compiled function to most basic.
So how do we know that this is the surefire function that gets the closest to the absolute truth about the nature of numbers? According to Philosophy, we don’t know! Perhaps some day, some super genius Mathematician (who probably has no life) will find a number theory that’s even more basic and will still abide by the original rules we observe through the Order of Operations. But given how the primitive recursive function, the Turing Machine, and other foundational proofs about numbers all seem to be functionally equivalent, this logic is the best humanity’s got, and as far as we know, will most likely stay that way until the end of time.
But what about Division and Subtraction operations?
As you can tell, I have tried to convince you that Division and Subtraction don’t exactly matter to the Order of Operations, since the most basic calculator in the world can’t even solve for them without adding another pile of numbers, and you would be right in thinking that that’s crazy! You’re such a believer in PEMDAS, after all.
But what if I told you that Multiplication and Division are functionally equivalent, and Addition and Subtraction are functionally equivalent?
These are more rules that you learn in elementary school that even adults often forget!
Subtraction is functionally equivalent to adding the negative. For example,
5 – 3 = 5 + (-3) = 2.
Division is functionally equivalent to multiplying by the reciprocal. For example,
15 ÷ 5 = 15 * (1/5) = 3.
And in case you forgot that there was an invisible R in PE(R)MDAS (it’s okay, it tends to get irrational), radicals are functionally equivalent to raising a base to the reciprocal power. For example,
√16 = 16^(1/2) = ±4. (Note: these operations may produce more than one evaluations, unless you’re in Engineering, I guess)
Okay. So this post isn’t the proof that will end all Order of Operation meme disputes. This won’t get us any closer to getting all of us on the same page in solving said problems. But what it does provide is proof of how the Order of Operations come to one solution, if you follow the rules properly.
So, what does 6÷3(2+1) equal?
Solve grouped terms first: 6÷3(3).
When in doubt, change divisors to multiplying by the reciprocal: 6 (1/3) (3).
By the commutative property of multiplication, you can solve the remaining terms in this example in any direction, since all of them are multiplication:
2 (3) (from left to right), OR
6 (1) (from right to left)
So the final answer is 6.
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