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Your argument is circular in nature
No it isn't. The order of operations rules are derived from the definitions of the operators - there's nothing circular about that.
The right answer is dependent upon the order that you do the operations
And so to get the correct answer, you have to do the operations in the correct order, yes.
So you canβt use the argument that the order of operation matters
But you just said it did when you said "The right answer is dependent upon the order that you do the operations"!
if you do the math not using the correct order of operations, the answer comes out incorrectly
Exactly! Glad you agree with me. :-)
we could just do away with order of operations all together and explicitly use parentheses everywhere
Except I showed here that most people put brackets in the wrong place to begin with.
So thereβs nothing inherently correct about our current order of operations
I already gave the proof of why it is. We already know that 2x3=2+2+2 by definition, so therefore 2+2x3=2+6=8. If you did addition first you'd get 2+2x3=4x3=12, which is demonstrably wrong. Welcome to how the order of operations rules were derived to begin with.
I will say that I like it and that it makes sense to me
It makes sense because it's been proven to be correct.
Ok, but why does 2 * 3 = 2 + 2 + 2 mean that we should prioritize multiplication and division over addition and subtraction? I don't understand this argument. This seems to be your central point, but I don't see how math would be broken if we flipped the order? It just feel like a story you are overly attached to. I'll take a source if you got one.
Take 2 + 3 * 4 for example. If you use our normal order of operations. We would interpret this as (2 + (3 * 4)) = 14. This is correct and fine. If you flip the order of operations it becomes ((2 + 3) * 4) = 20. This is also correct and fine. All the order of operations does is dictate how we interpret the equation. As I listed in my post there are even notations where equations can be written without considering the order of operations like RPN. In RPN ((2 + 3) * 4) becomes 4 2 3 + *
And so yeah, if everyone agreed to flip the order of operations, we would need to re-write a lot of equations, but we could do it and there wouldn't be an issue. Heck we could all agree to switch to RPN and then we could get rid of the order of operations all together, because it just a way to interpret equations that are ambiguous due to infix notation. I am not saying we should, I am saying we could and nothing would break.
Ok, but why does 2 * 3 = 2 + 2 + 2 mean that we should prioritize multiplication and division over addition and subtraction?
Because multiplication is shorthand for addition, and if you don't expand it before doing the addition you get wrong answers. Let me show you...
Multiply first (i.e. correct)
2+3x4=2+3+3+3+3=14 - the right answer, by the very definition of 3x4=3+3+3+3
Now let's see what happens if we do the addition first...
2+3x4=5x4=5+5+5+5=20 - which we know is the wrong answer! (because we already know the right answer is 14, because we already know that the actual original fully expanded expression was 2+3+3+3+3, so the rules of Maths have to guide us back to getting the same thing the original author started with, or it all breaks down! The author took 2+3+3+3+3 and wrote it as 2+3x4, so the rules of Maths have to make sure when we see that we get back to 2+3+3+3+3)
So the fact that we know multiplication is shorthand for addition, means we have to multiply before we add. Similarly, exponents are shorthand for multiplication (2Β³=2x2x2), so we have to do exponents before we do multiplication... which we have to do before we do addition! It all comes from what these very things have been defined as meaning in the first place.
Iβll take a source if you got one
Well, I just showed you by doing the Maths myself, which is one of the great things about Maths - some things you can prove it to yourself! :-) And that's another topic I wrote about here and here.
I still don't see where the correctness is coming from. Also, when I was asking for sources I was looking for other experts to backup your claims. Otherwise I can just counter source you with my previous posts.
2 + 3 * 4 only has a correct answer if you assume the current order of operations. Without order of operations, this equation is meaningless and has no value. This is why I'm saying your arguments are circular. You are saying that 14 is the correct answer because of the order of operations. And you're saying the current order of operations is correct because the answer is 14. These arguments don't stand on anything except for themselves. Am I missing something?
You are saying that 14 is the correct answer because of the order of operations
No, I am saying that is because that's the number that we started with and we wrote the expression based on the operator definitions.
Let me try this again, but be more explicit about each step...
Teacher is writing a test. Has decided the answer for this question is going to be 14. What can he make the question? Well, we can throw in some addition, so let's change it to 2+3+3+3+3. What else can we do?
Well, that 3+3+3+3 part there, we have a shorthand version of that in multiplication, that being 3x4, so now let's change it to 2+3x4.
So, now we've turned 14 into 2+3x4. None of that had anything to do with order of operations, just the definitions of the operators - + means addition, and x is shorthand for multiple additions.
So, we already know that the answer is 14, because that's what the teacher started with.
Now we need some rules of Maths to make sure that anyone who tries this question gets 14 as the answer.
And as we already saw, we have to do multiplication first or we arrive at the wrong answer - 20. And the reason we can only get the correct answer from doing multiplication first is because multiplication is shorthand for repeated additions, which was the first step the teacher made from having started from 14.
Similarly, we have defined exponents as being shorthand for multiplication, so for the very same reason we will need to solve exponents before we solve multiplication or we won't end up at 14.
And welcome to the order of operations rules! As I said, they are a natural consequence of the way that we have defined the operators. So, with the way we have defined addition, multiplication, and exponents, solving any such expression requires that we do them in the order E M A. In other words, we have to undo all the shorthand in the opposite way to how it was written to begin with, until we end up with just additions, and from there we arrive at the correct answer of 14.
Let's say though, way back in time, that instead they had defined exponents as shorthand for multiplication, and x as shorthand for exponents. Then, with these different definitions the order of operations rules would be M E A (because these definitions have exponents and multiplication the opposite way around to how we actually have defined them).
Do you see it now? If there's something you don't understand, then just ask me.
I think I understand what your argument is now. My problem is I just don't find it convincing. In fact, I find it pretty unrelated to the problem.
Sigh....
I looked back through the conversation to figure out what initially sparked this debate. You seem to take issue with my claim that the order of operations is based on consensus and that any order could be used. I still think this claim is true.
What I will say is I think that your rationale is probably the basis for the current order of operations that we use. So from a historical perspective I think it has some merit.
But it doesn't prevent us from using any other order of operations. Literally any order of operations can be used because it's not really a math thing, it's more of a reading an equation off the page thing. There exist systems (RPN) where the order of operations is not needed. If our entire world was built off of RPN we wouldn't even be having this debate. Order of Operations would not even exist.
I feel like you and I have been repeating ourselves a lot in this thread and if this doesn't convince you, I think we're at an impasse.
Feel free to reply if you want, but I'm probably going to stop at this point in time.
the order of operations is based on consensus and that any order could be used
No, as per example I gave, if you changed the order to addition first, you get a different answer (20 instead of 14), therefore demonstrably you can't use a different order of operations. You can only have a different order if you have different definitions of the operators to begin with... and again the order of operations would be derived from what definitions you used in that case.
I still think this claim is true
So you didn't understand the proof then.
your rationale is probably the basis for the current order of operations that we use
It's not my rationale - it's a mathematical proof. We started with 14. We therefore know any rules we make have to end up back at 14. Any rules which don't lead you back to 14 are demonstrably wrong and ruled out. That is the whole purpose of the rules of Maths in the first place - there is only 1 correct answer, and we have to have rules which can only give you that 1 answer when you follow them. It's the same thing the original authors of the order of operations would've done. There's no "consensus", there's just a Mathematician doing Maths and arriving at the rules which work. Then tells others what they are so that they don't have to go through working it out themselves (though some might if they want to confirm that what they've been told is correct. As I said, that's the beauty of Maths - you can do the Maths yourself and confirm that what you've been told is correct, like I just did).
But it doesnβt prevent us from using any other order of operations.
Of course it does. If you try using a different order of operations you no longer get 14 (as demonstrated when we do addition before multiplication with our current operator definitions).
Here's where I think we disagree.
I think without order of operations, 2 + 3 * 4 has no answer because the question is incomplete. You think that it does have an answer.
I think without order of operations, 2 + 3 * 4 has no answer... You think that it does have an answer
I started with the answer. I started with 14, and then I found different ways to write that using Maths definitions. The first different way to write that was 2+3+3+3+3. We know that's equal to 14 because...
- that's what I started with
- it's just a simple number line exercise anyway (which is what everything boils down to) - start at 0, jump 2, jump 3, jump 3, jump 3, jump 3. Where am I on the number line now? 14.
Then I went "what's yet another way we can write that using Maths definitions?". Well, 3x4 is shorthand for 3+3+3+3, so now we can also rewrite that as 2+3x4, and we know the answer to that is 14 because that's what we started with, and now, ok we're happy with that, we'll make that the question to use in the test.
Note that I haven't done anything using order of operations rules yet, I've only rewritten the answer using different Maths notation definitions.
And now, as we already know, any order of operations rules which don't get 14 as the answer are wrong, so we know that the only order of operations rules which get us 14 is M A... and the reason that they're the only rules which work is because multiplication is shorthand for addition, so we have to expand that before we do addition. In other words, the steps we have to take to solve it - the order of operations rules - have to be in the opposite order to which the expression was created. An analogy to this is if I take 3 steps East, then I take 3 steps West to get back to where I started (just like we do on a number line) - if I take 3 steps North then I end up somewhere else than where I started, which is the wrong destination if I'm wanting to get back to where I started.
You've lost me even harder now. How did we start with 14? Why couldn't we start with 77? Can we start with 2 + 3 * 4 + 5 = 14? Does the equation even matter in the situation?
It looks to me like you're jumping from floating island to floating island. I'm impressed. I don't know how you're doing it. And I certainly don't know how to follow you.
I don't know man. I'm beginning to think you are trolling me hard. I really don't know how to bridge this gap.
How did we start with 14?
It was just a number I picked for the proof - any number works!
Why couldnβt we start with 77?
We could! Let's go...
Teacher is writing a test. Decides the next question to be written will have 77 as the answer. How else can we write that? Let's start with 7+70. We know the answer to that is 77 because...
- it's what we started with
- we can show it's so on the number line - start at 0, jump 7, jump 70, where are we now? 77!
Now, we can also write that as a repeated addition. i.e. 7+10+10+10+10+10+10+10 - yet another way that we can write 77 (which we already know is the answer). What else?
We also know, by the definition of multiplication, that multiplication is short for repeated addition, so now let's rewrite that as 7+10x7, and we still know that the answer to this is 77 - all we have done is rewrite it in different ways based on the definitions of operators.
So now we need to look at what order of operations rules we need to make sure anyone doing this gets 77, because we already know the answer is 77!
So let's try addition first..
7+10x7=17x7=119. Nope, not 77, that doesn't work.
Now let's try doing multiplication first...
7+10x7=7+70=77. Yes! We got the right answer. Welcome to our first order of operations rule - multiplication before addition.
Can we start with 2 + 3 * 4 + 5 = 14
Let's see..
First let's pull out the 2...
14=2+12
Now let's pull out that 5...
14=2+7+5.
Can we make that 7 in the middle equal to the 3x4 you wrote? No, we can't. 3x4 by definition is equal to 3+3+3+3, which is equal to 12, which isn't equal to 7, so no, we can't write 14=2+3x4+5 by the definition of the operators of add and multiply.
Again, no order of operations in there, simply can we write this with our current Maths operator definitions, and the answer is no, we can't. i.e. the order of operations rules are derived from the definitions of the operators themselves. We can't write 14=2+3x4+5 while 3x4 is still defined to mean 3+3+3+3. To make your equation work, we would have to make 3x4=7, which is what + already means, and why would you make the multiply operator mean the same thing as the addition operator? That would be redundant.
It looks to me like youβre jumping from floating island to floating island
I'm going from proof to proof.
I donβt know how youβre doing it.
By knowing how arithmetic works. The rules of Maths work for all numbers and all operators - they were written for that specific purpose!
I really donβt know how to bridge this gap
Are you a visual learner? Do you want me to actually draw it up on a number line? Cos that's what it all boils down to - jumps on the number line. We're only dealing with 1 Dimensional Maths here.