The order of operations is a convention created by humans so as to ensure a consistent understanding of mathematical expressions. The reason for them being the way they are is merely because that’s what we’ve agreed upon.
To add, I believe regardless of order of operations if we used () for every part of the equation it would no longer matter, but that would get tedious so as you said humans agreed on a shorthand for consistency.
All order of operation rules are made up, but some of them are more useful than others. Rules that jive with associative and communtative laws are preferred.
None of them are made-up. They are all a natural consequence of the way things have been defined. e.g. 2x3 is shorthand for 2+2+2, so if you don’t do multiplication before addition then you get the wrong answer, hence the order of operations rules.
I wish I could have been in the room. You know someone was like, “Eh, just left to right. We’ll use parentheses to specify an order when it’s necessary.” Then someone else said, “What if we use this system of various rules instead.” If I were there, I would have killed that person to save mankind.
No, they have to be the way they are as a result of the way the operators have been defined. e.g. 2x3 is shorthand for 2+2+2, so if you don’t do multiplication before addition then you get the wrong answer, hence the order of operations rules.
The order of operations is made from consensus. Just like we all agree the first letter of the alphabet is “A”. You could make the order of operations whatever you like, you would just need to rewrite equations to reflect that change.
No it isn’t. It’s a natural consequence of the definitions in the first place. e.g. 2x3 is shorthand for 2+2+2, so if you don’t do multiplication before addition then you get the wrong answer, hence the order of operations rules.
Your argument is circular in nature. The right answer is dependent upon the order that you do the operations. So you can’t use the argument that the order of operation matters. Because if you do the math not using the correct order of operations, the answer comes out incorrectly.
For example, we could just do away with order of operations all together and explicitly use parentheses everywhere. While the system would be extremely painful to use, it would result in a consistent mathematics.
So there’s nothing inherently correct about our current order of operations. I will say that I like it and that it makes sense to me.
If you want a fun Wikipedia trip look up reverse polish notation (RPN). If I have to do away with the current order of operations, I might look to this as a possible alternative.
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 can’t 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 there’s 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 (2³=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.
I’ll 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 mean, that sounds like the reasoning behind the order of operations, to me. At least that’s the only reason I’ve ever been given.
The order of operations is a convention created by humans so as to ensure a consistent understanding of mathematical expressions. The reason for them being the way they are is merely because that’s what we’ve agreed upon.
To add, I believe regardless of order of operations if we used () for every part of the equation it would no longer matter, but that would get tedious so as you said humans agreed on a shorthand for consistency.
All order of operation rules are made up, but some of them are more useful than others. Rules that jive with associative and communtative laws are preferred.
Or come over to RPN and stop worrying about it.
None of them are made-up. They are all a natural consequence of the way things have been defined. e.g. 2x3 is shorthand for 2+2+2, so if you don’t do multiplication before addition then you get the wrong answer, hence the order of operations rules.
I wish I could have been in the room. You know someone was like, “Eh, just left to right. We’ll use parentheses to specify an order when it’s necessary.” Then someone else said, “What if we use this system of various rules instead.” If I were there, I would have killed that person to save mankind.
No, they have to be the way they are as a result of the way the operators have been defined. e.g. 2x3 is shorthand for 2+2+2, so if you don’t do multiplication before addition then you get the wrong answer, hence the order of operations rules.
The order of operations is made from consensus. Just like we all agree the first letter of the alphabet is “A”. You could make the order of operations whatever you like, you would just need to rewrite equations to reflect that change.
No it isn’t. It’s a natural consequence of the definitions in the first place. e.g. 2x3 is shorthand for 2+2+2, so if you don’t do multiplication before addition then you get the wrong answer, hence the order of operations rules.
Your argument is circular in nature. The right answer is dependent upon the order that you do the operations. So you can’t use the argument that the order of operation matters. Because if you do the math not using the correct order of operations, the answer comes out incorrectly.
For example, we could just do away with order of operations all together and explicitly use parentheses everywhere. While the system would be extremely painful to use, it would result in a consistent mathematics.
So there’s nothing inherently correct about our current order of operations. I will say that I like it and that it makes sense to me.
If you want a fun Wikipedia trip look up reverse polish notation (RPN). If I have to do away with the current order of operations, I might look to this as a possible alternative.
No it isn’t. The order of operations rules are derived from the definitions of the operators - there’s nothing circular about that.
And so to get the correct answer, you have to do the operations in the correct order, yes.
But you just said it did when you said “The right answer is dependent upon the order that you do the operations”!
Exactly! Glad you agree with me. :-)
Except I showed here that most people put brackets in the wrong place to begin with.
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.
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.
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 (2³=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.
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?
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.
Order of operations proof - Associativity and Terms/Expressions