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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.
Ok, I think I got something with 77.
So we start with 77. One way to make 77 is (7 * 10) + 7. But there are several ways to get 77. You can also do ((3 + 4) * 10) + 7 = 77. There are several different permutations up to and including 77 * 1. Ok, but none of that has anything to do with order of operations. Using parentheses we can ignore order of operations altogether.
Another way to look at it is to ask the question does 3 + 4 * 10 + 7 equal 77? Assuming the normal order of operations, it does not. But there is some order of operations that will get us 77 as seen above. Do you believe the statement above is true?
Assuming we use a completely reversed order of operations, we can get 77 this way. (7 * 10) + 7. This implies that the order of operations alone does not give you the correct answer. The combination between the equation and the order of operations that gives you the correct answer in infix notation. And because you can get the correct answer with any order of operations given that the equation is written correctly, then it follows that the order of operations is not critical for computation. To put it plainly the order of operations does not matter. We could use any order of operations and still do math correctly.
but none of that has anything to do with order of operations
Yep, you nearly had it there. I've been saying this repeatedly in my proofs, but you've been missing what that means.
Using parentheses we can ignore order of operations altogether
Only if you put them in the right place! With your example (7x10)+7=77, if we move the brackets 7x(10+7)=490, which is a different answer... which demonstrates why we can't change the order of operations - you get different answers. The order of operations rules are there to make sure everyone gets the same answer from the same expression.
Also, using brackets isn't ignoring the order of operations rules - brackets are part of the order of operations rules.
does 3 + 4 x 10 + 7 equal 77
I already addressed a similar question like this from you in my previous post, and you've ignored my answer (sigh). I'll do this again and if you ignore my response again I'm giving up (it seems to me that you're just being disingenuous now, ignoring my responses to your examples which prove they don't work).
3+4x10+7
Recall that by definition - i.e. this is nothing to do with order of operations - 4x10=4+4+4+4+4+4+4+4+4+4
therefore...
3+4x10+7=3+4+4+4+4+4+4+4+4+4+4+7=50, which is not 77, so no, 3+4x10+7 does not equal 77, and again, that has nothing to do with order of operations, that is only using the definition of multiplication as repeated addition.
Assuming we use a completely reversed order of operations, we can get 77 this way. (7 * 10) + 7
Huh?? That's not a reversed order of operations - that's our current order of operations! Multiply before adding!
This implies that the order of operations alone does not give you the correct answer
WTH?! It PROVES that order of operations alone gives you the right answer. (7x10)+7=77. 7x10+7=77. Same answer, no brackets needed! Because of the order of operations rules.
And because you can get the correct answer with any order of operations given that the equation is written correctly
No, you can't. 7+10x7 is written correctly, and we know the answer is 77. Let's try a different order of operations and do addition first. 7+10x7=70x7=490 which isn't 77, so it's wrong. There is only one order of operations which gives the correct answer of 77, and that's multiplication before addition and that's because multiplication is defined as being shorthand for addition!
it follows that the order of operations is not critical for computation
I've shown you, repeatedly, that in fact it is. 490 is not equal to 77.
put it plainly the order of operations does not matter
Only if getting the right answer "does not matter".
We could use any order of operations and still do math correctly
In what world exactly are 490 and 77 the same answer?
Okay. Well then it sounds like we agree.
It seems like we agree that we can't change the order of operations without changing the equations, otherwise we won't get the same answer.
I was never arguing against this.
All I was saying was that we could use any order of operations we want. It's just a way of interpreting the equation. We don't have to use the current order of operations. We could use a different order of operations. As long as we update the equations we use as well. So it doesn't really matter which order of operations we use.
For some reason I was getting pushed back from you on this. Now it seems like you have no problem with it.
So I think we agree also. I'm very confused how we got here in the first place.
Edit:
By the way, you can stop expanding x * 3 = x + x + x. I understand how multiplication works. I just don't find it a very convincing argument for why we have to use the current order of operations that we are using today. I think it has interesting historical note. I wouldn't be surprised if it was the basis for why we have the current order of operations. But I don't think it limits the order of operations we could use.
Well then it sounds like we agree
If you think that then you haven't understood anything I've said
It seems like we agree that we canβt change the order of operations without changing the equations
No, we canβt change the order of operations without changing the definitions of the operators. We have to do multiplication before addition because multiplication is shorthand for addition. If you wanted to have a "different" order of operations, where we didn't do multiplication before addition, then multiplication can't be shorthand for addition anymore. To have a "different" order of operations, you could swap the definitions around, so that addition is shorthand for multiplication, and then yes, you would be doing addition before multiplication, but that's the only way you can change the order of operations - by changing the definitions of the operators to begin with... and you would still end up with what we have now, except you're calling addition "multiplication" and calling multiplication "addition". All you would've done in reality is swapped the names around.
All I was saying was that we could use any order of operations we want
No, we can't. And yes, you're only saying that - you haven't actually tried it. You gave me some examples which I proved don't work, and yet you're still saying the same thing whilst doing absolutely no Maths at all to back up that claim - it's just words, and I'm showing you that it doesn't work.
Itβs just a way of interpreting the equation.
As with anything in Maths, there is a right way, and a wrong way. Only one way works. You might as well say "we could interpret 1+1 as equal to 3". Try doing some actual Maths using 1+1=3 and let me know how far you get.
As long as we update the equations
The definitions.
So it doesnβt really matter which order of operations we use.
It really does, otherwise you just get wrong answers. Again, you haven't actually tried doing it a different way, you just keep saying we could do it a different way (even though actually trying to do it a different way proves it doesn't work).
Iβm very confused how we got here in the first place
Because you're still saying the same thing you said to begin with - that we can "choose" to use a different order of operations. No we can't. The order of operations rules come directly from the definitions of the operators. If we define multiplication as being a shorthand form of addition, then that means we have to do multiplication before addition, it's that simple, yet you continue to not see it. Doing addition before multiplication only gives wrong answers.
By the way, you can stop expanding x * 3 = x + x + x.
No, you can't - they're the same thing! That's like saying "By the way, you can stop writing Maths as Mathematics". One is just an abbreviation of the other, and you're failing to see how multiplication being an abbreviation of addition means there's only one right answer and we can only get that answer by doing multiplication before addition, hence the order of operations rules!
I understand how multiplication works
Apparently not, or you'd understand why the order of operations rules are what they are.
2+3x4=2+3+3+3+3=14. That's it, that's the only correct answer by definition (I didn't use any order of operations rules there, just the definition of multiplication as shorthand for addition). Now, show me "a different order of operations" which still gives 14 as the answer. This is what your claim is, so prove it!
I just donβt find it a very convincing argument
It's a proof. If you get the wrong answer with a different set of order of operations, then you can't use that order of operations. We can only use order of operations which give the right answer (noted that you ignored my point about 490 vs. 77 - you keep ignoring every time I point out why what you said is wrong).
I wouldnβt be surprised if it was the basis for why we have the current order of operations
It is!!
But I donβt think it limits the order of operations we could use
IT DOES. This is what you're failing to understand - the order of operations rules come from the definitions of the operators themselves, and only one way works. Literally the only way to change the order of operations rules is to change the operators themselves. i.e. their definitions. As long as multiplication is defined as being shorthand for addition, you'll still have to do multiplication before addition, there's literally no other way to get the right answer, hence there is only one set of order of operations rules which works - all other variations only give the wrong answer.