this post was submitted on 03 Dec 2023
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My mom's a mathematician, she got annoyed when I said that the order of operations is just arbitrary rules made up by people a couple thousand years ago
It's organized so that more powerful operations get precedence, which seems natural.
Set aside intentionally confusing expressions. The basic idea of the Order of Operations holds water even without ever formally learning the rules.
If an addition result comes first and gets exponentiated, the changes from the addition are exaggerated. It makes addition more powerful than it should be. The big stuff should happen first, then the more granular operations. Of course, there are specific cases where we need to reorder, or add clarity, which is why human decisions about groupings are at the top.
Yeah, but that's why I like to buff my base attack before I invest in multipliers and armor penetration!
The "big stuff" is stuff that is defined in terms of something else. i.e. exponents are shorthand for repeated multiplication... and multiplication is shorthand for repeated addition, hence they have to be done in that order or you get wrong answers.
"Wrong answers" only according to our current order of operations, math still works if you, for example, make additions come first (as long as you're consistent about it).
OFC it is a convention and to change it you would have to change all expressions ever written all at the same time, to avoid confusion between competing standards. I'm not arguing that it should be changed, only that there is no 'high truth' behind it.
No, according to arithmetic.
No, it doesn't - order of operations proof. The only way it could work with addition first is if we swapped the definitions of addition and multiplication around... but then we still have the same order of operations, all we've done is swapped around what we call addition and multiplication!
There is when it comes to order of operations.
Let's assume for a minute addition comes first. We know 2+3 is 5, and 5x4 is the same as 5+5+5+5=20. What is the issue with that?
That it's wrong. If I have 1 2 litre bottle of milk, and 4 3 litre bottles of milk - i.e. 2+3x4 - how many litres of milk do I have? Without even doing the arithmetic, just count it up and tell me how many litres there is.
If we change how equations are parsed so addition comes before multiplication, 2+3x4 is not the equation required to solve that problem. 2+(3x4) is the equation needed. You can't change how equations work and then expect all equations to work the same after the change.
If your argument is that this will add parentheses where we didn't need them before, that's valid and its the reason we do it this way in the first place. But that doesn't mean there is anything fundamentally wrong with having a different system of writing equations in which operations are executed in a different order.
Our whole system of writing equations is just a convention, and yes, it is a good and easy to understand and use way of writing math. But there is no fundamental truth behind it, only that it is simpler for the majority of use cases.
Noted that you didn't answer my question - the answer is I have 14 litres of milk. 2+3+3+3+3=14 litres. When you did "arbitrary addition first", you got 20, which is wrong, which is why no other order of operations rules work than the ones we have.
In actual fact the point is that they will except for what ever your new notation is. e.g. if we instead defined + to mean multiply, and x to mean add, then we would do + before x, and again, that would be the only order of operations which works. i.e. the only order which gives us 14 litres.
No, and if you did that, you would again arrive at only one order of operations rules which works, cos I still have 14 litres, and the Maths in this new system still has to give an answer of 14 litres, not 20.
Nope, it's all rules, found in any Maths textbook, and if you don't obey the rules you get wrong answers (like you did when you got 20).
Yes there is - I have 14 litres, and only 1 set of order of operations rules gives that answer.
If you follow the rules of Maths then it is correct for every use case. That's why they exist in the first place.
I think you misunderstand my argument. I could use still math to solve a real-world problem with an altered order of operations. You could still do anything you can do with regular math, if you had a different order of operations. You could make a programming language that parses your inputted expressions with a different order of operations and still use it to calculate collisions or render a 3d scene or do anything else that involves math. Do you need me to calculate something, to prove it to you?
The order of operations is just part of a system of notation and any system of notation that exists in the world is inherently arbitrary. The same way the way that how we draw the number 3 or the number 5 has no inherent meaning behind it other than the convention of how we interpret it, the order of operations is nothing more than a standard part of the notation. Again, I'm not saying that we should or could change it, as there would be no way to indicate which convention we are using and the standard order of operations works perfectly fine.
No, you demonstrably didn't understand mine, which is, what you are saying is impossible, but you're still saying it's possible.
No, you can't. You already tried to do addition first in 2+3x4 and found out why it doesn't work. Ever since then you've been ignoring that result and pretending that there's some other way to make it work. No, there isn't. As long as multiplication is defined in terms of addition (i.e. 3x4=3+3+3+3) then it's impossible to get a right answer unless you do multiplication before addition.
No, you can't. Again, you already proved you can't.
Go ahead - I'm not holding my breath. I already told you why it literally can't work. But note that adding brackets isn't changing the order of operations - brackets are already part of the order of operations. Writing 2+3x4 as 2+(3x4) is exactly the same thing.
BTW just to FURTHER prove your "addition first" doesn't work, look at this example...
3x4+2=3x6=18. But earlier you did 2+3x4=5x4=20 - not even the same answer in an "addition first" world! Welcome to why it's impossible to make addition-first work. But knock yourself out - you're welcome to try! 😂
No, it isn't. It's part of the rules of Maths. Notation is how you write it - underlying that is how Maths actually works. This is embodied in the rules of Maths.
Completely fixed, and a result of the way the operators are defined - that was the only "arbitrary" bit, deciding what the operators were and what they were going to mean, but once you did that then the order of operations rules were already written for you (having already been determined as soon as you made the definitions of the operators in the first place).
Again, not a convention, a rule of how to interpret it. You can't just decide to interpret 5 as four, or again, you end up with wrong answers. The rules of Maths prevent you from getting wrong answers. You found that out yourself when you tried to do addition first in 2+3x4.
It's only a wrong answer if you use the same expression you would with the standard order of operations. And I'm not saying we can randomly start interpreting 5 as four, just that there is no law of the universe that makes 5 look like that, and we could theoretically (not practically ofc) switch the definitions of the symbols 5 and 4 if we did it all at once and revised old math expressions to match the new standard. Just as there is no reason the letters "bike" mean what they do other than that's what someone decided to call it, there is no reason the order of operations is what it is other than that is how someone decided to write it.
Scratch doesn't even have an order of operations. You can still do math in it.
I'm not saying you can take any expression and get the same answer by doing addition before multiplication. I'm saying you can take any problem and get the correct answer by doing addition before multiplication. In your milk example, that means I would use the expression 2+(3x4) because 2+3x4 is no longer the correct expression needed to solve the problem.
(For an example of my distinction of the words "expression" and "problem", "(4x)+2" is an expression, and "I start with 2 litres of milk. For every dollar I spend, I get 4 more liters of milk. How much milk do I have?" is a problem.)
My argument also relies on a distinction between the language of modern math and the concept of doing math, defining math as the dictionary definition of "The study of the measurement, properties, and relationships of quantities and sets, using numbers and symbols". As you can see, this makes no mention of the notation commonly used in math. All I am saying is that you can still use numbers to solve problems with an altered order of operations, or by altering any part of the system of notation.
Perhaps seeing how I could solve a problem with a different order of operations will help illustrate my argument:
Problem: 2 cars approach an interchange at a 90 degree angle to each other. Car A approaches the station from 15 meters away at 30 meters/second and Car B approaches the station from 50 meters away at 20 meters/second. How fast is the distance between the cars decreasing?
Answer: the rate of change of the distance between the cars is approximately -27.777 meters per second.
As you can see, I used my altered math notation to find the correct answer. I can still solve a real-world problem with this notation, but the same expressions you would use before may not work now.
Really? You want to do that again? Ok, fine... If I have 1 2 litre bottle of milk, and 4 3 litre bottles of milk - i.e. 2+3x4 - how many litres of milk do I have?
The definition of 5 as being 1+1+1+1+1 has nothing to do with order of operations.
No, but there is a rule of Maths which defines it.
In other words everything would be the same as now but we just switched the notation around. I already said that to you a while back. Now you're getting it.
Got nothing to do with how it's written - Maths is written differently in many different countries, and yet the underlying order of operations rules are universal.
And if it's not the same answer then it's wrong. You're nearly had it.
And I told you you can't. Waiting on a proof from you. Start with 2+3x4 - show me how you can get the correct answer by doing addition first - it's a nice simple one. :-)
They're literally the same thing.
And I told you that it's impossible. Changing the notation doesn't change the Maths.
BWAHAHAHAHA! Nope! I see you putting brackets around the multiplication to make sure it gets done first - same as if you hadn't used brackets at all! It's the exact same notation we use now, just with some redundant brackets added to it! And, predictably, you left the addition for last.
Ok, let's take your example and do addition first (like you claimed can be done)...
15²+50²=15x15+50x50=15x65x50=48,750. But 15²+50² is 2,725 according to my calculator. Ooooh, different answers - I wonder which one is right... I wonder which one is right...???
Thanks for proving it can only be done by following the order of operations rules (just like I've been saying to you all along). Bye now.
I"m beginning to wonder if you are willfully misunderstanding my point. Or perhaps you have sunk so much time into this argument you assume I must be wrong. Take another look at my third and fifth paragraphs. I promise, I am not trying to say what you think I'm trying to say.
All I did was use the expression necessary to evaluate correctly with the altered order of operations. There are, in fact, times when you can remove brackets that you would otherwise need, for example (x+4)(x-2) would no longer need brackets. The fact that "old" expressions often have to be written with new brackets to evaluate correctly with an altered order of operations is something I fully understand. The presence of brackets where there would be none otherwise does not invalidate my point.
What? I never wrote 15²+50². That is an expression you copied incorrectly. Your incorrectly copied expression has little relevance to the problem at hand.
If we were doing math with an altered order of operations, the expression 2+3x4 is just simply wrong. 2+(3x4) is the expression you need. If you try to do math the same as it is with the regular order of operations, it will not work. But that does not mean math with an altered order of operations is useless. It is still math. It can still be used to "study of the measurement, properties, and relationships of quantities and sets using numbers and symbols".
I fully understand that to correctly evaluate an expression written with a certain order of operations in mind, you need to use that order of operations. If someone wrote an expression with a different order of operations in mind, you could solve it with a different order of operations and still get what the author of the expression intended. For example, I write the equation a+2xa-2 with my order of operations, expecting you to use the same order of operations, and tell you to simplify. If you get 3a-2, that is wrong, because you used an order of operations different than the one I intended to be used to solve the problem. Imagine, for a moment, an alternate universe where everyone uses a different order of operations and a+2xa-2 simplifies to a^2-4. All I am trying to say is that that their math, with a different order of operations, would be no less useful then our math.
In summary, my only claim is that you can still use a different order of operations to manipulate numbers and solve real world problems.
I wrote and evaluated all of those expressions in my last comment with a different order of operations in mind, and was still able to come to the correct answer.
I wasn't going to reply any more, but I see now you don't understand terms either, so one more time for old time's sake (and maybe you might finally get it)...
You know teachers don't get paid for helping students outside class time right?
No assumption needed. What you are proposing is literally impossible. I've been saying that all along.
Ok...
And so far you haven't been able to show it works for any expression at all! Not even one expression! Just like I said would happen.
And I said you can't, and you haven't! All you did was put brackets around the multiplication to make sure we were still following the only order of operations that works! You have still not shown an actual instance where one can actually do addition first and get a right answer, not one! The idea that one could use addition first as an "alternate order of operations" is thus pure fantasy, just like I've been saying all along. It's literally impossible.
Yes it would! (x+4) is one term - that's what the brackets means - "these things are all together". If you remove that, because "addition first", it's now two terms, so the whole expression is two terms (instead of one), x, and 4(x-2) (which is a mistake people make when they write 8/2(2+2) as 8/2x(2+2) - just turned 2 terms into 3 terms and changed the answer!). Every example you've done so far you've used brackets to escape from having to do addition first, and the very same thing would therefore apply here - no brackets, no escaping "addition first" approach, brackets before addition leads to x+4(x-2)=x+(4x-8) =5x-8, which is not the product of (x+4) and (x-2).
No, the fact that you've not been able to show a single instance of where addition before multiplication would work does. You can't show "a way to solve this in an addition first world" when it's literally impossible for an "addition first world" to exist in the first place.
...and I removed the brackets to show that addition first doesn't work (since you keep putting in brackets to revert "addition first" back to the only order of operations that actually works).
And you've still not shown how. Every example you've used so far you've put in brackets to your (supposed) "addition first" so that we were evaluating it using the only order of operations that works. In other words, no, you can't use "addition first" to “study of the measurement, properties, and relationships of quantities and sets using numbers and symbols” - you used the regular order of operations to do it! You haven't shown a single example of where addition first could be used to do it.
You need to use an order of operations that gives a correct answer, of which there is only one - a fact you keep trying to avoid.
No it wouldn't, cos now you're ignoring terms as well. As per my earlier working out, it would simplify to 5x-8 unless you also changed the definition of terms. Do you see yet why it's impossible to have an "alternate order of operations"?
And you've completely failed to show a single instance where this is true - which is what I've been saying all along, it's impossible to have another set of order of operations that works. You keep pre-supposing it's possible, but then add brackets to the multiplications so that we follow the actual correct order of operations, the only order of operations that works.
And you've still failed to solve a single problem using addition first, because it's still a fact it's literally impossible to do so.
by using the only order of operations that works. i.e. multiplication before addition.
Now I really am done - I'm not going any further down this rabbit hole of whatever other Maths you may not understand either (this post it was Terms - who knows what's next)...
I'm not surprised. Here's the proof of the order of operations rules. Also, the equals sign wasn't invented until the 16th century, so only 500 years old at most (the earliest references to order of operations are over 400 years ago).
That proof for the order of operations sure seems to rely a lot on our current order of operations...
Doesn't use order of operations at all. It only uses the definitions of the operators. i.e. 3x4=3+3+3+3 by definition. i.e. nothing to do with order of operations.
If I have 1 2l bottle of milk, and 4 3l bottles of milk, how many litres of milk do I have? It can be solved by simply adding them up - again, nothing to do with order of operations here, just simple addition. Now, write it out as a mathematical expression which uses multiplication, and tell me which order of operations gets you the right answer. Voila! Welcome to how we worked out what the order of operations rules had to be.
2+(4x3) gives the right answer, with addition coming before multiplication
If we rewrote all of Maths so that addition came before multiplication, then no, 2+3x4 would no longer mean what it does now (because + and x would have to mean something different to what they do now in order for the order to be swapped). The order of operations rules come directly from the definitions. You can't just say "we'll do addition first" without having defined what addition is now, nor multiplication. In a world where addition comes before multiplication, that means multiplication is no longer shorthand for addition (because that's the very thing which means we have to do multiplication before addition, so it can't be true anymore if now we're doing addition first).
Let's take an imaginary scenario where we now use x for add, and + for multiply. That would indeed mean that + has to be done before x, but note that + now means multiply. That means your "addition first" 2+(3x4) is what we currently mean by 2x(3+4) which is 14. Now take away the brackets (since I don't use brackets when adding up the milk! Just 2+3x4). Your addition-first 2+3x4 is equivalent in our multiplication-first world to 2x3+4 which equals 10 - the wrong answer! So now you've created a world where we have to add brackets to things just to get the right answer! Why would you even want to do that when it works the way it is? The whole point to having order of operations rules is to not have to add brackets!