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
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
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!
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.
The big stuff should happen first, then the more granular operations
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.
“Wrong answers” only according to our current order of operations
No, according to arithmetic.
math still works if you, for example, make additions come first
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!
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.
You can’t change how equations work and then expect all equations to work the same after the change
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.
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
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.
Our whole system of writing equations is just a convention
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).
But there is no fundamental truth behind it
Yes there is - I have 14 litres, and only 1 set of order of operations rules gives that answer.
only that it is simpler for the majority of use cases
If you follow the rules of Maths then it is correct for every use case. That’s why they exist in the first place.
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
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!
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.
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.