Math, at it’s basis, doesn’t have an order of operation, as I’ve illustrated in my previous comment by breaking the left-to-right rule, doing addition before multiplication and ignoring brackets until the very end.
It only exists as a method of teaching students because it works. It’s simple and easy to remember.
The rest is me explaining how basic properties work:
Instead, mathematicians have long derived the basic properties that are supposed to be taught to students later on and is pretty much the first thing you learn in mathematical analysis in uni.
Those are:
commutative: a+b=b+a | same for mult
associative: (a+b)+c=a+(b+c) | same for mult
distributive: a×(b+c)=a×b+a×c
identity: a+0=a | a×1=a
inversion: a+(-a)=0 | a×(1/a)=1
This is what math is. Every equation is solved using those properties. Every theorem can be broken down into those actions. (Technically speaking, you can break it down even more - into addition only)
This is why in GEMA, BODMAS, etc, you have multiplication and division before addition and subtraction. Because (a×b)+c=a×(b+c) isn’t a property that exists. Try it. The sides won’t always be equal.
And those properties are also the reason why you don’t have to abide by an order of operations. Commutative and associative properties directly contradict them without making the solutions incorrect.
My uni math was more extensive than expected(read ‘practically useless’ but also occasionally fun) and no two math profs simplify in any relatively similar style whatsoever(This was beyond a headache before wolframAlpha). My personal experiences demand I still strongly lean toward any universally agreed upon convention. Don’t you think with how opinionated and picky mathematicians are, it’s still better to keep to a language everyone can participate in? It just seems almost conflicting with the spirit of math to not somewhat favor a most direct base or foundation if only for convenience. I suppose you could argue we DO have wolframAlpha, so who cares, and I guess I just feel it’s important that math stay teachable via human to human interaction and so any of the Acronyms seem much more helpful than not
Well, in my case, the order of operation in uni wasn’t brought up even once. But it was also a prestigious one with notoriously challenging math courses, so I may be a little out of touch in that regard. (Let me brag, ok?)
No, I do not think those conventions are needed. Because they aren’t fundamental. You don’t really know math until you understand how PEMDAS or w/e came to be and why it is the way that it is.
Not following those conventions doesn’t automatically make your solution incorrect. That’s the most important thing.
It shouldn’t matter how you solve as long as it is a correct solution.
There may, indeed, be inconsistencies in how things are written out. Whether 2x is the same as 2×x, for example. It’s common practice that it isn’t, but it’s also often not important.
If you write out the solution, people will understand what you mean by simply following it.
Compare:
6÷(2+4) = 6÷2(1+2) = 6÷2÷3 = 1
And
6÷(6÷3×(1+2)) = 6÷2(1+2) = 6÷2×3 = 9
They are written in the same manner, but those are 2 different equations to begin with, with their respective correct solution. For the same reason why 2x and 2×x may be the same or not. (Replace 1+2 with x, you’ll get 6/2x vs. 6x/2)
It’s not a matter of order of operations, but a matter of context. Whether juxtaposition took place or not. In real research, 2x always has a context.
Besides, the equations aren’t usually written out that way, aren’t they? You would do this (except for the dot in multiplication, unless it’s needed)
Let me just, ahem
1-2+3/(3+3)×2+3×6/3 = 1-2+3/(3+3)×2+1×6 = 1-2+3/(3+3)×2+6 = 7-2+3/(3+3)×2 = 7-2+3/(6+6) = 7-2+(1/2+1/2) = 5+(1/2+1/2) = 5+1=6
Ahh, yes, DMAMDSBA :P
Let’s just say BODMAS/PEMDAS isn’t all end-all be-all. They’re good, but there’s also better
For those interested, see: basic number properties
Idk what basic number properties are, but isn’t GEMS the new best/simple standard?
Not really.
Math, at it’s basis, doesn’t have an order of operation, as I’ve illustrated in my previous comment by breaking the left-to-right rule, doing addition before multiplication and ignoring brackets until the very end.
It only exists as a method of teaching students because it works. It’s simple and easy to remember.
The rest is me explaining how basic properties work:
Instead, mathematicians have long derived the basic properties that are supposed to be taught to students later on and is pretty much the first thing you learn in mathematical analysis in uni.
Those are:
This is what math is. Every equation is solved using those properties. Every theorem can be broken down into those actions. (Technically speaking, you can break it down even more - into addition only)
This is why in GEMA, BODMAS, etc, you have multiplication and division before addition and subtraction. Because (a×b)+c=a×(b+c) isn’t a property that exists. Try it. The sides won’t always be equal.
And those properties are also the reason why you don’t have to abide by an order of operations. Commutative and associative properties directly contradict them without making the solutions incorrect.
My uni math was more extensive than expected(read ‘practically useless’ but also occasionally fun) and no two math profs simplify in any relatively similar style whatsoever(This was beyond a headache before wolframAlpha). My personal experiences demand I still strongly lean toward any universally agreed upon convention. Don’t you think with how opinionated and picky mathematicians are, it’s still better to keep to a language everyone can participate in? It just seems almost conflicting with the spirit of math to not somewhat favor a most direct base or foundation if only for convenience. I suppose you could argue we DO have wolframAlpha, so who cares, and I guess I just feel it’s important that math stay teachable via human to human interaction and so any of the Acronyms seem much more helpful than not
Well, in my case, the order of operation in uni wasn’t brought up even once. But it was also a prestigious one with notoriously challenging math courses, so I may be a little out of touch in that regard. (Let me brag, ok?)
No, I do not think those conventions are needed. Because they aren’t fundamental. You don’t really know math until you understand how PEMDAS or w/e came to be and why it is the way that it is.
Not following those conventions doesn’t automatically make your solution incorrect. That’s the most important thing.
It shouldn’t matter how you solve as long as it is a correct solution.
There may, indeed, be inconsistencies in how things are written out. Whether 2x is the same as 2×x, for example. It’s common practice that it isn’t, but it’s also often not important.
If you write out the solution, people will understand what you mean by simply following it.
Compare:
6÷(2+4) = 6÷2(1+2) = 6÷2÷3 = 1
And
6÷(6÷3×(1+2)) = 6÷2(1+2) = 6÷2×3 = 9
They are written in the same manner, but those are 2 different equations to begin with, with their respective correct solution. For the same reason why 2x and 2×x may be the same or not. (Replace 1+2 with x, you’ll get 6/2x vs. 6x/2)
It’s not a matter of order of operations, but a matter of context. Whether juxtaposition took place or not. In real research, 2x always has a context.
Besides, the equations aren’t usually written out that way, aren’t they? You would do this (except for the dot in multiplication, unless it’s needed)
I’m still not getting your side of all of this. Are you basically saying that the order of operations should be reduced to two rules:
Or are you saying we don’t need order of operations at all?
I’m saying it’s not needed at all.
It’s good for teaching kids quickly, but it’s not a real rule you have to follow.