To be fair, it’s completely arbitrary, and all of math would be easier to understand, although slightly more verbose, if the only rule of order of operations is “always use parentheses to denote order, there are no implied parentheses”.
lazy mfs from centuries ago who were mortified by the thought of having to write ( and ) too much (lord what i wouldn’t give to hop in a time machine and show them lisp) should not be dictating our mathematical notation in this century. Explicit grouping is always more obvious to the reader.
Even then it’s still a quick mistake to make. If I’m not paying attention I could easily make a mistake like this, because I’m used to reading things left to right.
It’s also not that hard to just write it in a far less confusing way in many cases.
In this simple case, 4 x 2 + 2 or 2 x 4 + 2 would have been superior choices because both people reading left to right and people following pemdas correctly would get it right, and only people mis-remembering pemdas would be confused.
It’s not logic, it’s that what it means. 2*4 literally means 2+2+2+2. Just like 8/2 means how many times you can add 2 to itself until you get 8.
That’s what it means.
So why use braces? Because in more advanced maths you have more complex expressions that can’t be express in just multiplication which often occur in algebra or beyond.
For example what does 2 * ( a + 3) actually mean? Like why do we need to do the addition first. Its because we don’t know how many times we need to repeat the addition until we know what a means.
Let’s say a are points on an axis, and at some point it is worth three the. At that moment that expression is 2 * (3 +3) = 2 * 6 which is equal to 2+2+2+2+2
But in the next moment a might be 1
Right?
What’s arbitrary are the labels on the rules. The rules themselves aren’t arbitrary.
Maybe for very simple calculations like this one, but for more complex ones parenthesis actually make them much harder to read and write. If you’ve ever built a complex functions in Excel you know how difficult it gets because for 90% of the excel operations require parenthesis which means it works exactly like you’d want math to work. Just yesterday I had to do a more complex index match search in excel and excel corrected my parenthesis, because when your function is supposed to end with 5 parenthesis good luck keeping track of how many parenthesis you actually need to write out. Similarly if a week later I would have to change something inside that same function it’s going to take a lot more time to deconstruct the formula because of the abundance of parenthesis.
And the addition of parenthesis in math is entirely unnecessary because the nature of most operators already dictates the order of operations. Exponents are just multiplications and multiplication are just additions. 23 is the same as 2 x 2 x 2 is the same 2 + 2 + 2 + 2. If you take the example in the image then 2 + 2x4 transposed into additions is 2 + (2 + 2 + 2 + 2), parenthesis added to indicate what used to be the multiplication. Why people get it wrong is because they don’t understand the nature of those operators and so they do (2+2)x4 which is how they get (2+2)+(2+2)+(2+2)+(2+2) = 16. The order is clear, you can’t do addition before you do multiplication, because multiplication is a certain form of addition, and you can’t do multiplication before you do exponents, because exponents are a certain form of multiplication. The inverse functions maintain the same order of the function they’re inverting, meaning you can do subtraction before division and you can’t do division before rooting. No need for parenthesis for the natural order of operations. Parenthesis serve a purpose when you need to denote exceptions to the natural order of operations, like (2+2) x 4.
It’s not a “natural” order of operations. Why in the world would you think that we more often add before multiplying instead of vice versa? That’s such a weird claim
Did you just read the last sentence and not the rest of the comment? I went pretty in depth about what I mean by it. I don’t think we more often add before multiply, I know we must solve multiplication before doing addition and vice versa is the wrong way to do it, unless there’s something else, like parenthesis, stating a different order of operations.
To be clear, it’s the standard order of operations (PEMDAS) that is arbitrary. The expression in the post, assuming PEMDAS, is not arbitrary. There’s only one correct answer.
Also, I dunno man. The window from where math is complicated enough to have multiple different operators to where expressions get too complicated to be easily readable with just parentheses to denote order should be passed by like, early to mid highschool, if not junior high. Point being, frankly if you’re struggling with PEMDAS, your either still a high schooler, or you probably should be.
By the end of highschool you’ve mostly stop dealing with numbers and moved on to algebra, which foregoes the confusion of PEMDAS. a+bc is very obvious.
always use parentheses to denote order, there are no implied parentheses
I completely agree on this, and yes, this is what I always do, cuz… well, we’re human, we make mistakes, parentheses makes things easily visible, thus cutting down on mistakes.
Still, I do know operation order, as a rule I mean. In simple calcs like these, making a mistake is almost impossible. Thus, people that answered 16 probably just don’t know the order… that is something you learn in 1st, 2nd grade, it’s not quantum mechanics we’re talking about here.
lazy mfs from centuries ago who were mortified by the thought of having to write ( and ) too much (lord what i wouldn’t give to hop in a time machine and show them lisp) should not be dictating our mathematical notation in this century.
We only do that cuz we’re not sure how the compiler will interpret the operation order, and there’s waaaay too many versions and different languages to actually remember how each of them interprets math operation order. So, we do a safe bet, put parentheses on everything. Hell, I do it as well, I just can’t be bothered to remember if C interprets it like this, Python like that, Rust like… god knows what. They should, in theory, know math operation order, but let’s face it, we all do it cuz we’ve been faced with bugs that are a direct result of the compiler not intepreting things as it should.
That being said, yes, I do agree that prentheses on everything, even math on paper, is the way to go. Plus, even people that don’t know operation order, will learn it a lot qucker if you just show them how easy things become once you start using prentheses.
I will literally commit hate crimes against all of humanity if I had to write brackets around all operations in math. Surely remembering 6 things is easier than writing out brackets 100 times a day
Polish notation ftw. + 2 * 2 4, no parentheses needed and no ambiguity. (Though makes it harder to see at first glance where is the cut between the to terms of the operation.)
To be fair, it’s completely arbitrary, and all of math would be easier to understand, although slightly more verbose, if the only rule of order of operations is “always use parentheses to denote order, there are no implied parentheses”.
lazy mfs from centuries ago who were mortified by the thought of having to write
(and)too much (lord what i wouldn’t give to hop in a time machine and show them lisp) should not be dictating our mathematical notation in this century. Explicit grouping is always more obvious to the reader.That’s true, but it’s not that hard either.
Even then it’s still a quick mistake to make. If I’m not paying attention I could easily make a mistake like this, because I’m used to reading things left to right.
It’s also not that hard to just write it in a far less confusing way in many cases.
In this simple case,
4 x 2 + 2or2 x 4 + 2would have been superior choices because both people reading left to right and people following pemdas correctly would get it right, and only people mis-remembering pemdas would be confused.Multiplication is a notation which means add some number by itself a number of times.
5 x 3 = 5 +5 + 5
2 * 4 = 2 + 2 + 2 +2
So when you see some like 2 + 4 * 2 it literally means. 2+4+4
By that logic it could just as well be 2 + 4 * 2 = (2 + 4) + (2 + 4) = 12. You still need to know to multiply first, or it’s arbitrary
Order of operations proof - simple version
It’s not logic, it’s that what it means. 2*4 literally means 2+2+2+2. Just like 8/2 means how many times you can add 2 to itself until you get 8.
That’s what it means.
So why use braces? Because in more advanced maths you have more complex expressions that can’t be express in just multiplication which often occur in algebra or beyond.
For example what does 2 * ( a + 3) actually mean? Like why do we need to do the addition first. Its because we don’t know how many times we need to repeat the addition until we know what a means.
Let’s say a are points on an axis, and at some point it is worth three the. At that moment that expression is 2 * (3 +3) = 2 * 6 which is equal to 2+2+2+2+2
But in the next moment a might be 1
Right?
What’s arbitrary are the labels on the rules. The rules themselves aren’t arbitrary.
Maybe for very simple calculations like this one, but for more complex ones parenthesis actually make them much harder to read and write. If you’ve ever built a complex functions in Excel you know how difficult it gets because for 90% of the excel operations require parenthesis which means it works exactly like you’d want math to work. Just yesterday I had to do a more complex index match search in excel and excel corrected my parenthesis, because when your function is supposed to end with 5 parenthesis good luck keeping track of how many parenthesis you actually need to write out. Similarly if a week later I would have to change something inside that same function it’s going to take a lot more time to deconstruct the formula because of the abundance of parenthesis.
And the addition of parenthesis in math is entirely unnecessary because the nature of most operators already dictates the order of operations. Exponents are just multiplications and multiplication are just additions. 23 is the same as 2 x 2 x 2 is the same 2 + 2 + 2 + 2. If you take the example in the image then 2 + 2x4 transposed into additions is 2 + (2 + 2 + 2 + 2), parenthesis added to indicate what used to be the multiplication. Why people get it wrong is because they don’t understand the nature of those operators and so they do (2+2)x4 which is how they get (2+2)+(2+2)+(2+2)+(2+2) = 16. The order is clear, you can’t do addition before you do multiplication, because multiplication is a certain form of addition, and you can’t do multiplication before you do exponents, because exponents are a certain form of multiplication. The inverse functions maintain the same order of the function they’re inverting, meaning you can do subtraction before division and you can’t do division before rooting. No need for parenthesis for the natural order of operations. Parenthesis serve a purpose when you need to denote exceptions to the natural order of operations, like (2+2) x 4.
It’s not a “natural” order of operations. Why in the world would you think that we more often add before multiplying instead of vice versa? That’s such a weird claim
Did you just read the last sentence and not the rest of the comment? I went pretty in depth about what I mean by it. I don’t think we more often add before multiply, I know we must solve multiplication before doing addition and vice versa is the wrong way to do it, unless there’s something else, like parenthesis, stating a different order of operations.
To be clear, it’s the standard order of operations (PEMDAS) that is arbitrary. The expression in the post, assuming PEMDAS, is not arbitrary. There’s only one correct answer.
Also, I dunno man. The window from where math is complicated enough to have multiple different operators to where expressions get too complicated to be easily readable with just parentheses to denote order should be passed by like, early to mid highschool, if not junior high. Point being, frankly if you’re struggling with PEMDAS, your either still a high schooler, or you probably should be.
Or we can all learn polish notation
By the end of highschool you’ve mostly stop dealing with numbers and moved on to algebra, which foregoes the confusion of PEMDAS. a+bc is very obvious.
I completely agree on this, and yes, this is what I always do, cuz… well, we’re human, we make mistakes, parentheses makes things easily visible, thus cutting down on mistakes.
Still, I do know operation order, as a rule I mean. In simple calcs like these, making a mistake is almost impossible. Thus, people that answered 16 probably just don’t know the order… that is something you learn in 1st, 2nd grade, it’s not quantum mechanics we’re talking about here.
We only do that cuz we’re not sure how the compiler will interpret the operation order, and there’s waaaay too many versions and different languages to actually remember how each of them interprets math operation order. So, we do a safe bet, put parentheses on everything. Hell, I do it as well, I just can’t be bothered to remember if C interprets it like this, Python like that, Rust like… god knows what. They should, in theory, know math operation order, but let’s face it, we all do it cuz we’ve been faced with bugs that are a direct result of the compiler not intepreting things as it should.
That being said, yes, I do agree that prentheses on everything, even math on paper, is the way to go. Plus, even people that don’t know operation order, will learn it a lot qucker if you just show them how easy things become once you start using prentheses.
I will literally commit hate crimes against all of humanity if I had to write brackets around all operations in math. Surely remembering 6 things is easier than writing out brackets 100 times a day
Polish notation ftw. + 2 * 2 4, no parentheses needed and no ambiguity. (Though makes it harder to see at first glance where is the cut between the to terms of the operation.)