USE GEMS USE GEMS!
The P in PEMDAS just means resolve what’s inside the parentheses first. After that, it’s just simple multiplication with adjacent terms, and multiplication and division happen together left to right.
6÷2(1+2)
6÷2(3)
3(3)
9
This is actually a generational thing. Millennials were taught “PEMDAS”:
- Parenthesis
- Exponent
- Multiplication
- Division
- Addition
- Subtraction
But younger generations have been taught “BEDMAS” instead:
- Brackets
- Exponent
- Division
- Multiplication
- Addition
- Subtraction
Notably, Division and Multiplication are swapped on PEMDAS and BEDMAS, to make this “both happen at the same time” more straightforward. But that only applies if the entire operation can happen at the same time.
For instance, let’s say
6/2(3)compared to6÷2(3). At first glance, they both appear to be the same operation. But in the former, the6dividend would be over the entire2(3)divisor. Which means you would need to simplify the divisor (by resolving the multiplication of2•3) before you divide. So the former would simplify to6/6=1, while the latter would divide first and become3(3)=9.Technically, if you wanted to be completely clear, you would write it using multiple parenthesis as needed. For instance, you would write it as either:
(6÷2)(3)=9or6÷(2(3))=1to avoid the ambiguity. Then it wouldn’t matter if you’re using PEMDAS or BEDMAS.I am a millennial born in the 1990, they taught me BEDMAS.
But in the former, the
6dividend would be over the entire2(3)divisor.I have never heard of or seen an example of anyone using / and ÷ in different ways. If you want multiple terms in your divisor, either write it as a large fraction with all relevant terms in the dividend or divisor, or use parentheses. This just seems like sloppy notation to me.
It’s 9 if you actually understand PEMDAS
I’m guessing confusion is coming from those taking PEMDAS literally as that order? Rather than PE(M|D)(A|S), like it’s supposed to be?
The ÷ symbol is a bane of mankind
Uh oh, here we go! Before the Fediverse’s favourite mathematical charlatan comes to play, let’s lay out a few facts:
- This is an unusual way of writing down this expression: you would not normally mix in-line division (written with ÷) and multiplication written without a symbol. It’s written this way on social media to for engagement bait.
- Because of this, a perfectly valid reply is to ask “can you put in some brackets to make it clear” :)
- A strict, standard reading of the order-of-operations as abbreviated by PEMDAS, BODMAS, etc, is to perform multiplication and division in the order that they occur. This would mean the evaluation goes like this:
- 6÷2(1+2)
- Perform addition inside the brackets: 6÷2(3)
- Perform the first multiplication or division: 3(3)
- Perform the remaining multiplication: 9
- Occasionally, PEMDAS is interpreted as indicating that multiplication must be done first because the M occurs before the D. This is not usually how it is taught, but rarely it happens. This would give you 1 but, to be clear, in most places this is wrong. I myself was taught BODMAS and, in fact, do division first in all circumstances.
- Much more commonly, though, the actual practical order in which mathematicians, teachers and students all evaluate expressions is a little different, in that it evaluates symbol-less multiplication (also known as “juxtaposition” which just means “writing two things next to each other” or, in discussions about this topic in particular, “implicit multiplication”) before anything else. This is done because writing two things next to each other creates a tightly-bound visual unit.
It’s rare for this last point to be mentioned explicitly as a violation of the order-of-operations. It usually only becomes relevant well after those conventions are spelled out (which is typically done in late primary school or early high school) after children start learning algebra and how to write algebraic expressions: using letters to represent unknown quantities, omitting the × symbol. Exam boards and textbooks are usually quite careful to avoid writing problems in which this unstated rule actually matters.
It’s important to realise that the order in which we evaluate a mathematical expression is a matter of convention. After establishing how to add, multiply, subtract or divide two numbers, it is a separate question which operations should happen first when more than one is written together. This is why we need to teach students the order of operations - they can’t just work it out themselves. Having said that, it certainly makes a lot more sense to do multiplication before addition, and exponentiation before multiplication, because each of these operations is (typically: you can define them in different ways if you’re a masochist) defined in terms of the previous one. This means that if you have an expression involving all three, and you first turn all the exponentiation into multiplications, you are left with a simpler expression that means the same thing. This only happens if evaluating exponentiation is the first thing you’re supposed to do. However, it would be a mistake to think this means that there is any mathematical necessity about this: what a sequence of squiggles on paper means is entirely up to the people reading and writing the squiggles; as long as they agree, the person reading the squiggles will get the same answer as intended by the person writing them. There’s a good, lengthy write-up here
This means that while what I was taught is “wrong” according to how it is usually taught (including today in the same country), this wrongness is better understood mathematically as “unusual” - something that needs to be worked out by communication and consensus rather than by dictating one right and another wrong.
You do get some people with very strong opinions about this, which is not always correlated with their actual knowledge. If the aforementioned charlatan turns up, I’ll explain…
BODMAS
- Brackets
- Of
- Division
- Multiplication
- Addition
- Subtraction
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:
- 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)
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:
- Multiplication is stronger than Addition. (When they fight, Multiplication wins).
- Grouping Symbols override everything. (Parentheses/Brackets/Fractions are the “bosses” that force you to do things out of order).
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.
Well, Patrick IS an idiot … so it checks out?
I hate math, my teacher taught is as first in last out and to this day I still get confused. The answer is 9 right?
Yes, at least by the most common agreed on convention. Almost any mathematician, programming language, search engine or spreadsheet software will say it’s 9. It is for all intents and purposes the right answer.
There is no right answer. It just depends on convention. It’s like color vs colour, neither has been shouted down from the heavens to be the only way to write something, it depends on culture.
It is one though, you gotta do multiplication first
