What makes non-PEMDAS answers invalid?
It seems to me that even the non-PEMDAS answer to an equation is logical since it fits together either way. If someone could show a non-PEMDAS answer being mathematically invalid then I’d appreciate it.
My teachers never really explained why, they just told us “This is how you do it” and never elaborated.
In: 834
PEDMAS isn’t required. It’s always possible to write out a complex algebraic expression that isnt ambiguous about which operation to do first without PEDMAS. It might require a lot of brackets (and the understanding that everything inside brackets goes first) but it’s always possible.
What makes a non-PEDMAS answer invalid is that without it, 1+1×2 can either be 3 or 4 depending on which operation you do first. Its written ambiguously. I could write (1+1)x2 or 1+(1×2) to clarify, or we could agree that with PEDMAS rules, I always mean 1+(1×2). If I meant the other one, id have to revert to using brackets again.
PEDMAS was invented because mathematicians are inherently lazy and dont want to write so many brackets. It’s kind of a mathematician’s shorthand that is taught to be the right way to do it. It makes math a lot less ugly and cumbersome too, so I dont mind.
Edit: Here’s a (https://www.youtube.com/watch?v=y9h1oqv21Vs) from MinutePhysics explaining what I mean, courtesy of u/Necoras
P – Parentheses are an explicit prioritization
E – Exponents are basically repeated multiplication
(MD) – Multiplication is repeated addition and Division is just an inverted multiplication
(AS) – Addition is the most basic operation and Subtraction is just an inverted addition
So it makes sense to do the repeated version of an operation before you do the basic version of the operation
OK, there are many good answers, but let me point out one more thing.
It has been mentioned that PEMDAS is a convention, not a “empirical rule” or “truth”. We agree on it.
Furthermore, it has been pointed out that it is necessary to agree on PEMDAS (or any other rule for priority of operands, which is what PEMDAS is), when two people want to talk to each other in math (or you want to talk to your future self, in that case it is evenly important!).
In a nutshell, when you change the rules, the result will change. It’s not wrong, it’s just that your formula means something different. The very simple and obvious example has already been given:
(2*3) + 4 is not the same as 2*(3+4). Both interpretations of 2*3+4 are “correct”, you just can’t switch back at forth at random. And if you add parenthesis, you can basically enforce operant priority without any rule except, solve parenthesis first. It just gets ugly. But it gets ugly – anyway 🙂
When you work in math (or in my case, logic, which is often pretty much the same, but… ah well anyway), when you talk higher mathematics, one of the first things to figure out when you read a paper/book/… is: what notation rules (including priority of operators) has been used.
Because there is no “standard”. Seriously, there isn’t. There are conventions, and schools that use conventions, and areas of maths (or logic) that prefer those or those conventions. Sometimes it’s continent based, but it also changed dramatically over time. Heck, even the symbols Gottlob Frege used 120 years ago are completely outdated, while what he wrote about is foundations of modern mathematics…
So, just be happy that for your everyday kitchen math, PEMDAS is quite sufficient because that’s what the world in general agrees upon. 🙂
It’s a bit like driving on the left or right side of the road.
In some countries they drive on the left. In some countries they drive on the right.
You *could* functionally change the rules and it wouldn’t really make any difference to safety or quality of life. But the rules are set and agreed upon so that everyone knows what everyone else is doing.
PEMDAS is like deciding (as a country) that people drive on the right side of the road. The rules are set, the infrastructure is built, and everyone who learns to drive learns the same rules so it’s consistent.
If someone independently decided to then drive on the *left* things would get ugly *really* quick lol. Same with math.
So in this example non-PEMDAS would be like driving on the left.
There’s no real reason one is better than the other (although I’m sure many will argue that whichever side they’re used to is the ‘better’ one lol) it’s just what was agreed upon. Same with PEMDAS. It *could* be switched to no real detriment as long as *everyone* made the switch together at the same time.
Which would be a hysterically bad time haha
It’s not strictly “required”, it’s just a defect of the common notation.
If you use other notation, such as reverse Polish notation, operator precedence is not an issue.
For example, 1+2*3 becomes very different if you ignore the operator precedence, 7 if you do it right, otherwise 9.
However, in RPN, you first write the operands, then the operator. This means that the above expression would be:
2 3 * 1 + or if you prefer, 1 2 3 * +, which both can be read from left to right, both yielding the same result.
So, it’s not strictly necessary, just a bug in our way to write math.
Your question sort of misunderstands what math is.
Math is not *really* actually about numbers.
Instead, math is an extremely precise and rigorous system for communicating abstract concepts.
Scientists who are talking about precise notions need a way to transfer those notions to each other without any ambiguity so that nothing is lost in translation.
The place where we start with that is with numbers, because they are a pretty easy model that almost everybody can understand.
So your question kind of puts the cart before the horse; the only thing that’s really special about PEMDAS is that it is one specific system that everybody has agreed upon to use. That way if I have a numerical calculation that I need to communicate to you, I can do so and be absolutely sure that you’ll get the same output from the process that I did.
So you’re kind of right, in the sense that given some such mathematical expression, if you did some other chain of operations and got some other answer, it would be a perfectly valid answer *if* that particular order of operations had been the one that everybody had universally agreed upon.
The reason your teachers never said anything other than “use PEMDAS” is that most of them were not terribly mathematically sophisticated and didn’t know this answer themselves.
So for me as a mathematician, all of these PEMDAS-related memes that come around on Facebook and so on are incredibly infuriating. Every single one of them represents an attempt at communication that has been made as inscrutable as possible just to fuck with people, so that whoever can come up with the “right” answer can feel morally superior to the others or something. That kind of ignores anything that’s actually good or useful about math.
I think the main point that everyone else is trying to get at but maybe not quite communicating clearly is that; there is only 1 correct answer to any of these given problems, one way to “do” math if you will.
PEMDAS does not describe the way in which math answers are calculated, it describes the way in which math is written out so that other humans can understand what they are reading.
I’m English, we agree to read left to right. That doesn’t mean it’s the “right” way to read; in Arabic for example they read right to left. Either method is fine, as long as everyone agrees to which order words should be read in.
Math is the same way. You need to decide what order to calculate (“read”) in. PEMDAS is the order that has been agreed to, so mathematicians “write” in that order.
If some random scientist decided they wanted to use a different order, anything they wrote would be nonsense to anyone else reading their math, in the same way that if someone decided to write English right to left would produce nonsense.
Edit: changed Japanese to Arabic as an example of a right to left language.
This is like asking why it’s wrong to drive on the other side of the road. There’s nothing inherently wrong with it, but everybody already agreed to do it the other way, so you won’t get along well with them doing it your way
PEMDAs exists for notation purposes only. It helps to clarify order of operations when reading a formula that somebody else has written down without knowing any context to the formula.
In reality, order of operations in math is really dependent of the context for the problem you’re trying to solve. Logic will dictate the order in which you ask questions and therefore, solve pieces of the formula. When presenting equations, scientists will include the logic sequence in their papers and show the individual components of the formula before presenting the entire thing.
For example. Assume you’re holding a party. You want to know how much to budget for chairs. Each chair costs $3 to rent for the day and each chair suitable for 2 people (they’re loveseats). You have 3 families of 4, 4 families of 3, and 30 singles attending.
How many chairs will you need? 4+4+4+3+3+3+3+3+1+1…., or (3X4)+(4X3)+(30X1) = 54.
How many chairs will you need? 54/2 = 27
How much money should you budget? 27X$3 = $81
If I were to write the whole thing out, then it would be ((3X4)+(4X3)+(30X1))/2*$3, and PEMDAs would help you to know the order of operations without knowing my logic sequence so you would get $81 instead of 9 or 117 or something.
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