# Is Order of Operations, PEMDAS Arbitrary?

• B

## Main Question or Discussion Point

Is there any deep reason why math follows PEMDAS order of operations?

It seems like it's totally arbitrary to me. Like couldn't we read things right-to-left instead of left to right and couldn't we do subtraction before multiplication?

Why does it matter? Is it just a made up "rule" that math "authorities" decided on?

berkeman
Mentor
Is there any deep reason why math follows PEMDAS order of operations?

It seems like it's totally arbitrary to me. Like couldn't we read things right-to-left instead of left to right and couldn't we do subtraction before multiplication?

Why does it matter? Is it just a made up "rule" that math "authorities" decided on?
This is not a good thread start, IMO. You are asking our members to waste their time trying to guess what you are getting at, so that you can post your pre-conceived ideas.

berkeman
Mentor
Post links to the reasons behind the currently standard order of operations, and post links to peer-reviewed math journal articles that argue that the order should be changed.

symbolipoint
Homework Helper
Gold Member
This is not a good thread start, IMO. You are asking our members to waste their time trying to guess what you are getting at, so that you can post your pre-conceived ideas.

Not a bad question. The Order of Operations makes sense, at least for me, but I do not know why. The question is equivalently, how do we connect numbers using operations and how do they mean what they do?

symbolipoint
Homework Helper
Gold Member
Is there any deep reason why math follows PEMDAS order of operations?

It seems like it's totally arbitrary to me. Like couldn't we read things right-to-left instead of left to right and couldn't we do subtraction before multiplication?

Why does it matter? Is it just a made up "rule" that math "authorities" decided on?
The order of operations is important because mathematics or arithmetic needs to be precise, or unambiguous.

Is there any deep reason why math follows PEMDAS order of operations?

It seems like it's totally arbitrary to me. Like couldn't we read things right-to-left instead of left to right and couldn't we do subtraction before multiplication?

Why does it matter? Is it just a made up "rule" that math "authorities" decided on?
Yes, it is a completely made up rule that math authorities decided on. There are many other rules we could have invented, but PEMDA's is the one we take now.
All the things you say are completely possible: reading from right-to-left or doing substraction before multiplication. I think it's important we tell students that PEMDA's are arbitrary.

That does not mean they're not important or useful however. If I give $3+4-2\times(2+4)$ to you and to an arbitrary different person, it would be good to come up with the same answer. Why? If person A claims to have computed the landing speed of an aircraft he could have made mistakes. So ideally, a person B would need to check it. If person A has totally different conventions from person B, person B will have a lot of difficulties with checking this and might be very prone to errors. So it is useful to have some convention. Whether this is PEMDA's or "substraction always first" doesn't matter, but apparently PEMDA's won out and that's what everybody uses now. I have never been raised in a system where we just read left-to-right, but I do imagine many rules would look very very complicated as opposed to PEMDA's...

fresh_42
Mentor
Yes, it is a completely made up rule that math authorities decided on.
And it is really and truly amazing that the entire globe agreed on it without starting a math war or even needed the UN for assistance. Those math authorities are tighter crosslinked than one might expect. This Al-Gebra network is really far spread.

And it is really and truly amazing that the entire globe agreed on it without starting a math war or even needed the UN for assistance. Those math authorities are tighter crosslinked than one might expect. This Al-Gebra network is really far spread.
Yes, it's very amazing. Let me think a bit on how that would happen. First of all, the entire PEMDA rules and others depend very crucially on the notation people used. The notation for even something simple like polynomials $3x^2 + 5x + 6 = 0$ were very horrible with the ancient Greeks. Gradually however, the notation improved. Authors who invented new notations of course also set out rules for interpreting them. After a while, you saw that some notations won and others were ignored. With these notations that won, came of course the conventions too. So after a while we had the same notation as we have now with more or less the same convention. Not all of the conventions were fixed however, that happened much more recently.

ProfuselyQuarky
Gold Member
The order of operations is important because mathematics or arithmetic needs to be precise, or unambiguous.
Yes, it is a completely made up rule that math authorities decided on. There are many other rules we could have invented, but PEMDA's is the one we take now.
I have thought about this, too, because the order of operations used when working a problem affects the answer greatly, obviously (actually, in algebra 2, the teacher assigned 5 complex expressions to simplify nearly every week so that the class wouldn't forget PEMDAS).

If we just happened to use a different rule (not PEMDAS), does that mean our answers to all those problems would be different?

I have thought about this, too, because the order of operations used when working a problem affects the answer greatly, obviously (actually, in algebra 2, the teacher assigned 5 complex expressions to simplify nearly every week so that the class wouldn't forget PEMDAS).

If we just happened to use a different rule (not PEMDAS), does that mean our answers to all those problems would be different?
Yes, it would mean that. Or at least, it would mean that we would need to use a different notation system.

Mathematically and logically, there is no problem. Something like $2 + 3\cdot 4$ is actually an abbreviation. Logically it has no meaning until we assign it one. We can assign it $(2+3)\cdot 4$ or $2 + (3\cdot 4)$. We chose the latter.

Even something like $2+3+4$ has no meaning initially. We can either assign it to be $2+(3+4)$ or $(2+3)+4$. Luckily, both give the same answer. But this would not be the case if you used $-$ instead of $+$.

ProfuselyQuarky
Gold Member
Yes, it would mean that. Or at least, it would mean that we would need to use a different notation system.

Mathematically and logically, there is no problem. Something like $2 + 3\cdot 4$ is actually an abbreviation. Logically it has no meaning until we assign it one. We can assign it $(2+3)\cdot 4$ or $2 + (3\cdot 4)$. We chose the latter.

Even something like $2+3+4$ has no meaning initially. We can either assign it to be $2+(3+4)$ or $(2+3)+4$. Luckily, both give the same answer. But this would not be the case if you used $-$ instead of $+$.
I always thought that people never invented math, rather, math has been discovered. How else could it so perfectly parallel our world? So, like anything else, I had the impression that there should always be a right and wrong way to order operations.

But, never mind, thinking like this too much would probably do more harm than good (for me at least).

I always thought that people never invented math, rather, math has been discovered. How else could it so perfectly parallel our world? So, like anything else, I had the impression that there should always be a right and wrong way to order operations.

But, never mind, thinking like this too much would probably do more harm than good (for me at least).
You're right, people did discover math, not invent it. In the same way that people didn't invent love or friendship or hate, they discovered it. Order of notations however are a language. Something to communicate with to other people. So why people didn't invent love, they did invent many words for love or hate. In the same way, people did invent many different notation systems to communicate math with.

You're studying abstract algebra, so you should know what an operation is. An operation is a function. For example, addition is a function $f:\mathbb{R}\times\mathbb{R}\rightarrow \mathbb{R}$. This means that addition can only take two numbers and output a third one. We cannot take in three numbers or $4$ numbers. So something like $2+3+4$ is not defined. Rather, something like $2+(3+4)$ (formally $f(2;f(3,4))$) is defined, and the same for $(2+3)+4$ or $f(f(2,3),4)$. Those end up being the same thing.

Strictly speaking, the order of notations is not even needed. It's just not a part of math actually. If $g:\mathbb{R}\times\mathbb{R}\rightarrow \mathbb{R}$ is multiplication and $f$ is addition, then I can safely write $2+3\cdot 4$ as $f(2,g(3,4))$. Or I can write more complicated expressions such as $f(g(f(2,g(5,4)),4),f(2,g(3,4)))$. Everything in usual math I can write with this $f$ and $g$. Brackets and order of notations are never actually needed. You will however agree with me that notations such as $f(g(2,3),g(3,4))$ are hard to read. That's why we abbreviate them in a more convenient system that is easier to read. But then we need to make an arbitrary choice - order of notations. So order of notations are just not a part of math, they're just meant to make life easier on us.

fresh_42
Mentor
I always thought that people never invented math, rather, math has been discovered.
Eureka! Another Platonist! Welcome to the club! Many if not most scientist share this opinion.

How else could it so perfectly parallel our world? So, like anything else, I had the impression that there should always be a right and wrong way to order operations.
There is a convenient and a long-winded way of notation. On the long run the convenient way will be accepted and the other one will be forgotten. In some cases (Newton's and Leibniz' notation of differentiation) more than one survives and are used by personal taste or whatever in a special the situation is suitable. The fact that the entire globe uses the same PEMDAS rule is simply a strong evidence for its convenience.
You could try and solve a schoolbook exercise and write it in two ways and see what suits better. Could you imagine what it'll mean to sites like PF if each time someone starts a thread a long prologue has to take place to achieve an agreement on notation? And each single answer would have to be translated into the code of this agreement! Or if each scientific publication uses its own convention?
There are plenty of good reasons our convention isn't at stake. And can you imagine how many senseless hours I had to spent while tutoring to get rid of this ugly, terrible, useless and error provoking notation 5⅜ where it is all of a sudden the "+" we drop and not the "*"?

Other than micromass I would have answered: No, the math or the answer to a question won't change. Only the way to write it down.

ProfuselyQuarky
Gold Member
That all makes absolute sense. Thanks @micromass and @fresh_42 :)
Eureka! Another Platonist! Welcome to the club!
So when's initiation? I'll bring a dodecahedral cake

fresh_42
Mentor
There is a good reason why scientific papers nowadays are mainly published in English. (Basically the same reasons as those for PEMDAS.)
In former times it had been Latin, and until WWII even German. But it has always been one preferred language all others were assumed to be able to read.
A friend of mine once requested a thesis from an author. It arrived per conventional mail - in Chinese! We couldn't even say whether it was in Mandarin or Cantonese.

ProfuselyQuarky
Gold Member
There is a good reason why scientific papers nowadays are mainly published in English. (Basically the same reasons as those for PEMDAS.)
In former times it had been Latin, and until WWII even German. But it has always been one preferred language all others were assumed to be able to read.
A friend of mine once requested a thesis from an author. It arrived per conventional mail - in Chinese! We couldn't even say whether it was in Mandarin or Cantonese.
That must have been frustrating. What did you guys do?

Needless to say, it’s a good thing that English is the language predominantly used now, or else having to learn Latin or some other language would surely become a deterrent for students wanting to study a field in science or math…

Yes, it is a completely made up rule that math authorities decided on. There are many other rules we could have invented, but PEMDA's is the one we take now.
All the things you say are completely possible: reading from right-to-left or doing substraction before multiplication. I think it's important we tell students that PEMDA's are arbitrary.

That does not mean they're not important or useful however. If I give $3+4-2\times(2+4)$ to you and to an arbitrary different person, it would be good to come up with the same answer. Why? If person A claims to have computed the landing speed of an aircraft he could have made mistakes. So ideally, a person B would need to check it. If person A has totally different conventions from person B, person B will have a lot of difficulties with checking this and might be very prone to errors. So it is useful to have some convention. Whether this is PEMDA's or "substraction always first" doesn't matter, but apparently PEMDA's won out and that's what everybody uses now. I have never been raised in a system where we just read left-to-right, but I do imagine many rules would look very very complicated as opposed to PEMDA's...
What you have said is very interesting, I have never thought of it like this before. To me, the equation you have $3+4-2\times(2+4)$, is reduced to a + b - c, where c = $2\times(2+4)$, and since c = $2\times(2+4)$, I will now compute c first. I see c as: a x b, where a = 2, b = 2 + 4, so now I will compute b first... And after this downward scaffolding is completed, I retrace my steps back up.
It makes sense that you say this is entirely arbitrary, but I do not believe it is random. Yes some people came up with PEMDAS initially, but not because they draw it out of a hat, but because PEMDAS makes sense and works consistently.
Just like many other laws and rules we have, they were all initially thought-out as theories, and then repeatedly proven to be true, and believed to never will be false, then they became laws. These include Newton's Laws of Motion, Galileo's Laws, and so on.

symbolipoint
Homework Helper
Gold Member
Yes some people came up with PEMDAS initially, but not because they draw it out of a hat, but because PEMDAS makes sense and works consistently.
Really good.

FactChecker
Gold Member
I can't help but think that there are human factor reasons that PEMDAS is dominant. Clearly P is first because it is used to override any other order. I notice that the more rare and complicated operations are done before the more common ones. As though they are rare, have the smallest scope, and we are getting them out of the way first.

Drakkith
Staff Emeritus