In math, why do we have to do the order of operations in the order we do?

It’s just agreed upon. I don’t think it matters other than what is agreed upon, to share formulas across the globe.

Because that’s how things are expressed in our mathematical notation. I think the order of operations is just a method of showing how mathematical symbols work; it fits our definitions. I don’t see it as a hierarchy, or an “order”.

Parentheses, brackets, etc. group things, so all operations within the must be done before applying outside operations. That’s not to say you can’t perform an operation outside of a set of parentheses beforehand.
Ex: 5–6+(8–4). You can do 8–4 or 5–6 first; it won’t make a difference. (I mean, you could even do -6+8 first because of addition’s associative property.)

Exponents only apply to the number it is to the upper-right of, so that’s why it’s “higher up” on the “order”.
Ex: 3+6^2. The ”^2” only applies to the 6 because that’s how we express exponents. Similarly, other functions (logarithms, sines, etc.) also follow a similar format. They only apply to their arguments.

Multiplication and division are the same operation, and addition and subtraction are the same operation. Multiplication does not need to come before division and addition does not need to come before subtraction. I’d say multiplication “comes first” because multiplying adds another dimension. Multiplying is made up of addition. I’m sure there’s a better explanation out there—I just can’t think of one.

@talljasperman It’s more than that. If you do it any other way you get the wrong answer.

There has to be a rule or people would not know which to do first. You can’t just do math in a different order, the answer will be different. I don’t know how they agreed on the order, but some group of mathematicians decided somewhere in history and it stuck I guess.

There is a magical logic about it. I know there is, because math itself is kind of magic. Some one was smart enough to know what it is.

^^It’s a language. Like having grammatical rules.

Mathematics is not much more than a concrete way of expressing an abstract notion that we call “cause and effect”. It is clear that much of what we know about the physical universe is based at least in part on “cause and effect”.
Math is a way of demonstrating in a totally objective way, the vague concept of “cause and effect”
If you don’t follow the sequence, then you can not demonstrate the preceding cause with the subsequent effect.

But how did they determine what the sequence should be? It can’t be a matter of simple agreement. If every one agreed that 1 + 1 = 3, that still wouldn’t make it right.

@Dutchess_III
Incorrect
Because in mathematics, 1 (that is a single thing) plus 1 (that is an identical thing) equals 2 (that is the combination of two identical things). The only required convention is the name of the things (one and two, or pick any convention you like) Beyond that, it is pretty self evident.

If you do it any other way you get the wrong answer.

No it’s just a convention as @talljasperman wrote.

1+2*3

If we agreed that addition and subtraction preceded multiplication and division, the right answer would be 9.

But the agreement is that multiplication and division precede, so the answer is 6.

But that would be wrong. I mean, how can they use math to solve complex physics problems if they come up with the wrong number. How can they use math to explain the universe if the answers are wrong.

Please forgive me if I’m coming up as obtuse. I’d really like to understand what you guys are saying.

If you don’t follow the proper order of operations you are not doing it correctly. There is nothing agreed upon about it. If that was the case nothing we engineer or build would work. It is simply a procedure based on natural laws that must be followed to get the correct answer.

Nope. You’re confusing the underlying math with the way we write it.

If we agree we are writing in base 10 then
10 + 10 = 20

If we agree we are writing in base 2 then
10 + 10 = 100

Both answers are correct.

Osoraro and Rocket Guy and Raggy, where the hell are you??

@ARE_you_kidding_me, that’s my point. It wouldn’t work just because some Great Apes got together to decid that’s what the answer should be, and the way to get the answer, whether right or wrong. But…I think I’m missing something deeper….

@jaytkay How does writing in base 2 = 10 + 10 = 100?

@jaytkay In no way am I talking about nomenclature. Base2,8,10 or 16… the math is exactly the same and the order of operations still matters.

@Dutchess_III
Base 2 is binary
10 = 2
100 = 4

I hate binary. It goes against the Bible. Shit.

So, 1000=8? Or 6?

110 = 6
1000 = 8

Counting in binary:
00 = 0
01 = 1
10 = 2
11 = 3
100 = 4
101 = 5
110 = 6
111 = 7
1000 = 8
.
.

the order of operations still matters.

Yes it does. But there is no reason that multiplication has to go first.

If we all agreed that from this day forth, that the order of operations is addition before multiplication then
1+2*3 = 9

I’m done. Isn’t Mariah our math wizard? Where is she?

I guess @ARE_you_kidding_me described my thoughts in a way that made more sense. There was probably no council to determine which preceded each other because it’s just axiomatic; self-evident, as @josie said. The order of operations is a result of the way mathematical symbols work.
There actually is one thing about ordering that I think is a convention: we read from left to right. So when you have something like 6/3*2, you must divide first. This simplifies to 4, not 1.

”...a long time ago, people just decided on an order in which operations should be performed. It has nothing to do with magic or logic. Some people decided to adopt a way, and it has stuck ever since. It just makes communication a lot easier.

”Another way of saying this is that rather than being inherent in the structure of mathematics, the concept of ‘order of operations’ is a matter of mathematical notation. Order of operations refers to which operations should be performed in what order, but it’s just convention.”

The Math Forum @ Drexel

*******************************************************************************************************

“It is clear that one person did not invent the rules but rather current practices have grown gradually over several centuries and are still evolving.

“Prompted by the need for such conventions, the basic rule that multiplication has precedence over addition appears to have arisen without much disagreement as algebraic notation was being developed in the 1600s.”

Vanderbeek, Greg, ‘Order of Operations and RPN’ (2007).MAT Exam Expository Papers.Paper 46 University of Nebraska – Lincoln

Thanks @jaytkay. That last answer is hopefully settling this.

just curious… What base leads to 1+2*3=6 (as you said above). ;-)

So that we get the sums right

Mariah and LuckyGuy.

What base leads to 1+2*3=6

Sorry, that was a dumb math error on my part.

XD I get off the internet for one evening…

I agree that it is a manmade convention. If we were to agree upon a different order of operations, we would have to phrase our equations differently than we do now, such that they end up having the same results as the ones we use now. For example, if we did addition before multiplication, the formula for a straight line, rather than the current:

y = mx + b

where m is slope and b is y intercept and (x,y) is any point on the line, we’d either have to be more specific with our parentheses:

y = (mx) + b

or we’d state the formula in a completely different but equivalent way. I’m trying to come up with what it would be, but trying to alter the order of operations that is so thoroughly ingrained in my brain is proving to be a huge challenge. I’ll keep thinking about it.

Furthermore, many of the rules of algebra that we can use now would not apply anymore. For example, the distributive property:

a(b+c) = ab + ac

would not hold anymore, as shown:

a(b+c) != a * (b+a) * c

We’d have to formulate these rules differently, again either by using parentheses more:

ab+c = (ab) + (ac)

or by stating them differently…...again, very challenging for me to try to retrain my brain well enough to come up with an example.

If we followed all the new rules, we would get the same results for classic physics problems, etc. You just have to be consistent with whatever system you’re using.

To elaborate a little more….the order we’ve chosen is not completely arbitrary, it makes a large amount of sense, and trying to state even a simple formula like y = mx+b with a “reversed” order of operations is proving to be very difficult….

Multiplication is basically bulk additions in the sense that 4*2 = 2 + 2 + 2 + 2…
Exponentiation is bulk multiplications: 2^4 = 2 * 2 * 2 * 2

We start with the “biggest” operation – exponentiation – and then add on smaller pieces, the multiplications, and then the smallest pieces – additions. This is like counting out an amount of money by starting with big bills, moving to smaller bills once necessary, and then finally change. Doing it in the reverse order just doesn’t lend itself to easy computation.

^^^ Thank you.

An outstanding answer, @Mariah. Still, I’m curious: did the standard order of operations develop in parallel with the growth & development of both number theory and abstract algebra?

I don’t know the history – sorry.

@Mariah This is like counting out an amount of money…
Does this mean that we are stuck with a generation of fast food clerks who can’t do math without the help of the POS computer?

I think it just comes down to, as others said, we just decided that’s the way to do it, for no other reason than for us all to be able to come to the same answer.

Math would be totally different if humans didn’t have 10 fingers . It would change reading an abacus and Roman numerals. So I would say that some of mathematics is discovered and some invented.