It's something we agreed on. Like you pointed out,

1+2x3

would have more than one possible answer unless there was a convention. It would be ambiguous what you actually meant. So people created "order of operations" rules to make it possible to write math without the ambiguous confusion that would happen if there was no agreed convention.

There is no rule of nature.

There is a rule with how you write math. The rules of order of operations can make some equations look simpler. But you have to make sure that you read the equations the same way that they were written.

"Because that's the rule" is only half the point. It's the rule for a reason.

Multiplication is repeated addition. It can be unpacked into addition: 5 x 3 means 5 + 5 + 5 2 + 5 x 3 means 2 + 5 + 5 + 5

The same relationship is true for exponents and multiplication - exponents are repeated multiplication: 4^3 means 4 x 4 x 4 4^3 x 7 means 4 x 4 x 4 x 7

Subtraction is reversed addition. Adding and subtracting are done in the same step - last.

Division is reversed multiplication. Multiplying and dividing are done in the same step, just before addition/subtraction.

Exponents (and logarithms, where applicable) are done in the same step, just before multiplication/division.

The only thing that's really convention for the sake of itself is parentheses, because we needed a way to demonstrate that certain parts of an equation had to be solved out of order. So anything in parentheses is done first, because that's what they are, by definition.

There are some really interesting low level algorithms that take place in computers and calculators, to turn equations into long strings of logic gates, in a similar sort of method. Pretty much anything that can be computed can be unpacked like this, further than most of us would ever notice.

It's "just a convention", but it's something so ubiquitous that pretty much many mathematician independently came to the same thing.

Historically, it used to be context-dependent, sometimes you add first. For example, look at this picture: https://en.wikipedia.org/wiki/File:JakobBernoulliSummaePotestatum.png . Can you guess which line you add first and which line you multiply first? If you understand the math, you can figure it out, but if not, this can be confusing.

Eventually, various mathematicians made up various different conventions on how to write and interpret formulas. Even though they produce different convention, one thing are shared among them all: multiplication takes priority over addition (unless superseded by other stuff like brackets and lines). And there is a good reason for that. Multiplication distributes over addition, so it's a lot easier to write an expression as a sum of product than a product of sum. In other word, a lot more formulas would end up requiring you to do all the multiplications first to get a bunch of products, and then doing addition to add up those products. So it is a lot simpler to make that the default: by requiring that multiplication takes priority over addition, you don't have to put brackets everywhere.

So even though it's "just a convention", there is a mathematical reason behind the convention.

The order of operations is merely a common standard agreed upon by mathematicians. There is nothing innate with mathematics that requires that standard, though having a standard of some form is a requirement for the common infix notation.

There are other options, such as pre-fix and post-fix notation, also known as Polish and reverse-Polish notation. With pre-fix notation, you'd write `1+2*3` as `+ 1 * 2 3` or `+ * 2 3 1` and `(1+2)*3` as `* 3 + 1 2` or `* + 1 2 3`. With these, you don't even need to define left-to-right or right-to-left as its implied by the definition - though you have to agree whether its pre-fix or post-fix.

Because a bunch of mathematicians got together and came up with a order of operations so that there would be no confusion when looking at a math equation.

It is this way is because all those math nerds said it should be that way. That's it. Left side of the number line is negative because the math nerds got together and said that is how it should be. Much of math and science has shit like this where 'by convention' i.e. nerds got together and made a decision, something is done a certain way.

Multiplication is just shorthand for repeat addition, as such the 2 values are linked together, you can imagine an invisible parenthesis.

1 + 2 x 3
= 1 + (3 + 3) , or 1 + (2 + 2 + 2)
= 7

1 + 5 x 10

The above is similar to saying you are adding $1 to a 5-pack of $10s (or a 10-pack of $5s), which equals $51 and not $60.

If you instead meant to have a 10 packs each containing both a $5 & $1, so $60 total, your expression should be:

(1 + 5) x 10

Because that is what the mathematics world agreed upon. It doesn't really matter as everyone is consistent and uses the same order of priority. The examples are really just poorly formatted. As a general rule it is good practice to use parentheses since it makes things clearer--especially when using the x for multiply...

Leaving out the parentheses in such a case is just asking for trouble since many people are not aware that operators have an order of precedence.

If you have an ant, and two groups of three ants. You have seven ants. 1+2x3

If you add the single ant to the group (1+2) then you have effectively given the single ant the equivalency of three ants, making nine ants total. (1+2) x 3

It is exactly just a convention we agreed on. Once agreed on, everyone uses it and it works. It could have been decided that it should work that addition and subtraction should work first instead of multiplication and division and then everything would have to be rewritten.

One way to sort of get around this is to use something called REVERSE POLISH NOTATION, which places operators and operands explicitly in the order we want to execute them. You place the operands first, and then the operator after them so;

In your first example, to get 7, you would write

2 3 x 1 +

which means take the operand 2 and the operand 3, then apply the multiplication operator them. Now take the result (6) and use that as the new operand and take the operand 1 and apply the addition operator and you get 7

In your second example;
1 2 + 3 x

which means take the operand 1 and operand 2 and apply the addition operator to them, then take the result (3) and the operator 3 and apply the multiplication operator to them which equals 9

So if you want to, you can use that (many calculators actually use that), BUT it is simpler to just memorize PEDMAS (or BEDMAS is you like "brackets" over "parenthesis") and stick to the conventions we always use.

TLDR: Yes, everyone just agreed to use that order.

Wait until you learn the English order of adjectives!

edit: not just English. TIL! :-)

Most of these answers are incorrect. It isn’t PEMDAS simply because some mathematicians decided that’s how it should be, PEMDAS is there because that’s how math works naturally.

For example, 1 + 3 x 4 = 13. This is objective, not subjective depending on what mathematicians agreed to follow. This is because 3 x 4 is just addition on its own, ie 3 x 4 = 3 + 3 + 3 + 3 or 4 + 4 + 4, both equal 12. This is also why multiplication is reversible.

Mathematicians DID just use multiplication in order to shorten the equation though, it’s much easier to write 3 x 4 rather than adding 4 3s together.

To put this in a real life term, let’s say you are having a party, you invited 3 people and each of them is bringing 3 friends. So in essence each of your friend groups total 4 people, 1 + 3 x 4 = 13 total people, which is correct for the attendance. 1 (yourself) + 3 (groups of friends) x 4 (people per friend group) = 13 total people. If you added 1 and 3 first you’d get 16 total people and your party would be a let down with less people coming.

Mathematicians can’t just change it by all agreeing and saying add the 1 + 3 first and still result in a correct answer.

>or is it just a convention we just agreed on?

Pretty much. It's called PEMDAS.

You could, in your daily life, go against it BUT you would need to be consistent with it. So does everyone.

You could do it with numbers (the symbols) just as long as the flow of logic is consistent.

As others have said, it is just convention. The convention relies on an implied order for summation and multiplication, and that can get you into trouble if you're not careful to provide the explicit order. For example, when for part of your problem, you come up with the equation X = 2Y + 4. In another part, you discover that Y = X + 4. You substitute X+4 for Y in the first equation and you come up with X = 2X + 4 + 4. This is wrong, but a student early in their learning might not immediately understand why. I should have used brackets/parenthesis around the substitute term when I replaced the variable. X = 2[X+4] + 4

If the implicit order of operations (the "DMAS" part of "BEDMAS") is causing you problems, I suggest eliminating it before you start working on the problem. Translate the problem from an implicit order of operation to an explicit order of operations: BRACKET ALL THE THINGS!

X = 2Y + 4 <-- Nah.

[X] = [ [2 * Y] + [4] ] <-- Yes!

It might be a crutch, like counting on fingers to add or subtract, but it was the only thing that worked for me. I still use it when I'm not completely sure that Excel and I agree on the implicit order in a formula.

In case it are not clear what me are talking about. Why does us decided that the top one are right, while the lower one are wrong.[punctuation sic]

Get it? Yes, we've just gradually agreed on a way to communicate. Math is just language.

It's just a convention we agreed upon. But I prefer not to rely on it, especially while programming. Use parentheses to make things clear. Write (a * b) + c. Not a * b + c.

I think a lot of these answers are losing the forest for the trees.

What are the two most pointless symbols in mathematics education?

The symbols for multiplication (×) and division (÷).

Why? Because they disappear when you get to algebra.

Multiplication instead becomes the default of two values are next to each other without an operator, then they are multiplied:

Instead of 5 × x, we just write 5x.

All division turns into fractions.

Now think about something like a polynomial. For example,

5x^2 + 4x + 3 = 0

Is much easier to write than

5 × x^2 + 4 × x + 3 = 0

Which is also easier to write than

5 × (x^2 ) + 4 × x + 3 = 0

Now consider how the order of operations makes expressing the terms of a polynomial straightforward:

exponent first, so the variable can be raised to the appropriate power given for each term.

Multiplication next, so each term can be multiplied by a constant.

Addition last, for the final combination of fully calculated terms.

PEMDAS makes writing a polynomial straightforward and helps the notation capture meaning.

Obviously, there is more to math than polynomials and as other answers have touched on, there are lots of reasons why multiplication comes before addition.

I would also like to add that subtraction doesn't exist and it would be better if we stopped teaching it once negative numbers are introduced, but I do think that ship might have sailed.

PEMDAS and BODMAS are like money and language. To be exact they're quite literally mathematical grammar.

When I learned how to write that it would have been:

1 + (2x3) = 7

or

(1+2) x 3 = 9

But I'm old.

Hey, I went to the shops today and bought 3 eggs and 2 loaves of bread. Eggs cost me $2 each and the bread was $5. In total I paid $16.

3 x 2 + 2 x 5 = 16

We do the multiplication first.

However, if we didn't, and we just worked from left to right, then this would happen:

3 x 2 + 2 x 5 = 40 (eggs then bread)

Or:

2 x 5 + 3 x 2 = 26 (bread then eggs)

It just doesn't matter what order I buy my eggs and bread if we do multiplication first.

In any math that matters, to expect others to rely on pemdas would be very very bad form.

Formulas are always written with brackets so there can be no misinterpretation. After all, past the 3rd grade, no one trying to trip anyone up with ambiguous syntax.

Multiplication is a shorthand way of addition. This means 1+2×3 is actually 1+2+2+2 or if you want 1+3+3. It is simply a way to compile or group a series of additions together.

This means in order to reverse it you have to first degroup the multiplication then do the adding.

On one hand, we simply decided to use it that way. It is just a convention we agreed upon.

-

On the other hand, when people were deciding on the convention, surely they had *some* reason, right?

I think they did, and that there is some underlying logic to it.

The mathematical operations have some sort of inherent order to them, basically about how 'strong' (in a sense) they are.

• You may have heard that multiplication is repeated addition. (e.g. 3x4 = 3+3+3+3 by definition).
• Exponentiation (taking a 'power') is repeated multiplication. (e.g. 3^4 is 3x3x3x3 by definition).
• So, in a sense, by being repeated versions of another operation, Exponents are stronger than Multiplication, and Multiplication is stronger than Addition (and Addition is stronger than Counting, for what it is worth).

We don't have to do operations in the order of most 'strong' to least 'strong', but it feels natural to do so, and so we formally choose and decide to do so.

So do exponents (aka powers) first, then multiplication, then addition.

But we've forgotten some operations, so let's add them in. Division is just as 'strong' as multiplication, because is the opposite, and can therefore undo exactly what multiplication causes. Subtraction is similarly the same strength as addition. So, in standard ways of listing the order of operations (like BODMAS/PEMDAS/etc), we obey that ordering.

I'll reiterate that despite this reasoning, it isn't lot as if we are logically forced to use this convention. It is just a fairly natural one to use.

It is a convention, but there is a reason to prefer this convention to one where you add first, which is that multiplication distributes over addition, but not the other way around:

(1 + 2) x 3 is (1 x 3) + (2 x 3), but 1+ (2 x 3) is not (1 + 2) x (1 + 3).

That means that a lot of expressions where sums and products appear, like polynomials, would require using more parentheses if we were to write in a system that adds first.

Performing products first also keeps things consistent when working with units in equations: 50 meters + 10 meters = 60 meters, "meters" can be seen as a thing that gets multiplied by 50 and 10 and 60. If you added first you'd need parentheses to write that.

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remember what 1+2x3 actually MEANS - multiplication is a shorthand for multiple additions, and further up pemdas exponents are multiple multiplications; so you simplify back down the line.

1+2x3^3=1+2x(3x3x3)=1+((3+3+3)+(3+3+3)+(3+3+3))+((3+3+3)+(3+3+3)+(3+3+3))=55

The other reason is the practical considerations - if you have 5 boxes of a dozen eggs and 2 extra eggs, you have 5x12+2 eggs; the standard case for multiplication will be rows + columns + leftovers, and there are FAR fewer cases where you care adding more rows or columns instead of adding in the leftovers, so it's easier to note the special times when you are adding more columns than the times when you are adding the leftovers.

Math is a language, and order of operations is simply grammar to that language. Some times the order of operations makes logical sense, like performing parentheses and exponents first. But some operations, like adding/subtracting, multiplying/dividing see to be able to work regardless if the order. I mean, we do get an answer every time right? Jist different answers. So, how do we know what the "right" answer is? Well, as I said before, we use order of operations in the same way we use gramercy to dictate specific rules in language to help meaning in written word come across without confusing the intent.

Every mathematical operations. Can be analogies tk a real world scenario. A formula that reads, 2+5x3 can give an answer of 17 or 21 depending on whether the order of operations occurs or not.

The way to see what's happennign is to analyze what the problem is saying. This problem is saying, im adding two apples to 5 bags worth or apple, each holding 3 apples. Or 2 apples to 3 bags of apples, each containing 5 apples. That's the principle of of multiplicity. In this analogy we see the answer is 17 and that it makes sense.

Addition and subtraction are a lower ordered operation because they deal with single dimensional operations. Single number of apples vs. Multiplication. And division which deal with higher dimensional math, grouping or fractions. You can't add two single apples to packs of apples containing multiple apples and magically end up with more than what you actually have.

This break in logic is what is happenni g when you add or subtract before multiplying or dividing.

BODMAS - you’ve never been taught this ?

Okay. So this is what I just figured out by trying to do a couple equations like these.

4 + 3 x 8 + 6 - 8 / 2

We can make sense of how the order works by trying to look at numbers separately. So 4 alone is 4. It’s absolutely 4. But if you look at the next number, it’s 3, but accompanied with x (3 times) So it’s 3x8. So the actual number there would be 24. So we put 24 instead of 3x8. Next, there is 6. Which again is a complete number by itself, so we leave it alone. Next comes 8, but it is accompanied with 8/2, that means it’s actually 4, not 8 alone. Once we’re done changing all the numbers to complete values, we do the addition and subtraction

Probably because at its core, multiplication is just short form addition. Instead of writing 2+2+2 we can just write 3×2, and similarly instead of 3+3 we can write 2×3.

Convention.

We could fully parenthesize everything, but it gets annoying to write polynomials if you have to put parens around everything, and it turns out that polynomials are super useful in math, so we set up a convention to make them easier to write purely for convenience.

If it happened to be that we most often wanted to do addition first instead of multiplication, then we would have set up the convention so that addition has a higher precedence so we wouldn't have to write a lot of parens for that thing we do all the time.

That's literally it. Mathematical notation simply exists to be a concise way of writing what we do most often to save us writing. It's nothing to do with math.

This is why it's super annoying when people insist that some equations like 6/2(1+2) are ambiguous. Literally the entire reason we set up these rules is to make sure that there are no ambiguities, so unless you're willing to accept that these super smart people that set up these rules just did a crap job of it and you found a flaw, the situation here is simply that you just don't know the rules they set up.

It’s because you’re not really multiplying. You’re repeat adding.

1 + 2 x 3 is really 1 + 2 +2 + 2.

Now if you throw some parentheses around the one and the two, you change the story.

(1 + 2) x 3 is really (1 + 2) + (1 + 2) + (1 + 2) which is indeed 9

It is a naming convention. Entirely just syntax up help denote things. Also fun thing, Prefix and postfix notation do not have this confusion. With these you can also write them out in trees to help show even better exactly what is going on. It's used a lot in discrete mathematics to describe different tree traversals.

I have 2 bags of 3 apples each + 1 additional apple. That's always 7 apples Rewriting the equation doesn't magically produce 2 extra apples

It's just a convention, you could instead write out each operation explicitly with brackets, this convention allows you to not need the brackets. The convention is far from perfect, leading to viral math problems of type 6 / 2(2 + 2) = ? Does that translate into this:

   6
--------
2(2 + 2)

or does that translate into that?

6
- (2 + 2)
2

You can argue about the right answer until the end of the universe, but the fundamental failure is that there is confusion to begin with, meaning that PEMDAS is not clear and unambiguous enough. In notation with the horizontal bar you notice there is no confusion, none of them are ever going to become viral math problems people fail to answer right.

Multiplication is like a crate/set or something. 1 Apple plus 2 crates with 3 apples each. 1 + 2x3 ... = 7.

We don't always do that.

If you have (1 + 2) x 3 for example, then you add before multiplying,.

It's just a convention to make it easier to read and write maths. We've agreed that if nothing else is specified, you multiply first. And if you want something else, you can toss a pair of parentheses around it to override that rule.

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The same reason we write sentences the way we do. Arbitrary tradition seeking to remove ambiguity while also conveying information with as little effort as possible.

"Put the cookies in the box on the counter" can mean, "put the cookies into the box which is currently on the counter", or it can mean, "put the cookies which are currently in the box on to the counter". So we rely on context clues to give us the correct interpretation. Because maths can exist in a bit of a context less void, the correct way to write an equation removes any ambiguity, but failing that, we rely on the most accepted context, which in this case is the order of operations.

The real reason why we do this is so that we can write DISTRIBUTION neatly.

(4+5)*6 = 4*6 + 5*6

See how I didn't have to write the right side of that with any parentheses at all?

If we had pure left-to-right order of operations, I'd have had to write

(4+5)*6 = (4*6) + (5*6)

which is much uglier.

Note also that * distributes over +, but + doesn't distribute over *:

4+(5*6) != (4+5) * (4+6)

So there's no advantage of less parentheses in assuming that + should be the first operation you perform.

Similarly, exponentiation distributes over multiplication. That's why the usual order of operations is PEMA:

P (exceptions before general rules)

E (^ distributes over *)

M (* distributes over +)

A

It is about setting down some ground rules so everyone arrives to the same solution.

It would be chaos if rules were arbitrary. Just imagine if one computer arrives to different math solution and you suddenly have 20% less money in your account.

It is like this in every field of science, including math, that you need to define the basic rules and frame of reference.

---

There was a NASA debacle, that after 10 months the Mars Climate Orbiter got destroyed. The reason was, that one of the engineers used imperaial measures instead of metric.

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Things that drive or contribute daily lives should have one possible right answer, ambiguity leads to issues and errors

It s just the writing convention so that the other party does the calculation correctly.

Usually the writer of the operation should place parentheses between the numbers but they don't always do that.

Many basic misconceptions regarding multiplications can be dispelled if you just remember that multiplication is a type of addition, written in shorter form.

If you have 1+2×3, you don't have one plus two, followed by times-three, but, broken down, you have one, plus three times two (or two times three)

Thus, 1+2×3=1+2+2+2 -how many times you have the number two? Three times. Or, 1+2×3=1+3+3. -how many times you have the number three? Two times. Both sets of addititon thus equal 7.

For the same reason, you can't "multiply" irl an actual object or set of nultiple ojects by zero, because in that case your definition is erroneous.

It's a convention, but it's a convention people have had a chance to argue about for well over 1000 years. For the mathematics developed over the last thousand years, it's a better convention. If one is only looking at arithmetic on integers, though, the advantages aren't so clear.

Any math at the high school algebra level or beyond requires extensive manipulation of unknown quantities (variables). Almost always the thing you want when manipulating a variable is for there to be only one copy of it in an equation. In order to convert an equation into this form you gather terms and factor (i.e., you apply the distributive law in reverse), ending up with a single multiple of your variable in your manipulated equation. Work with linear equations, polynomials, or integrals relies very heavily on doing this sort of thing, and it's just awkward to write out the result if your convention is that addition is performed first.

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This entire thread is blowing my mind because it reminds me that we just made math up? I mean, we did though right? Like we could’ve easily decided to always start with subtraction, couldn’t we have?

Because multiplication is a shorthand of repeated addition.

2 × 3 is 3 + 3, 3 × 5 would become 5 + 5 + 5, etc.

So 1 + 2 × 3 will give you 1 + (3 + 3). That 1 is just a simple number, it doesn't tell you anything about how many times you're supposed to add 3.