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.

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'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.

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.

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.

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.

Finally an 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.

Wait until you learn the English order of adjectives!

edit: not just English. TIL! :-)

That’s fun, but there’s actually reason to believe that adjective ordering really is based on more than simple convention. In particular it seems to hold across many different languages.

https://medium.com/ontologik/the-universal-adjective-ordering-mystery-not-a-mystery-c10614e50761

Edit: The Chomsky Universal Grammar stuff in that article is bullshit though.

To put that in even simpler terms, imagine you have 5 boxes of 12 eggs, and an extra 2. That's 2 + 5 x 12. But if you add first, you'd have a lot more complicated way of trying to express that. It's not a law of nature, but there is a practical use case that made selecting multiplication first a clear choice.

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.

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.

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.

Did this convention come about before or after the usage of parenthesis in mathematical notation? It seems like they cover the ambiguity problem pretty well.

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.

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.

Why couldn’t you just write it 5 x 12 + 2 and solve it all left to right? Or just stick with parentheses as the operator. (5x12) + 2. Could math not be simplified in this way or would it fall apart with more complex equations?

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.

You can and should. A large function in algebra (including basic mathematics) is in making a long problem easier to understand. The problem you wrote could also be written 12 + 12 + 12 + 12 + 12 + 2, but writing it as 5 x 12 + 2... or (5 x 12) + 2... makes it understandable at first glance. This is all that is meant by the common math problem "Simplify this equation."

Agreeing on a convention to write math problems a certain way just removes ambiguity and makes it possible for any person familiar with the relatively simple Order of Operations rule to understand any other problem written by someone. My biggest problem with the order of operations "riddles" that swarm Facebook from time to time is that they'll intentionally write them in a confusing way in order to trick people. It's the same as posting a scribbled rough draft of a paragraph and then making fun of people who misinterpret it.

"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.

Parentheses do fix ambiguity problems beautifully. But they can also be a total pain if you have to write out a whole mess of them again and again and again.

The reason we have the multiply-then-add rule (rather than the other way around) is because "Add up a list of values-multiplied-by-quantity" is a super common kind of scenario - and this convention lets us shortcut away lots of parentheses from these often-encountered problems.

For example, imagine adding up the value of one penny, two nickels, four dimes, and three quarters... Writing out "(1×1) + (2×5) + (4×10) + (3×25)" is amazingly unambiguous and perfectly legible... but simply writing out "1×1 + 2×5 + 4×10 + 3×25" (and maybe leaving some whitespace for extra clarity) saves the work of two parentheses per list-item. Maybe that isn't a big deal when writing out a single list of only four coin denominations... but if you have to do hundreds of similar such problems then those extra strokes will definitely add up.

If parentheses are used most of this confusion immediately evaporates. A person can still remember how to do math decades after forgetting the "order of operations."

If someone wanted the answer to 1+2x4+3 I'd ask them why they wrote it out in a way that is so easy for typical humans to misinterpret.

May not be helpful, but my point is, avoid needless confusion if possible. One does this with parentheses. I'm not sure I can think of an example other than a math class where it would be advantageous to avoid parentheses. Long live parentheses.

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.

Rad! Thanks!

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.

Writing them is slightly tedious but typing them is a royal pain. Capital case, lower case, capital case, lower case ad nauseum.

>Why couldn’t you just write it 5 x 12 + 2 and solve it all left to right?

Well, what if you are writing down the eggs you see, as you see them?

What if you spot 2 eggs, so you write that down, and then spot 2 cartons of a dozen, so you write that down.

Oh, dang, now you've written 2+ 5x12, and under your convention, that isn't what I meant (I've written 7x12, but I mean 2 more than 5x12).

-

Or, what if you have a formula like: "D = 2*x/y + 5*z^2"

You might be working in a lab, , and your assistant tells you "I've measured x & y, they are 7 and 3.5 respectively."

You're waiting for them to measure z, and figure that you'll simplfiy your equation, and you work out the left term happens to be 4 (2*6/3.5=4).

So you write "D = 4 + 5*z^2" to keep your equations easy to read and lined up.

Oh, dang again! That isn't what you meant, you need to change the order around!

-

I think it is better to have a convention where the order you write addition in doesn't matter, so that you can add things as you discover them/calculate them, rather than needing to reorder your equation every time.

You *could* do it, but I think if society did, we'd have more errors than using our current convention.

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.

I bet there's a more precise explanation than just "we agreed to it." Without looking it up, it seems it's something having to do with this:

Consider this formula:

• 4 x 8 + 9 - 2 x 7 + 9 ÷ 3

We know that the multiplication sign between 4 and 8 only acts between those two numbers. And the multiplication sign between the 2 and 7 only acts on those two numbers and the division sign only acts on the 9 and 3. BUT, the +4 and -2 and +5 could literally be anywhere else in the formula and nothing would change. Basically, the exact location of the multiplication and division symbols between two numbers matter, whereas the exact location of the numbers added and subtracted doesn't matter.

I don't think this is just because we decided to do it this way as a convention. It's because the multiplication and division signs are unique in that they specifically imply that a calculation should happen between the two numbers on either side of the operation.

Maybe a way to think about it is that multiplication and division essentially transforms the adjacent numbers. Examples:
- 3 x 8 (three, eight times is 24; or eight, three times is 24).
- 45/3 (45 split into 3 is 15).

Numbers that are added or subtracted are more just independent from the rest of the numbers as they can appear anywhere (they don't necessarily need to be added to the adjacent number).

Adding on since I don't have enough for a parent comment, but multiplication is doing an addition multiple times. So the example could also be written as 1+2+2+2.

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?

[removed]

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.

While taking a course in Group Theory for my mathematics degree, the author of the book declared that parenthesis are unnecessary and redundant.

> I don't think this is just because we decided to do it this way as a convention.

It is 100% because it's decided as convention.

>BUT, the +4 and -2 and +5 could literally be anywhere else in the formula and nothing would change.

Elaborate?

>multiplication is doing an addition multiple times.

Only for integer multiplication, though.

That must have been a fun passing grade to earn.

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.

The numbers that are added or subtracted within a given formula are not subtracted from the number adjacent to them. They are just added or subtracted from the overall series of numbers.

Conversely, the multiplication and division symbols strictly indicate that the multiplication or division must occur between the numbers on either side of the multiplication or division symbol -- so you can't just move those numbers around that are on either side of those symbols.

This being the case, it's necessary to first do the multiplication and division calculations so those operators work first between the two numbers on either side and not along with some other number that is added or subtracted.

As long as the multiplication and division is done, the order doesn't matter on addition and subtraction.

You my friend, are mixing cause and effect.

I know what you're saying but I don't think so.

I feel for you.

I know so because its 100% just because we agreed on using this ruleset. Try thinking about your reasoning if we mixed it up and reversed the order of operations

I still write parentheses when I don’t need to sometimes, but as I get more comfortable with order of operations it does make things simpler not to have all the nested parentheses in complicated equations.

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.

>the multiplication and division symbols strictly indicate that the multiplication or division must occur between the numbers on either side of the multiplication or division symbol

Because of the convention of the order of operations. If we instead changed that to say that addition/subtraction is before multiplication/division, then it would be just as valid to say that

2+3 x 4+8 x 7+2

could be arranged as

4+8 x 7+2 x 2+3

Which is exclusively because we decided to do it this way as a convention, not anything inherent.

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

This comment provides a similar explanation to what I'm saying:

https://www.quora.com/Is-there-a-logical-explanation-for-the-mathematical-order-of-operations/answer/Stephen-McBride-13?ch=10&oid=300169584&share=4e8656ca&srid=YAyd&target_type=answer

See also here: https://www.reddit.com/r/mathematics/comments/k2nfui/comment/gdvfjla/?utm_source=share&utm_medium=web2x&context=3

This comment provides a similar explanation to what I'm saying:

https://www.quora.com/Is-there-a-logical-explanation-for-the-mathematical-order-of-operations/answer/Stephen-McBride-13?ch=10&oid=300169584&share=4e8656ca&srid=YAyd&target_type=answer

See also here: https://www.reddit.com/r/mathematics/comments/k2nfui/comment/gdvfjla/?utm_source=share&utm_medium=web2x&context=3

It'd be less convenient to do it addition-first, but the system would still work and be consistent.

Hell there's even Polish notation, which you'd write

(1+2) x 4

as

x+124

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.

So why not just have it be like SDPAEM? (subtract, divide, parentheses, add, exponent, multiply)? It doesn't make sense that the order is entirely arbitrary.

It seems to me that some arbitrary decisions were made, like to have addition before subtraction, or whether to have division before multiplication, but it seems clear the choice (for example) to have multiplication and division before addition and subtraction is not merely arbitrary and rather, is based on multiplication and division having a greater order of magnitude in their effect compared to addition and subtraction. Same with exponents being before multiplication and division.

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.

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.

This is the answer

Again, you can have a system that works perfectly well with multiplication prior to addition. There is no "inherent rule in nature" as OP phrased it guiding this.

Consider this formula:

4 x 8 + 9 - 2 x 7 + 9 ÷ 3

BUT, the +4 and -2 and +5 Where did the 5 came about?

Can you elaborate? I feel like the logic should apply to all numbers. Hell even in the complex sphere you should be able to do multiplication through iterations of addition.

That was just an error. I intended it to say 4 x 8 + 9 - 2 x 7 + 9 ÷ 3 + 5

There seems to be disagreement on this, and not just by me (see my sources).

Well, how would you describe pi*e in terms of iterative addition?

If I were to try, I'd say e+e+e+(pi-3)e, but that's just shifting the multiplication somewhere else, no?

Okay, do

i*i

using addition.

No, the rules apply through the real numbers. It's just easier to visualize in integers.

Let's take 1.4 × 2.5. Both are non-integers.

You can still interpret this as taking 1.4 and adding it to itself 2.5 times: 1.4 + 1.4 + 0.7 = 3.5

Another example, take pi, an irrational number. pi × pi = pi + pi + pi + (.1415...)pi = 9.8696...

Sure, but you're kicking the can, no? Define that last term in pi^(2),

(.1415...)pi

using only your repeated addition definition.

It's simply a fraction of pi. You don't need to think of the parenthesis as multiplication per se, just that it's a modifier that represents the leftover.

You can say "half of 4 is 2", "a quarter of 12 is 3". In a similar vein, .1415...th of pi is 0.44...

Fractions are inherently using multiplication, though. I'm not saying you can't visualize it as repeated addition, just that it's not equivalent to repeated addition, outside of integers (or at least having one of the numbers being an integer).

No, but it IS still equivalent to repeated addition. It just gets more complicated in how that's represented symbolically. The fundamental operation doesn't change just because we use less clean symbology of rational and irrational numbers. The concept is still the same. Multiplication is an operation upon addition.

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

If it is equivalent, then please give me the exact value of 0.75 x 0.44, only using addition.

Parentheses very quickly become unreadable when you have too many of them.

3(5x^3+2)^2 becomes 3*(((5*(x^3))+2)^2) without order of operations to do the implicit grouping for you. It's not incomprehensible, but it's much harder to read. Longer equations would be awful.

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

You're not wrong, but most of the confusion with order of operations happens at the multiplication -> addition level. At least in my experience. Like 5x^2 is really obvious what it's supposed to be to most people (if you're using actual super script, anyway).

Although that said, I understand that this very well may be because once you start doing more complicated math that actually requires a lot of parentheses and exponents and stuff you've already used the order of operations so many times it starts to become second nature, so it might just be that those are more obvious because the people that are encountering them are already well practiced.

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.

just do long multiplication. this reduces the problem to repeated integer multiplications.

its also equivlent to his example of adding up the integer parts then you deal with the fractional part by multiply everything so that the fractional part is integral, doing the repeated addition then dividing the answer at the end by the factor you multiplied by. this is equvalent to shifting the decimal point. the point doesn't change how you approach the algorithm, and the answer in the end is still a rational fraction, not really a "single number" 0.25 isnt simpler than 1/4, its just another way to write it.

that trick wont work for irrationals though. but they aren't calculable no matter what you do so...

this is really trivial for binary multiplication, as it just reduces to shifting the number up one place across itself and adding the digits to itself IF the digit in that place is a 1; as you either add 0xthat (which is no addition) or 1xthat, (which is no multiplication). binary addition is also simple, it reduces to looking if they digits are different, in which case its 1, otherwise its 0 and u carry a 1 if theyre both 1.

    1011 (11)
   x0110 (6)
   -----
   1011  (22)
  1011   (44)
  ----
 1000010 (66)

>give me the exact value of 0.75 x 0.44, only using addition.

as for your question... we can agree integer multiplcation is trivial and can always reduce to repeted addition so i wont write it out but...

4x5 + 4x70 + 40x5 + 40x70 = 3300; put decimal place in correct 2x2=4 point position = 0.33

all of this works with rational numbers and they reduce to repeated addition. his example with pi is bogus for different reasons, and that's just because irrational and especially transcendental numbers like pi are just uncalculable; so you'd end up doing this long multiplication process forever; infact you wouldnt even get to the point of multiplying pi, because you're still working out what pi is so you can multiply it. all calculations that have ever been done are on rational numbers. "real" numbers are just symbols we manipulate and replace with approximations when a calculation is needed. they aren't real.

or you can just pretend its 22/7.

I don't think we disagree that doing multiplication prior to addition makes sense intuitively.

My point is that there's nothing forcing us to do it that way, and we could have a well defined system where we add and subtract first. If you disagree with that, then fair enough.

That is, indeed, the whole point. You practice them so that they become second nature when you do more complicated math.

you're trying to be clever using those decimals, but the concept is still the same.

First let's convert those decimals into symbology that's easier to work with (fractions):

0.75 = 3/4

0.44 = 11/25

0.75 x 0.44 = 3/4 x 11/25

So therefore using ADDITION to evaluate the multiplication of the fractions,

11+11+11 = 33 for the numerator

25+25+25+25 = 100 for the denominator

The fraction 33/100 simplifies to 0.33 in decimal form.

>Like 5x^2 is really obvious what it's supposed to be to most people

In the spirit of the original question, you could argue: why is it obvious that 5x² means 5*(x²) and not (5*x)²?

Indeed people often get confused over -x²: is that (-1)(x²) or (-1x)²?

Because in a group, we are only dealing with a single associative binary operation, in which case parenthesis are indeed unnecessary.

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

I wasn't trying to be clever, I was just choosing random numbers.

You're right, though, and you definitely can use long multiplication to multiply any two rationals using only addition (though not sure if/how you can reduce to the lowest terms).

I got lost/distracted and started arguing something I didn't mean to initially. My initial argument that I still hold is that you can't define multiplication as simply repeated addition, and to further clarify I mean strictly that if you multiply a and b you add a to itself b-1 times.

The thing that is confusing to me though is that math is precise - especially when considering massive undertakings like space flight for example

There are laws of physics and math… so if the reasons why the order of multiplication vs addition (despite leading to potentially VASTLY different conclusions) are somewhat arbitrary… why does it still work?

The example in the original post lead to two different answers, and that was an extremely basic example.

How does this work with hundreds of different calculations?

The math itself is fundamental. It would work no matter what convention we chose. The only arbitrary thing we're deciding on here is how to WRITE math so other humans know what you mean when there's more than one possibility.

(more than one possibility for what math you're trying to describe, NOT more than one way that the math itself could go!!!)

But we arrive at two different answers depending on which path we choose

Like if it was a math test should the teacher mark both correct?

Sorry just trying to make sense of how both answers are right when there should be a right answer you know what I mean?

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.

Both answers aren't right. On a test only one of those is right, and one is wrong. When you write:

1+2x3

That means "multiply two x three, then add one". There is only one right answer to this. It's 7. 9 is wrong.

If what you actually meant was add 1+2, then multiply by 3, than you have to write it as (1+2)x3, and then the only right answer is 9.

The rules we're talking about here are only about how to write the math so that it means what you intend it to mean (like multiply 2 by 3 then add 1, vs add 1+2 then multiply by 3). But for whatever you write, there's only one right answer.

Yes I get that

The question is though

When you have thousands of calculations, again in the instance of like a rocket launch, both are correct as long as everyone is on the same page?

And is the order between adding and multiplying is mostly meaningless as long as everyone is using those same calculations?

I'd like to delve deeper to clarify this concept. If we're talking PURELY numbers, PURELY mathematics, there IS NO order of operations. The math just *IS*.

For example, 1+2+3 = 6 doesn't happen left to right, right to left, or whatever order at all. 1+2+3 *IS* 6 and it *IS* 12-6 and it *IS* 18/3. All of these terms are just different ways of representing the number 6. They are equivalent, one in the same. Another way to think of it is that ALL the operations are done *at the same time*.

However, human beings aren't instant and we lose track of things all the time. Therefore, we devised the order of operations as an INTERPRETATION of how we observe the math actually works.

But I do want to clarify, this order of operations isn't just an "agreed upon standard" arbitrarily. It's following observed rules. The reason why multiplication comes before addition is because it is already a form of repeated addition.

7 + 3 x 4 is the same as 7+(4+4+4).

If we try it purely left to right, it doesn't make sense.

7+3x4 is not the same as 10x4 because the 3 is *modifying* the 4.

The 3 and the 4 are not separate terms. They are telling us that 4 happens 3 times (or that 3 happens 4 times). The 7 is just another term being added to the evaluation of 3 times 4. So we MUST evaluate 3x4 first.

Here's an illustration:

7 apples + 3 baskets of 4 apples. You don't add 7 apples to 3 baskets. You'd get what, 10 apple/baskets? That doesn't make sense. You add the baskets first (12 apples total), then add the extra 7 apples for a total of 19 apples.

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.

Exactly. The number 3 in 3 x 5 is not an absolute number but represents the "number of groups". So, it's like 3 groups of 5 (of whatever object) and not "3 objects" and "5 objects".

In your example, you're talking about if the 'agreed language' of math was different, the other answer would be correct. But, if the required answer for this scenario was 7, the actual calculation would be written differently.

It's like if you have a room full of people speaking English, and one dude who only speaks French. It's not that French is wrong, but they're not going to be understanding each other.

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Just to be sure, it is -1(x²) right?

I live in excel for work... Color coded parentheses ftw

... Seriously though, I probably use more than I need to, but they reduce ambiguity to a point that any loss of immediate readability is a sacrifice worth making imo

Pfft easy

i + -1 + -i

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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.

Right……

But like if the math adds up then what makes that one French person wrong?

Not sure what point you think you made mate

My question was about the math not about culture

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

As a financial engineer, I write a lot of mathematical code. I, too, use more parentheses than I need to, but they reduce ambiguity to the next person reading the code. Long love parentheses!

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.

---

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

Nothing at all makes the French person wrong, if the math adds up. I'm not talking about culture at all, it's just an example, like launching a rocket mentioned earlier in the thread. You seem a little angry, you ok?

To put it another way, forget about the equation. The "required" answer to launch the rocket in the example at the start of this thread is 7, not 9. Could we conceivably write and read math in a different order? Yeah absolutely, but if we write math the same and two people read it differently, then there's too much rocket fuel and it explodes on launch. That's why we have a standard for reading back these equations so that we can all get to the same answer.

it is

In maths you use this: [ ] and this: { } as second and third parentheses. It´s not so confusing then.

Nope. Subtraction is adding the additive inverse and doesn't distribute over multiplication. The reason why we have PEMA is because Exponentiation distributes over Multiplication and Multiplication distributes over Addition and Parentheses break those rules so you have to do the Parentheses first.

Fun fact: (with some assumptions about operators) only one of the cyclic permutations of a RPN (or PN) string is valid, so you can be sneaky and define the value of any string as the value of the only parsable-as-RPN cyclic permutation of it, leading to funny stuff like 1 + 2 - 3 = 2, 5 * 6 - 7 + 8 * 9 = 368. Another fun fact: all strings with N numbers and N-1 binary operators are valid under that scheme, such as - + 8 7 * 6 5 * * 4 3 = 169. Proofs are an exercise to the reader.

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.

And the answer to the "why" is because exponentiation distributes over multiplication, and not the other way around, just like multiplication distributes over addition.

xy^(2) = x*(y^2) = x*y^2 != (x*y)^2 = (xy)^(2) = x^(2)y^(2)

x*(y^2) != (x*y) ^ (x*2)

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.

No, there -is- something forcing us to do it this way: * distributes over + but + doesn't distribute over *. So if you want to write the distributive property a*(b+c) = a*b+a*c you don't have to use ANY parentheses if you do * before +. And there's no reason why you would try to do it the other way because a+(b*c) != (a+b) * (a+c).

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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That's exactly how I'd do it, but i kinda understand your point. To do a set of non-whole integers multiplied, one requires a way of splitting 'the last plus', and the only way my Imagination allows me to would be multiplication

Yup. Programming languages or technical formulas end up having so many parenthesis that most editors support color coding or matched pair highlighting so you can sort out which is which. And you'd need more if every operation had to have parenthesis around it to clarify which order it's supposed to be done in. If you kept the left-to-right convention (despite throwing other conventions which are no more arbitrary away), you could reorder some things to remove a bit of the confusion. But it wouldn't help nearly as much as every symbol having an order of operations so you skip as many parenthesis as possible while remaining unambiguous.

Convention. It's like the alphabet. The alphabet isn't required to be in ABC ordering by a fundamental force of nature but rather just some particular ordering for better communication.