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BurnOutBrighter6 t1_iyax6la wrote

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.

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orangezeroalpha t1_iybjwhl wrote

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.

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DecentChanceOfLousy t1_iyc04yn wrote

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.

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Cypher1388 t1_iyc76hc wrote

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

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the_running_stache t1_iyc8r0r wrote

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!

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DecentChanceOfLousy t1_iydvi3w wrote

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.

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PobreCositaFea_ t1_iyc9ded wrote

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

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MoobooMagoo t1_iyc0zud wrote

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.

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SirX86 t1_iyc20rc wrote

>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?

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Kalirren t1_iycayo7 wrote

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)

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No-Eggplant-5396 t1_iyed1g0 wrote

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.

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DecentChanceOfLousy t1_iyc1nqo wrote

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

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ohyonghao t1_iybrk08 wrote

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

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TwiNighty t1_iyc2of0 wrote

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

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yogert909 t1_iybunl4 wrote

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.

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Gigantic_Idiot t1_iybpgt8 wrote

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.

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Way2Foxy t1_iybsn7w wrote

>multiplication is doing an addition multiple times.

Only for integer multiplication, though.

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Sereaph t1_iybyrjj wrote

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...

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Way2Foxy t1_iybyvh8 wrote

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

(.1415...)pi

using only your repeated addition definition.

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Sereaph t1_iybz0eq wrote

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...

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Way2Foxy t1_iybz77k wrote

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).

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Sereaph t1_iybzgmy wrote

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.

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Way2Foxy t1_iyc036j wrote

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

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Phrygiaddicted t1_iyc18ek wrote

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.

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Sereaph t1_iyc1wc1 wrote

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.

3

Way2Foxy t1_iyc2s6k wrote

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.

0

thatnotsorichrichkid t1_iybxyrj wrote

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.

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Way2Foxy t1_iyby7kg wrote

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?

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thatnotsorichrichkid t1_iyd6k6r wrote

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

1

00PT t1_iyb6jhc wrote

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.

1

x1uo3yd t1_iybj9mt wrote

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.

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JimAsia t1_iybm38j wrote

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

4

Spadeninja t1_iyc313a wrote

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?

0

BurnOutBrighter6 t1_iyc3kd5 wrote

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!!!)

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Spadeninja t1_iyc3ukk wrote

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?

1

BurnOutBrighter6 t1_iyc4yuq wrote

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.

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Spadeninja t1_iyc5chh wrote

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?

−1

John_Vattic t1_iyc64td wrote

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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Spadeninja t1_iyc89h3 wrote

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

−4

John_Vattic t1_iyc8ya7 wrote

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.

5

nemplsman t1_iyboolz wrote

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).

−6

Way2Foxy t1_iybskl4 wrote

> 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?

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unfamous2423 t1_iybtbu3 wrote

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

0

Way2Foxy t1_iybutw9 wrote

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

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nemplsman t1_iybt6t0 wrote

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.

−1

Way2Foxy t1_iybupyc wrote

>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

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nemplsman t1_iybvric wrote

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

1

Way2Foxy t1_iybw2wm wrote

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

3

nemplsman t1_iybwwmd wrote

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.

0

Way2Foxy t1_iybxmkz wrote

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.

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nemplsman t1_iyby6xr wrote

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

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Way2Foxy t1_iyc1lfu wrote

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.

1

Kalirren t1_iycbj65 wrote

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).

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Tsjernobull t1_iybtn2x wrote

You my friend, are mixing cause and effect.

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nemplsman t1_iybtu77 wrote

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

−1

Tsjernobull t1_iybudfi wrote

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

3

ezhikstumani t1_iybxrsl wrote

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?

1

nemplsman t1_iyby4kc wrote

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

1