Submitted by **TheManNamedPeterPan** t3_z8c5vf
in **explainlikeimfive**

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**nemplsman**
t1_iyboolz wrote

Reply to comment by **BurnOutBrighter6** in **ELI5 why we first multiply, then add** by **TheManNamedPeterPan**

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).

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**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.

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**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.

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**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:

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

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**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

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**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.

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**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.

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**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.

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**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

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**nemplsman**
t1_iybvr4e wrote

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

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

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**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?

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**nemplsman**
t1_iyby4kc wrote

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

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