nemplsman t1_iybt6t0 wrote on November 30, 2022 at 5:42 AM

Reply to comment by Way2Foxy in ELI5 why we first multiply, then add by TheManNamedPeterPan

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.

Way2Foxy t1_iybupyc wrote on November 30, 2022 at 5:59 AM

>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

nemplsman t1_iybvric wrote on November 30, 2022 at 6:10 AM

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

Way2Foxy t1_iybw2wm wrote on November 30, 2022 at 6:14 AM

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

x+124

nemplsman t1_iybwwmd wrote on November 30, 2022 at 6:24 AM

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.

Way2Foxy t1_iybxmkz wrote on November 30, 2022 at 6:32 AM

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.

nemplsman t1_iyby6xr wrote on November 30, 2022 at 6:39 AM

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

Way2Foxy t1_iyc1lfu wrote on November 30, 2022 at 7:23 AM

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.

Kalirren t1_iycbj65 wrote on November 30, 2022 at 9:46 AM

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).