Submitted by TheManNamedPeterPan t3_z8c5vf in explainlikeimfive
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).
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?
unfamous2423 t1_iybtbu3 wrote
As long as the multiplication and division is done, the order doesn't matter on addition and subtraction.
Way2Foxy t1_iybutw9 wrote
Which is exclusively because we decided to do it this way as a convention, not anything inherent.
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.
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
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
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
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.
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.
nemplsman t1_iyby6xr wrote
There seems to be disagreement on this, and not just by me (see my sources).
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.
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).
Tsjernobull t1_iybtn2x wrote
You my friend, are mixing cause and effect.
nemplsman t1_iybtu77 wrote
I know what you're saying but I don't think so.
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
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
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?
nemplsman t1_iyby4kc wrote
That was just an error. I intended it to say 4 x 8 + 9 - 2 x 7 + 9 ÷ 3 + 5
Viewing a single comment thread. View all comments