The "MDAS" mnemonic is correct, but it isn't a 1,2,3,4 order of those items.
It is Multiplication and Division, in order from left to right
Addition and Subtraction, in order from left to right.
Yes, it's "My Dear" "Aunt Sally"
You would not say "addition is before subtraction" and calculate 1-2+3 as
1-(2+3) = 1-5 = -4. You would calculate 1-2+3 = -1 +3 = 2.
To elaborate further, there is no other option. The axioms of
arithmetic say the subtraction operator is the inverse of addition,
and the addition operator must commute, that is,
1-2 must be the same thing as 1+(-2) and
1+(-2)+3 must be the same as 3+(-2)+1, both must be 2
Any other interpretation would mean we could not do Algebra.
This is
completely parallel to saying "multiplication before division" and
calculating 2/2*2 as 2/(2*2) = 2/4 =1/2. You would calculate 2/2*2 = 1*2 =2.
Again, the division operator is the inverse of multiplication, and the
multiplication operator must commute, that is,
2/2 must be the same thing as 2*(1/2) and
2*(1/2)*2 must be the same as (1/2)*2*2, both must be 2
It makes sense when we look at the whole big picture.
Pull out any scientific calculator that knows order of operations, or open
Excel and try 2/2*2. You will get 2.
Strictly speaking, the order of left to right should make no
difference (assuming multiplications are done before addition) because
the addition operators must commute among themselves and the
multiplication operators must communite among themselves.
In practice, because the number of digits representing a number is
very finite, if you multiply two large numbers, you can loose a lot of
digits. If you then divide by a large number, you may get a less
accurate answer than if you divided one of the large numbers first.
