Simple Math Problem

[markem]

PEMDAS and BODMAS are mnemonics, not mathematical rules. Language is fickle and you can't have two letters occupying the same position in a word, so some order must be chosen. This is the ambiguity of these two mnemonics.

It helps to know that any division can be rewritten as a multiplication without loss of meaning and multiplication can be rewritten as division. The same is true for with regards to addition and subtraction. Remember too that multiplication is a shorthand for addition and division for subtraction.

So all this can be reduced to PEMA or PEDS / PODS. PEMA is the dominant form, probably because multiplication and addition are commutative, associative and multiplication has the distributed rule. PEMA stands for parentheses, exponentiation, multiplication, addition. Note that PEMA has no ambiguity since there is only one operation at each evaluation step (or level, if you know about the precedence pyramid: https://www.teacherspayteachers.com/...yramid-2358878). (Note that the difference between PEMDAS and BODMAS is really just selecting which item at a specific level is listed first, and thus ambiguous).

If we use multiplication for division and addition for subtraction, the solution is unambiguous (aka deterministic). Equivalent maths should give the same answer.

8 / 2(2+2) can be rewritten as
8 / 2 * (2+2) can be rewritten as
8 * (1/2) * (2+2) can be rewritten as
8 * (1/2) * 4 which reduces to
16

How the order of the operations in the final reduction step do not matter because multiplication is both commutative and associative:

(8 * (1/2)) * 4 == 8 * ((1/2) * 4) == 4 * (1/2) * 8

Note that I use (1/2) to indicate one-half, which could just as easily be replaced by 0.5, but that may complicate understanding for some.

The "DM" and "MD" of the mnemonics do not dictate a required ordering, they are just a handy memory device. There are not 6 cases to remember, just 4, since two cases can be recast in terms of other cases.

Here are some useful links:
http://mathforum.org/library/drmath/view/57199.html
http://thomas.tuerke.net/on/tech/?thread=1521859580

Since MD are at the same level of precedence, some method must be found to denote which order the operations will occur, or non-determinism results. Having a mathematical system that yields different results because of ambiguity is not good and would reduce the value of much of mathematics.

The "consensus" of about the last 100 years (yes, resolving this is a newish problem) is that operations at the same level of precedence are unambiguous when evaluated "left-to-right". Note that this is not a rule of mathematics, but a guide for resolving ambiguity.

Precedence plus the left-to-right guide removes the ambiguity. PEMA has no need for the left-to-right guide so is much simpler to get right.

The issue of PEMDAS and BODMAS (why no PEDMAS?) comes about through misunderstanding of the meaning of the mnemonics, not because math is non-deterministic. The underlying rule is PEMA. Any equation resolved with either of the mnemonics must be equivalent to PEMA or the answer is incorrect by mathematical rule, which cannot be changed. Note that you can reduce PEMA to PEA without introducing non-determinism.

So the real problem is NOT that the equation is ambiguous or opaque. It clearly is not under PEMA. The problem is a misunderstanding about the mnemonics being a mathematical rule rather than just a memory device.