@realcaseyrollins
@caseypicks

Sure, so in math, any math, the answer to any question depends a great deal on context. In any math setting, the rules change drastically and you can never answer any math problem without context, typically for most people and everyday math you are oblivious to this as the context is assumed. Let me give an example to explain what I mean.

5*5?

You would assume the answer is 25, but without context you really don't know if that's true. If you assume we are doing grade school algebra (what we would call the Ring of real numbers under standard multiplication) you'd be right. But in reality you dont really know that, i could be doing factor math and 5 and 5 are two identical vectors of length 5 along degree 0, and the * in that case would represent the cross product, not standard algebraic multiplication.. So if we assume this to be vector math the answer would actually be 0.

So the important part here is that the answer depends on the rules of the operators and the numbers you deal with. Those rules can, and often are, defined however you wish. With that said, ultimately, there **is** some assumed standards around a notation, and which standard we use is assumed, so lets talk about the standards assumed in ordinary algebraic operations like we are talking about here.

So what it comes down to is we are asking if there is a standard regarding order of standard operations in algebraic arthmatic, the answer is, yes and no.

The standard rings and most of the core definitions of the operations do not define any ordering since they are defined in isolation. In other words they only define how the individual operators work and don't really define the notation around combining them. So in the strict formalization of the operators there is no order of operations.

However there is still consensus on use, for the most part, with only a few outliers, here is the order from most consensus to least consensus on how this is handled (and some references).

1) Most common standard employed by math professionals is also the least surprising it can be summarized simply as:

> Don't be ambiguous

This basically means it is unacceptable to ever use the ÷ or / to represent division. It is always done using a horizontal bar with a top and bottom that explicitly defines what is above and what is below the bar. This avoids the problem entirely. This is basically the standard, it dictates that the problem, as stated, uses invalid characters for division, problem solved.

2) The next most common answer you will get is that division and multiplication have equal precidence and should be considered in the order they are done.. so 4÷5*6 would be the same as (4÷5)*6. So by this standard the answer is 1.

This is by far the most common way to answer the question that isnt a cop-out. It has been established as the standard from multiple sources as a way to handle the operators as single-line text.

Here is a PDF from a educational source expressing these rules rather explicitly: http://www.mathcentre.ac.uk/resources/uploaded/mc-ty-rules-2009-1.pdf

3) However the standard in #2 is not absolute (and thus why it isnt used). for example while this standard is still the more common in Physics circles there are some standards-setting boards that still set contradictory rules for their own journal submissions. Several physics journals explicitly define multiplication as having higher precedence than division. In fact even Feynman himself used this standard, as does the Physical Review Physics journal.

So clearly there is no hard standard here even if there is some degree of consensus.