I understand that the order of operations is done as PEMDAS because exponentation is just repeated multiplication and multiplication is just repeated addition and you could even make division and subtraction into multiplication and addition respectively and consequently you could really write everything in terms of addition WHEN YOU'RE DEALING WITH NATURAL NUMBERS.
e.g. 2+3*4^2
= 2+3*(4*4)
= 2 + 3*(4+4+4+4)
= 2 + (4+4+4+4) + (4+4+4+4)+ (4+4+4+4)
but what's the explanation for why we still use PEMDAS for rational numbers?
Hmmm, I think there's a mistake in my reasoning actually. By going from line 1 to line 2 in my example I am assuming that the exponent applies to the 4 which means I'm assuming that the exponent should be performed first.
Nevertheless, is the idea that exponents and multiplication can be broken down into addition (when dealing with naturals) related to why we choose PEMDAS as a convention?
Try for example .
When you use the definition of multiplication as repeated addition, .
So .
What would have happened if you had evaluated this left to right?
.
Does that make sense, when you consider that the multiplication was supposed to represent a repated addition of 7 three times?
The reason we use PEDMAS (or BODMAS if you're in Australia) is because it is the only reason that makes sense.
The point is that everything boils down to addition - if you expect to be able to perform multiplication, then you expect that you would need to be able to evaluate it using addition. Therefore, it makes sense to get everything to addition BEFORE trying to do any evaluating.
Right, but to know what to boil down to addition requires a convention, right?
P.S. To the mod, I posted this in Analysis because I would like to know the answer an Analysis book would give rather than Pre-Algebra textbooks which often just state the rules without explaining why.
Well no, right? Because I don't see any reason why another culture couldn't have decided to use the notation of 4+3*7 to represent 7 boxes of 7 marbles in each box and then still write 7*7 as repeated addition: 7+7+7+7+7+7+7
Therefore, it would still be broken down to addition.
I don't see any reason why another culture couldn't have decided to not use the Arabic numeral system. There's no reason why we must express numbers in base 10. We just do. It is how math has evolved in the Middle East, then the West, and eventually, everywhere. PEMDAS is a convention, although there is no reason why one has to do it that way, except that it is how it is.
Other than it is adequate?
Why did the world settle on VHS rather than BetaMax for a domestic video tape standard (since you probably barely remember video tape I will tell you that in this case the technically inferior solution dominated)?
There is no objective reason other than one gains an initial advantage in market penetration, then because doing what the majority do has an advantage an initial small random greater usage is amplified to near 100% usage.
Not every thing has a reason, or at least what we would recognise as one. Nor does the best solution invariably dominate - I presume you are typing on a qwerty keyboard, which for computer use is ludicrously non-optimal.
Another example of a convention where different culture have settled on different solutions is the side of the road that we drive on. I drive on the left, because everyone else here does so. Others drive on the right for much the same reason. The difference is due to an accident of history.
CB
Would you like someone to tell you that Newton flipped a coin to decide?
There is no reason for a convention- that's essentially what convention means. It's the same reason some nations drive cars on the right side of the street and others on the left. The only difference is that since different nations and different cultures have to be able to talk "mathematics" to each other, we all agree on the same convention rather than having separate conventions.