tensorproduct said:
Except it is. We can formulate sets of rules other than PEMDAS and then, as long as we apply those rules consistently, we will get results consistent within that rule-set. The choice of PEMDAS as our evaluation scheme is entirely arbitrary. It is one of a multitude of conventions, each of which is as consistent as another.
Both cookyt and myself have provided different evaluation rules based on the same sets of symbols, each of which is as arbitrary as PEMDAS.
If your talking about the expression (1 + 3 * 2 ^ 4 - 5) you used before then yet you get different answers depending on how you evaluate. I understand the point, but my point was that PEMDAS ensure consistent answers when you do more complex things such as substitution. From the example I've showed in my post right to left method gives different results when using substitution.
Now I guess I should point out that, when I made the substitution 3*5=5+5+5, I inherently used PEDMAS, but this furthers my point that PEDMAS is important because it makes more complex techniques easier. If you didn't use PEDMAS then using substitution would be much harder to the point of uselessness as it would also become bound to order of operation rules i.e. say you had the two equations
5*x = 10
x = y + 1
then by L-to-R rule, considering you don't use parenthesis, then you would always need to solve 5*x = 10 then x = y+1, i.e. a simple L-to-R rules also makes the equations dependent on the order you solve them in. In PEDMAS, the order you solve the equations don't matter.
Also another point to make PEDMAS makes things easier when expressing equation. With PEDMAS, equation are not dependent on how they are written i.e.
4*5 + 3 = 3 + 4*5
are the same. However with other order or operation rules, you would have to be careful how you would express equations. Granted, the same can be said for PEDMAS but I would say it easier to work with PEDMAS then to ensure I got my terms in the correct sequential order. While this is a simple example, for more complex express I would say the L-to-R would become more complicated and less easier to understand.
Relating the to sciences, PEDMAS system lends it self much better for expressing mathematical models, take physics for example. If where trying to find the total energy of something, say E. Then considering just kinetic energy and potential energy,
E = P.E. + K.E.
E = mgh + 1/2 m v^2
but under a L-to-R you can think of the adding the two energy in the first, but in the second express you can't cause the equation is invalid under the L-to-R rules and must be expressed in another way which doesn't simple look like the K.E. added P.E.
Also, under other order of operations systems, algebra would become much harder. Imagine try to solve the following for x under a left to right rules
3 + 5*x + 2*5 = 90
now under left to right you would have to solve it this way
8*x + 2 * 5 = 90
40*x + 10 = 90
40x = 80
x = 2
the problem with the methods is as you can see you always have to unwrap the expression to solve it and IMO its harder to solve using PEDMAS. Furthermore in the 2nd to 3rd step, to solve for x, you have to a R-to-L operation to multiple the 5 out ( or you could of divided by 5 ), but the point it becomes more complicated the simply L-to-R operations and you have to things of the expression as a whole rather then parts. For the PEDMAS method
3 + 5*x + 2*5 = 90
5*x = 77
x = 77/5
you can simply subtract 3 and 2*5 cause their their the last operations to be applied (PEDMAS kinda work backwards when you rearranging equations) . You can think of the terms by themselves and not worry about the equation as a whole.
Sorry for the long post but to sum up my point, for basic mathematics it is quite arbitrary from a purely abstract point but for more complex mathematics applied to science it ensure consistency and simplicity.
